Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf ·...

136
critical & supercritical Mnica Clapp Introduction Yamabes problem The variational problem The classical results The geometric structure Multiple solutions Punctured domains Symmetries Thin holes The end Elliptic boundary value problems with critical and supercritical nonlinearities. Part 1. Mnica Clapp Universidad Nacional Autnoma de MØxico Flagsta/, June 2012

Transcript of Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf ·...

Page 1: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Elliptic boundary value problems with criticaland supercritical nonlinearities. Part 1.

Mónica Clapp

Universidad Nacional Autónoma de México

Flagsta¤, June 2012

Page 2: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionThe problem

We consider the problem

(p)

∆u = jujp2 u in Ω,u = 0 on ∂Ω,

where

Ω RN is a bounded smooth domain, N 3, p = 2 := 2N

N2 is the critical Sobolev exponent, or p > 2 is supercritical.

Throughout this talk p = 2.

Page 3: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionThe problem

We consider the problem

(p)

∆u = jujp2 u in Ω,u = 0 on ∂Ω,

where

Ω RN is a bounded smooth domain, N 3,

p = 2 := 2NN2 is the critical Sobolev exponent, or

p > 2 is supercritical.

Throughout this talk p = 2.

Page 4: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionThe problem

We consider the problem

(p)

∆u = jujp2 u in Ω,u = 0 on ∂Ω,

where

Ω RN is a bounded smooth domain, N 3, p = 2 := 2N

N2 is the critical Sobolev exponent, or

p > 2 is supercritical.

Throughout this talk p = 2.

Page 5: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionThe problem

We consider the problem

(p)

∆u = jujp2 u in Ω,u = 0 on ∂Ω,

where

Ω RN is a bounded smooth domain, N 3, p = 2 := 2N

N2 is the critical Sobolev exponent, or p > 2 is supercritical.

Throughout this talk p = 2.

Page 6: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionThe problem

We consider the problem

(p)

∆u = jujp2 u in Ω,u = 0 on ∂Ω,

where

Ω RN is a bounded smooth domain, N 3, p = 2 := 2N

N2 is the critical Sobolev exponent, or p > 2 is supercritical.

Throughout this talk p = 2.

Page 7: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionWhy study the critical problem?

Some reasons for studying (2):

1 It is a simplied model for fundamental problems inDi¤erential Geometry, e.g.

the Yamabe problem, the prescribed curvature problem.

2 It gives rise to an interesting and challenging variationalproblem:

Usual variational methods cannot be applied due to thelack of compactness.

3 It has a rich geometric structure.

4 It has been an amazing source of open problems and newideas.

Page 8: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionWhy study the critical problem?

Some reasons for studying (2):

1 It is a simplied model for fundamental problems inDi¤erential Geometry, e.g.

the Yamabe problem,

the prescribed curvature problem.

2 It gives rise to an interesting and challenging variationalproblem:

Usual variational methods cannot be applied due to thelack of compactness.

3 It has a rich geometric structure.

4 It has been an amazing source of open problems and newideas.

Page 9: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionWhy study the critical problem?

Some reasons for studying (2):

1 It is a simplied model for fundamental problems inDi¤erential Geometry, e.g.

the Yamabe problem, the prescribed curvature problem.

2 It gives rise to an interesting and challenging variationalproblem:

Usual variational methods cannot be applied due to thelack of compactness.

3 It has a rich geometric structure.

4 It has been an amazing source of open problems and newideas.

Page 10: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionWhy study the critical problem?

Some reasons for studying (2):

1 It is a simplied model for fundamental problems inDi¤erential Geometry, e.g.

the Yamabe problem, the prescribed curvature problem.

2 It gives rise to an interesting and challenging variationalproblem:

Usual variational methods cannot be applied due to thelack of compactness.

3 It has a rich geometric structure.

4 It has been an amazing source of open problems and newideas.

Page 11: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionWhy study the critical problem?

Some reasons for studying (2):

1 It is a simplied model for fundamental problems inDi¤erential Geometry, e.g.

the Yamabe problem, the prescribed curvature problem.

2 It gives rise to an interesting and challenging variationalproblem:

Usual variational methods cannot be applied due to thelack of compactness.

3 It has a rich geometric structure.

4 It has been an amazing source of open problems and newideas.

Page 12: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionWhy study the critical problem?

Some reasons for studying (2):

1 It is a simplied model for fundamental problems inDi¤erential Geometry, e.g.

the Yamabe problem, the prescribed curvature problem.

2 It gives rise to an interesting and challenging variationalproblem:

Usual variational methods cannot be applied due to thelack of compactness.

3 It has a rich geometric structure.

4 It has been an amazing source of open problems and newideas.

Page 13: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionWhy study the critical problem?

Some reasons for studying (2):

1 It is a simplied model for fundamental problems inDi¤erential Geometry, e.g.

the Yamabe problem, the prescribed curvature problem.

2 It gives rise to an interesting and challenging variationalproblem:

Usual variational methods cannot be applied due to thelack of compactness.

3 It has a rich geometric structure.

4 It has been an amazing source of open problems and newideas.

Page 14: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionThe Yamabe problem

Let M be a compact Riemannian manifold.

Two metrics g and g on M are conformally equivalent ifthere exists a smooth function ρ > 0 such that g = ρg .

Page 15: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionThe Yamabe problem

Let M be a compact Riemannian manifold.

Two metrics g and g on M are conformally equivalent ifthere exists a smooth function ρ > 0 such that g = ρg .

Page 16: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionThe Yamabe problem

In dimension 2 one has the classical result:

Theorem (Klein-Poincaré uniformization theorem)Every surface admits a metric of constant curvature.

Problem (Yamabe)If (M, g), dimM 3, does there exist a metric g conformallyequivalent to g such that (M, g) has constant scalar curvature?

Page 17: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionThe Yamabe problem

In dimension 2 one has the classical result:

Theorem (Klein-Poincaré uniformization theorem)Every surface admits a metric of constant curvature.

Problem (Yamabe)If (M, g), dimM 3, does there exist a metric g conformallyequivalent to g such that (M, g) has constant scalar curvature?

Page 18: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionThe Yamabe problem

In dimension 2 one has the classical result:

Theorem (Klein-Poincaré uniformization theorem)Every surface admits a metric of constant curvature.

Problem (Yamabe)If (M, g), dimM 3, does there exist a metric g conformallyequivalent to g such that (M, g) has constant scalar curvature?

Page 19: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionThe Yamabe problem

Problem (Yamabe)If (M, g), dimM 3, does there exist a metric g conformallyequivalent to g such that (M, g) has constant scalar curvature?

Yamabe (1960) claimed there exists such a metric, but . . . Trudinger (1968) found a fundamental mistake inYamabes proof.

Theorem (Yamabe, Trudinger, Aubin 1976, Schoen 1984)The answer to Yamabes problem is YES.

If we write ρ := u22 and g := ρg , then the scalar

curvatures Rg of (M, g) and Rg of (M, g) satisfy

cN∆gu + Rgu = Rgu21 on M.

Page 20: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionThe Yamabe problem

Problem (Yamabe)If (M, g), dimM 3, does there exist a metric g conformallyequivalent to g such that (M, g) has constant scalar curvature?

Yamabe (1960) claimed there exists such a metric, but . . .

Trudinger (1968) found a fundamental mistake inYamabes proof.

Theorem (Yamabe, Trudinger, Aubin 1976, Schoen 1984)The answer to Yamabes problem is YES.

If we write ρ := u22 and g := ρg , then the scalar

curvatures Rg of (M, g) and Rg of (M, g) satisfy

cN∆gu + Rgu = Rgu21 on M.

Page 21: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionThe Yamabe problem

Problem (Yamabe)If (M, g), dimM 3, does there exist a metric g conformallyequivalent to g such that (M, g) has constant scalar curvature?

Yamabe (1960) claimed there exists such a metric, but . . . Trudinger (1968) found a fundamental mistake inYamabes proof.

Theorem (Yamabe, Trudinger, Aubin 1976, Schoen 1984)The answer to Yamabes problem is YES.

If we write ρ := u22 and g := ρg , then the scalar

curvatures Rg of (M, g) and Rg of (M, g) satisfy

cN∆gu + Rgu = Rgu21 on M.

Page 22: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionThe Yamabe problem

Problem (Yamabe)If (M, g), dimM 3, does there exist a metric g conformallyequivalent to g such that (M, g) has constant scalar curvature?

Yamabe (1960) claimed there exists such a metric, but . . . Trudinger (1968) found a fundamental mistake inYamabes proof.

Theorem (Yamabe, Trudinger, Aubin 1976, Schoen 1984)The answer to Yamabes problem is YES.

If we write ρ := u22 and g := ρg , then the scalar

curvatures Rg of (M, g) and Rg of (M, g) satisfy

cN∆gu + Rgu = Rgu21 on M.

Page 23: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

IntroductionThe Yamabe problem

Problem (Yamabe)If (M, g), dimM 3, does there exist a metric g conformallyequivalent to g such that (M, g) has constant scalar curvature?

Yamabe (1960) claimed there exists such a metric, but . . . Trudinger (1968) found a fundamental mistake inYamabes proof.

Theorem (Yamabe, Trudinger, Aubin 1976, Schoen 1984)The answer to Yamabes problem is YES.

If we write ρ := u22 and g := ρg , then the scalar

curvatures Rg of (M, g) and Rg of (M, g) satisfy

cN∆gu + Rgu = Rgu21 on M.

Page 24: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The variational problemThe functional

The solutions to problem

(p)

∆u = jujp2 u in Ω,u = 0 on ∂Ω,

p 2 (2, 2],

Ω RN bounded smooth domain, N 3, 2 := 2NN2 ,

are the critical points of

Jp(u) =12kuk2H 10

1pkukpLp , u 2 H10 (Ω),

where kuk2H 10 :=R

Ω jruj2 .

Page 25: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The variational problemThe functional

The solutions to problem

(p)

∆u = jujp2 u in Ω,u = 0 on ∂Ω,

p 2 (2, 2],

Ω RN bounded smooth domain, N 3, 2 := 2NN2 ,

are the critical points of

Jp(u) =12kuk2H 10

1pkukpLp , u 2 H10 (Ω),

where kuk2H 10 :=R

Ω jruj2 .

Page 26: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The variational problemThe functional

The solutions to problem

(p)

∆u = jujp2 u in Ω,u = 0 on ∂Ω,

p 2 (2, 2],

Ω RN bounded smooth domain, N 3, 2 := 2NN2 ,

are the critical points of

Jp(u) =12kuk2H 10

1pkukpLp , u 2 H10 (Ω),

where kuk2H 10 :=R

Ω jruj2 .

Page 27: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The variational problemThe graph of the functional

Jp(u) =12kuk2H 10

1pkukpLp , p 2 (2, 2].

Page 28: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The variational problemVariational methods

Variational methods: Follow the negative gradient ow toobtain critical points.

The problem is: The ow lines do not necessarily take usto a critical point!

They do if p < 2. But not necessarily when p = 2.

Page 29: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The variational problemVariational methods

Variational methods: Follow the negative gradient ow toobtain critical points.

The problem is: The ow lines do not necessarily take usto a critical point!

They do if p < 2. But not necessarily when p = 2.

Page 30: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The variational problemVariational methods

Variational methods: Follow the negative gradient ow toobtain critical points.

The problem is: The ow lines do not necessarily take usto a critical point!

They do if p < 2.

But not necessarily when p = 2.

Page 31: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The variational problemVariational methods

Variational methods: Follow the negative gradient ow toobtain critical points.

The problem is: The ow lines do not necessarily take usto a critical point!

They do if p < 2. But not necessarily when p = 2.

Page 32: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The variational problemsupercritical vs subcritical

In fact,

if p 2 (2, 2) variational methods give innitely manysolutions to problem

(p)

∆u = jujp2 u in Ω,u = 0 on ∂Ω,

whereas,

if p 2 there are domains Ω for which the problem hasno solution,

e.g. Ω = ball. the existence of solutions depends on Ω.

Page 33: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The variational problemsupercritical vs subcritical

In fact,

if p 2 (2, 2) variational methods give innitely manysolutions to problem

(p)

∆u = jujp2 u in Ω,u = 0 on ∂Ω,

whereas,

if p 2 there are domains Ω for which the problem hasno solution,

e.g. Ω = ball. the existence of solutions depends on Ω.

Page 34: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The variational problemsupercritical vs subcritical

In fact,

if p 2 (2, 2) variational methods give innitely manysolutions to problem

(p)

∆u = jujp2 u in Ω,u = 0 on ∂Ω,

whereas,

if p 2 there are domains Ω for which the problem hasno solution,

e.g. Ω = ball.

the existence of solutions depends on Ω.

Page 35: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The variational problemsupercritical vs subcritical

In fact,

if p 2 (2, 2) variational methods give innitely manysolutions to problem

(p)

∆u = jujp2 u in Ω,u = 0 on ∂Ω,

whereas,

if p 2 there are domains Ω for which the problem hasno solution,

e.g. Ω = ball. the existence of solutions depends on Ω.

Page 36: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The classical resultsNonexistence

Theorem (Pohozhaev 1965)Problem

(p)

∆u = jujp2 u in Ω,u = 0 on ∂Ω,

with p 2 does not have a nontrivial solution if Ω is strictlystarshaped.

Page 37: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The classical resultsExistence

Theorem (Kazdan-Warner 1975)If Ω is an annulus, i.e.

Ω = fx 2 RN : 0 < a < jx j < bg,

then (p) has innitely many radial solutions for every p > 2.

Page 38: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The classical resultsExistence in punctured domains

Theorem (Coron 1984)Let Ω be a bounded smooth domain, ξ 2 Ω and ε > 0. Then

(2,ε)

∆u = juj2

2 u in Ωε := ΩrBε(ξ),u = 0 on ∂Ωε,

has a positive solution for ε small enough.

Page 39: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The classical resultsExistence

Theorem (Bahri-Coron 1988)If H(Ω;Z/2) 6= 0, then (2) has a positive solution.

The proof relies on the fact that one knows all positivesolutions to the problem in RN .

It uses delicate estimates and sosticated tools from algebraic topology.

Page 40: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The classical resultsExistence

Theorem (Bahri-Coron 1988)If H(Ω;Z/2) 6= 0, then (2) has a positive solution.

The proof relies on the fact that one knows all positivesolutions to the problem in RN .

It uses delicate estimates and

sosticated tools from algebraic topology.

Page 41: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The classical resultsExistence

Theorem (Bahri-Coron 1988)If H(Ω;Z/2) 6= 0, then (2) has a positive solution.

The proof relies on the fact that one knows all positivesolutions to the problem in RN .

It uses delicate estimates and sosticated tools from algebraic topology.

Page 42: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The classical resultsExistence in contractible domains

Is it true that there is no solution if Ω is contractible?

Examples (Dancer 1988, Ding 1989, Passaseo 1989)There are nontrivial solutions in some contractible domains, e.g.

Annulus with a very thin tunnel

Page 43: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The classical resultsExistence in contractible domains

Is it true that there is no solution if Ω is contractible?

Examples (Dancer 1988, Ding 1989, Passaseo 1989)There are nontrivial solutions in some contractible domains, e.g.

Annulus with a very thin tunnel

Page 44: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometric structureMöbius invariance

If u is a solution to∆u = jujp2 u in Ω,u = 0 on ∂Ω,

then, for any Möbius transformation

φ : RN [ f∞g ! RN [ f∞g,

the functionuφ := jdetDφj

12 (u φ)

is a solution to(∆uφ =

22 uφ in φ1(Ω),u = 0 on ∂

φ1(Ω)

.

Page 45: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometric structureMöbius invariance

If u is a solution to∆u = jujp2 u in Ω,u = 0 on ∂Ω,

then, for any Möbius transformation

φ : RN [ f∞g ! RN [ f∞g,

the functionuφ := jdetDφj

12 (u φ)

is a solution to(∆uφ =

22 uφ in φ1(Ω),u = 0 on ∂

φ1(Ω)

.

Page 46: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometric structureMöbius invariance

If u is a solution to∆u = jujp2 u in Ω,u = 0 on ∂Ω,

then, for any Möbius transformation

φ : RN [ f∞g ! RN [ f∞g,

the functionuφ := jdetDφj

12 (u φ)

is a solution to(∆uφ =

22 uφ in φ1(Ω),u = 0 on ∂

φ1(Ω)

.

Page 47: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometric structureMöbius transformations

A Möbius transformation is a nite composition ofreections on planes and inversions on spheres.

φ(x)

x

x

φ(x)

Examples:

euclidean isometries, i.e. translations and linear isometries, dilations: x 7! λx , λ > 0.

Page 48: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometric structureMöbius transformations

A Möbius transformation is a nite composition ofreections on planes and inversions on spheres.

φ(x)

x

x

φ(x)

Examples:

euclidean isometries, i.e. translations and linear isometries, dilations: x 7! λx , λ > 0.

Page 49: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometric structureMöbius transformations

A Möbius transformation is a nite composition ofreections on planes and inversions on spheres.

φ(x)

x

x

φ(x)

Examples: euclidean isometries, i.e. translations and linear isometries,

dilations: x 7! λx , λ > 0.

Page 50: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometric structureMöbius transformations

A Möbius transformation is a nite composition ofreections on planes and inversions on spheres.

φ(x)

x

x

φ(x)

Examples: euclidean isometries, i.e. translations and linear isometries, dilations: x 7! λx , λ > 0.

Page 51: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometric structureMultiplicity in domains with spherical boundaries

Example (C.-Pacella 2008)If ∂Ω = union of two disjoint spheres, then problem (2) hasinnitely many solutions.

Page 52: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometric structureMultiplicity in domains with spherical boundaries

Examples (C.-Pacella 2008)If ∂Ω = union of two disjoint spheres, then problem (2) hasinnitely many solutions.

Proof.There exists an inversion which maps Ω onto an annulus:

Page 53: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometric structureMultiplicity in domains with spherical boundaries

Examples (C.-Pacella 2008)If ∂Ω = union of two disjoint spheres, then problem (2) hasinnitely many solutions.

Proof.There exists an inversion which maps Ω onto an annulus:

Page 54: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometric structurePositive entire solutions

Consider the problem in RN

(RN )

∆u =j u j22 u in RN ,u(x)! 0 as jx j ! ∞.

Page 55: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometric structureStandard bubbles

Theorem (Aubin, Talenti 1976, Gidas-Ni-Nirenberg 1979,Lions 1985)The standard bubble

U(x) = aN

1

1+ jx j2

! N22

is the only positive solution to (RN ), up to translations anddilations.

Page 56: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometric structurecauses blow-up in bounded domains

As for

if Ω is a bounded domain, there are trajectories t 7! ut inH10 (Ω), such that

J(ut )! c , rJ(ut )! 0, as t ! ∞,

but (ut ) does not converge to a critical point as t ! ∞!

Page 57: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometric structurecauses blow-up in bounded domains

As for

if Ω is a bounded domain, there are trajectories t 7! ut inH10 (Ω), such that

J(ut )! c , rJ(ut )! 0, as t ! ∞,

but (ut ) does not converge to a critical point as t ! ∞!

Page 58: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometric structurecauses blow-up in bounded domains

As for

if Ω is a bounded domain, there are trajectories t 7! ut inH10 (Ω), such that

J(ut )! c , rJ(ut )! 0, as t ! ∞,

but (ut ) does not converge to a critical point as t ! ∞!

Page 59: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometry of the problemcauses blow-up in bounded domains

They look like this:

Struwe (1984) showed that the lack of compactness issolely due to this phenomenon.

Page 60: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometry of the problemcauses blow-up in bounded domains

They look like this:

Struwe (1984) showed that the lack of compactness issolely due to this phenomenon.

Page 61: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

The geometry of the problemcauses blow-up in bounded domains

They look like this:

Struwe (1984) showed that the lack of compactness issolely due to this phenomenon.

Page 62: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsOur program

We now go back to the problem

(2)

∆u = juj2

2 u in Ω,u = 0 on ∂Ω,

in a bounded smooth domain Ω RN , N 3,2 := 2N

N2 .

QUESTIONS:

In those cases where existence is known, are there othersolutions?

How do they look like?

Page 63: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsOur program

We now go back to the problem

(2)

∆u = juj2

2 u in Ω,u = 0 on ∂Ω,

in a bounded smooth domain Ω RN , N 3,2 := 2N

N2 .

QUESTIONS:

In those cases where existence is known, are there othersolutions?

How do they look like?

Page 64: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsOur program

We now go back to the problem

(2)

∆u = juj2

2 u in Ω,u = 0 on ∂Ω,

in a bounded smooth domain Ω RN , N 3,2 := 2N

N2 .

QUESTIONS: In those cases where existence is known, are there othersolutions?

How do they look like?

Page 65: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsOur program

We now go back to the problem

(2)

∆u = juj2

2 u in Ω,u = 0 on ∂Ω,

in a bounded smooth domain Ω RN , N 3,2 := 2N

N2 .

QUESTIONS: In those cases where existence is known, are there othersolutions?

How do they look like?

Page 66: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsAnswers & methods

SOME ANSWERS:

1 In punctured domains (like those of Coron)

much progress has been made.

2 In other slightly perturbed domains &

3 in more general domains with nontrivial topology

there are a few recent results.

THE METHODS:

1 Lyapunov-Schmidt reduction,

which works very well for punctured domains.

2 Variational methods + symmetries,

the symmetries help us deal with the lack of compactness.

Page 67: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsAnswers & methods

SOME ANSWERS:

1 In punctured domains (like those of Coron)

much progress has been made.

2 In other slightly perturbed domains &

3 in more general domains with nontrivial topology

there are a few recent results.

THE METHODS:

1 Lyapunov-Schmidt reduction,

which works very well for punctured domains.

2 Variational methods + symmetries,

the symmetries help us deal with the lack of compactness.

Page 68: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsAnswers & methods

SOME ANSWERS:

1 In punctured domains (like those of Coron)

much progress has been made.

2 In other slightly perturbed domains &

3 in more general domains with nontrivial topology

there are a few recent results.

THE METHODS:

1 Lyapunov-Schmidt reduction,

which works very well for punctured domains.

2 Variational methods + symmetries,

the symmetries help us deal with the lack of compactness.

Page 69: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsAnswers & methods

SOME ANSWERS:

1 In punctured domains (like those of Coron)

much progress has been made.

2 In other slightly perturbed domains &

3 in more general domains with nontrivial topology

there are a few recent results.

THE METHODS:

1 Lyapunov-Schmidt reduction,

which works very well for punctured domains.

2 Variational methods + symmetries,

the symmetries help us deal with the lack of compactness.

Page 70: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsAnswers & methods

SOME ANSWERS:

1 In punctured domains (like those of Coron)

much progress has been made.

2 In other slightly perturbed domains &

3 in more general domains with nontrivial topology

there are a few recent results.

THE METHODS:

1 Lyapunov-Schmidt reduction,

which works very well for punctured domains.

2 Variational methods + symmetries,

the symmetries help us deal with the lack of compactness.

Page 71: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsAnswers & methods

SOME ANSWERS:

1 In punctured domains (like those of Coron)

much progress has been made.

2 In other slightly perturbed domains &

3 in more general domains with nontrivial topology

there are a few recent results.

THE METHODS:

1 Lyapunov-Schmidt reduction,

which works very well for punctured domains.

2 Variational methods + symmetries,

the symmetries help us deal with the lack of compactness.

Page 72: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsAnswers & methods

SOME ANSWERS:

1 In punctured domains (like those of Coron)

much progress has been made.

2 In other slightly perturbed domains &

3 in more general domains with nontrivial topology

there are a few recent results.

THE METHODS:

1 Lyapunov-Schmidt reduction,

which works very well for punctured domains.

2 Variational methods + symmetries,

the symmetries help us deal with the lack of compactness.

Page 73: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsAnswers & methods

SOME ANSWERS:

1 In punctured domains (like those of Coron)

much progress has been made.

2 In other slightly perturbed domains &

3 in more general domains with nontrivial topology

there are a few recent results.

THE METHODS:

1 Lyapunov-Schmidt reduction,

which works very well for punctured domains.

2 Variational methods + symmetries,

the symmetries help us deal with the lack of compactness.

Page 74: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsAnswers & methods

SOME ANSWERS:

1 In punctured domains (like those of Coron)

much progress has been made.

2 In other slightly perturbed domains &

3 in more general domains with nontrivial topology

there are a few recent results.

THE METHODS:

1 Lyapunov-Schmidt reduction,

which works very well for punctured domains.

2 Variational methods + symmetries,

the symmetries help us deal with the lack of compactness.

Page 75: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsAnswers & methods

SOME ANSWERS:

1 In punctured domains (like those of Coron)

much progress has been made.

2 In other slightly perturbed domains &

3 in more general domains with nontrivial topology

there are a few recent results.

THE METHODS:

1 Lyapunov-Schmidt reduction,

which works very well for punctured domains.

2 Variational methods + symmetries,

the symmetries help us deal with the lack of compactness.

Page 76: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsAnswers & methods

SOME ANSWERS:

1 In punctured domains (like those of Coron)

much progress has been made.

2 In other slightly perturbed domains &

3 in more general domains with nontrivial topology

there are a few recent results.

THE METHODS:

1 Lyapunov-Schmidt reduction,

which works very well for punctured domains.

2 Variational methods + symmetries,

the symmetries help us deal with the lack of compactness.

Page 77: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsIn punctured domains

The data:

Ω a bounded smooth domain in RN , ξ 2 Ω, ε > 0 small enough, Ωε := ΩrBε(ξ).

ProblemDoes problem

(2,ε)

∆u = juj2

2 u in Ωε,u = 0 on ∂Ωε,

have more than one solution for ε > 0 small enough?

Page 78: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsIn punctured domains

The data: Ω a bounded smooth domain in RN ,

ξ 2 Ω, ε > 0 small enough, Ωε := ΩrBε(ξ).

ProblemDoes problem

(2,ε)

∆u = juj2

2 u in Ωε,u = 0 on ∂Ωε,

have more than one solution for ε > 0 small enough?

Page 79: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsIn punctured domains

The data: Ω a bounded smooth domain in RN , ξ 2 Ω,

ε > 0 small enough, Ωε := ΩrBε(ξ).

ProblemDoes problem

(2,ε)

∆u = juj2

2 u in Ωε,u = 0 on ∂Ωε,

have more than one solution for ε > 0 small enough?

Page 80: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsIn punctured domains

The data: Ω a bounded smooth domain in RN , ξ 2 Ω, ε > 0 small enough,

Ωε := ΩrBε(ξ).

ProblemDoes problem

(2,ε)

∆u = juj2

2 u in Ωε,u = 0 on ∂Ωε,

have more than one solution for ε > 0 small enough?

Page 81: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsIn punctured domains

The data: Ω a bounded smooth domain in RN , ξ 2 Ω, ε > 0 small enough, Ωε := ΩrBε(ξ).

ProblemDoes problem

(2,ε)

∆u = juj2

2 u in Ωε,u = 0 on ∂Ωε,

have more than one solution for ε > 0 small enough?

Page 82: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsIn punctured domains

The data: Ω a bounded smooth domain in RN , ξ 2 Ω, ε > 0 small enough, Ωε := ΩrBε(ξ).

ProblemDoes problem

(2,ε)

∆u = juj2

2 u in Ωε,u = 0 on ∂Ωε,

have more than one solution for ε > 0 small enough?

Page 83: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Multiple solutionsIn punctured domains

Theorem (Ge-Musso-Pistoia 2010)

# of solutions to (2,ε) !ε!0

∞.

The solutions look like superpositions of standard bubbleswith alternating signs (bubble towers):

Page 84: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

SymmetriesThe energy functional

Recall that the solutions to

(2)

∆u = juj2

2 u in Ω,u = 0 on ∂Ω,

are the critical points of the functional

J(u) =12kuk2H 10

12kuk2

L2 , u 2 H10 (Ω).

Page 85: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

SymmetriesThe energy functional

Recall that the solutions to

(2)

∆u = juj2

2 u in Ω,u = 0 on ∂Ω,

are the critical points of the functional

J(u) =12kuk2H 10

12kuk2

L2 , u 2 H10 (Ω).

Page 86: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

SymmetriesThe energy functional

J has the mountain pass geometry:

but the rst mountain pass is never attained!!!

Page 87: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

SymmetriesThe energy functional

J has the mountain pass geometry:

but the rst mountain pass is never attained!!!

Page 88: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Symmetriesand variational methods

Let G O(N) be a group of linear isometries of RN .

The G -orbit of a point x is Gx := fgx : g 2 Gg. We assume that Ω is G -invariant, i.e.

Gx Ω for all x 2 Ω,

and look for G -invariant solutions u, i.e.

u is constant on each Gx .

They are the critical points of the restriction of

J(u) =12kuk2H 10

12kuk2

L2

to the subspace

H10 (Ω)G := fu 2 H10 (Ω) : u is G -invariantg.

Page 89: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Symmetriesand variational methods

Let G O(N) be a group of linear isometries of RN .

The G -orbit of a point x is Gx := fgx : g 2 Gg.

We assume that Ω is G -invariant, i.e.

Gx Ω for all x 2 Ω,

and look for G -invariant solutions u, i.e.

u is constant on each Gx .

They are the critical points of the restriction of

J(u) =12kuk2H 10

12kuk2

L2

to the subspace

H10 (Ω)G := fu 2 H10 (Ω) : u is G -invariantg.

Page 90: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Symmetriesand variational methods

Let G O(N) be a group of linear isometries of RN .

The G -orbit of a point x is Gx := fgx : g 2 Gg. We assume that Ω is G -invariant, i.e.

Gx Ω for all x 2 Ω,

and look for G -invariant solutions u, i.e.

u is constant on each Gx .

They are the critical points of the restriction of

J(u) =12kuk2H 10

12kuk2

L2

to the subspace

H10 (Ω)G := fu 2 H10 (Ω) : u is G -invariantg.

Page 91: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Symmetriesand variational methods

Let G O(N) be a group of linear isometries of RN .

The G -orbit of a point x is Gx := fgx : g 2 Gg. We assume that Ω is G -invariant, i.e.

Gx Ω for all x 2 Ω,

and look for G -invariant solutions u, i.e.

u is constant on each Gx .

They are the critical points of the restriction of

J(u) =12kuk2H 10

12kuk2

L2

to the subspace

H10 (Ω)G := fu 2 H10 (Ω) : u is G -invariantg.

Page 92: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Symmetriesand variational methods

Let G O(N) be a group of linear isometries of RN .

The G -orbit of a point x is Gx := fgx : g 2 Gg. We assume that Ω is G -invariant, i.e.

Gx Ω for all x 2 Ω,

and look for G -invariant solutions u, i.e.

u is constant on each Gx .

They are the critical points of the restriction of

J(u) =12kuk2H 10

12kuk2

L2

to the subspace

H10 (Ω)G := fu 2 H10 (Ω) : u is G -invariantg.

Page 93: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Symmetriesand variational methods

Let G O(N) be a group of linear isometries of RN .

The G -orbit of a point x is Gx := fgx : g 2 Gg. We assume that Ω is G -invariant, i.e.

Gx Ω for all x 2 Ω,

and look for G -invariant solutions u, i.e. u is constant on each Gx .

They are the critical points of the restriction of

J(u) =12kuk2H 10

12kuk2

L2

to the subspace

H10 (Ω)G := fu 2 H10 (Ω) : u is G -invariantg.

Page 94: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Symmetriesand variational methods

Let G O(N) be a group of linear isometries of RN .

The G -orbit of a point x is Gx := fgx : g 2 Gg. We assume that Ω is G -invariant, i.e.

Gx Ω for all x 2 Ω,

and look for G -invariant solutions u, i.e. u is constant on each Gx .

They are the critical points of the restriction of

J(u) =12kuk2H 10

12kuk2

L2

to the subspace

H10 (Ω)G := fu 2 H10 (Ω) : u is G -invariantg.

Page 95: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Symmetriesproduce compactness

Theorem (compactness)If

c < minx2Ω

(#Gx)c∞, c∞ :=1NSN/2,

then J satises the Palais-Smale (PS)Gc , i.e.

every sequence s.t.

un 2 H10 (Ω)G , J(un)! c , rJ(un)! 0,

contains a convergent subsequence.

CorollaryIf #Gx = ∞ for every x 2 Ω then problem (2) has innitelymany solutions.

Page 96: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Symmetriesproduce compactness

Theorem (compactness)If

c < minx2Ω

(#Gx)c∞, c∞ :=1NSN/2,

then J satises the Palais-Smale (PS)Gc , i.e.

every sequence s.t.

un 2 H10 (Ω)G , J(un)! c , rJ(un)! 0,

contains a convergent subsequence.

CorollaryIf #Gx = ∞ for every x 2 Ω then problem (2) has innitelymany solutions.

Page 97: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Symmetriesproduce compactness

Example (Kazdan-Warner)If G = O(N) and Ω = annulus, then problem (2) hasinnitely many radial solutions.

ExampleIf G = SO(2) and Ω = torus, then (2) has innitely manysolutions which are invariant under rotations.

annulus torus

Page 98: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

SymmetriesThe orbits must be innite

Example (Pohozhaev)If G = O(N) and Ω = ball, problem (2) does not have anontrivial solution!!!

Page 99: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeThe setting

The data:

Ω a bounded smooth domain in RN , M a closed submanifold of Ω, dimM N 2, ε > 0 small enough, Ωε := fx 2 Ω : dist(x ,M) > εg.

ProblemDoes problem

(2,ε)

∆u = juj2

2 u in Ωε,u = 0 on ∂Ωε,

have more than one solution for ε small enough?

Page 100: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeThe setting

The data: Ω a bounded smooth domain in RN ,

M a closed submanifold of Ω, dimM N 2, ε > 0 small enough, Ωε := fx 2 Ω : dist(x ,M) > εg.

ProblemDoes problem

(2,ε)

∆u = juj2

2 u in Ωε,u = 0 on ∂Ωε,

have more than one solution for ε small enough?

Page 101: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeThe setting

The data: Ω a bounded smooth domain in RN , M a closed submanifold of Ω, dimM N 2,

ε > 0 small enough, Ωε := fx 2 Ω : dist(x ,M) > εg.

ProblemDoes problem

(2,ε)

∆u = juj2

2 u in Ωε,u = 0 on ∂Ωε,

have more than one solution for ε small enough?

Page 102: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeThe setting

The data: Ω a bounded smooth domain in RN , M a closed submanifold of Ω, dimM N 2, ε > 0 small enough,

Ωε := fx 2 Ω : dist(x ,M) > εg.

ProblemDoes problem

(2,ε)

∆u = juj2

2 u in Ωε,u = 0 on ∂Ωε,

have more than one solution for ε small enough?

Page 103: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeThe setting

The data: Ω a bounded smooth domain in RN , M a closed submanifold of Ω, dimM N 2, ε > 0 small enough, Ωε := fx 2 Ω : dist(x ,M) > εg.

ProblemDoes problem

(2,ε)

∆u = juj2

2 u in Ωε,u = 0 on ∂Ωε,

have more than one solution for ε small enough?

Page 104: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeThe setting

The data: Ω a bounded smooth domain in RN , M a closed submanifold of Ω, dimM N 2, ε > 0 small enough, Ωε := fx 2 Ω : dist(x ,M) > εg.

ProblemDoes problem

(2,ε)

∆u = juj2

2 u in Ωε,u = 0 on ∂Ωε,

have more than one solution for ε small enough?

Page 105: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeSymmetries & sign changing solutions

To obtain sign changing solutions we consider domainswith symmetries. We assume

G O(N) is a nite group of linear isometries of RN , Ω and M are G -invariant, For simplicity, G acts freely on Ω,

i.e. gx 6= x for all g 2 G , x 2 Ω.

Page 106: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeSymmetries & sign changing solutions

To obtain sign changing solutions we consider domainswith symmetries. We assume

G O(N) is a nite group of linear isometries of RN ,

Ω and M are G -invariant, For simplicity, G acts freely on Ω,

i.e. gx 6= x for all g 2 G , x 2 Ω.

Page 107: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeSymmetries & sign changing solutions

To obtain sign changing solutions we consider domainswith symmetries. We assume

G O(N) is a nite group of linear isometries of RN , Ω and M are G -invariant,

For simplicity, G acts freely on Ω,

i.e. gx 6= x for all g 2 G , x 2 Ω.

Page 108: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeSymmetries & sign changing solutions

To obtain sign changing solutions we consider domainswith symmetries. We assume

G O(N) is a nite group of linear isometries of RN , Ω and M are G -invariant, For simplicity, G acts freely on Ω,

i.e. gx 6= x for all g 2 G , x 2 Ω.

Page 109: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeSymmetries & sign changing solutions

To obtain sign changing solutions we consider domainswith symmetries. We assume

G O(N) is a nite group of linear isometries of RN , Ω and M are G -invariant, For simplicity, G acts freely on Ω,

i.e. gx 6= x for all g 2 G , x 2 Ω.

Page 110: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeSymmetries & sign changing solutions

ExampleGn := group generated by the rotation of angle 2π

n about thez-axis in R3,

Ω is a torus of revolution about the z-axis, M Ω is a toroidal knot:

G5-inv. G8-inv. G10-inv.

Then Ωε := fx 2 Ω : dist(x ,M) > εg is Gn-invariant, but#Gnx = n < ∞.

Page 111: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeSymmetries & sign changing solutions

ExampleGn := group generated by the rotation of angle 2π

n about thez-axis in R3,

Ω is a torus of revolution about the z-axis,

M Ω is a toroidal knot:

G5-inv. G8-inv. G10-inv.

Then Ωε := fx 2 Ω : dist(x ,M) > εg is Gn-invariant, but#Gnx = n < ∞.

Page 112: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeSymmetries & sign changing solutions

ExampleGn := group generated by the rotation of angle 2π

n about thez-axis in R3,

Ω is a torus of revolution about the z-axis, M Ω is a toroidal knot:

G5-inv. G8-inv. G10-inv.

Then Ωε := fx 2 Ω : dist(x ,M) > εg is Gn-invariant, but#Gnx = n < ∞.

Page 113: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeSymmetries & sign changing solutions

ExampleGn := group generated by the rotation of angle 2π

n about thez-axis in R3,

Ω is a torus of revolution about the z-axis, M Ω is a toroidal knot:

G5-inv. G8-inv. G10-inv.

Then Ωε := fx 2 Ω : dist(x ,M) > εg is Gn-invariant, but#Gnx = n < ∞.

Page 114: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeA trick to produce sign changing solutions

Let τ : G ! Z/2 := f1,1g be a group homomorphism.

We look for solutions u which satisfy

u(gx) = τ(g)u(x) 8g 2 G , 8x 2 Ωε.

i.e. if τ(g) = 1, then

u(gx) = u(x) 8x 2 Ωε,

and if τ(g) = 1, then

u(gx) = u(x) 8x 2 Ωε.

Therefore, if τ is surjective, u changes sign.

Page 115: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeA trick to produce sign changing solutions

Let τ : G ! Z/2 := f1,1g be a group homomorphism. We look for solutions u which satisfy

u(gx) = τ(g)u(x) 8g 2 G , 8x 2 Ωε.

i.e. if τ(g) = 1, then

u(gx) = u(x) 8x 2 Ωε,

and if τ(g) = 1, then

u(gx) = u(x) 8x 2 Ωε.

Therefore, if τ is surjective, u changes sign.

Page 116: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeA trick to produce sign changing solutions

Let τ : G ! Z/2 := f1,1g be a group homomorphism. We look for solutions u which satisfy

u(gx) = τ(g)u(x) 8g 2 G , 8x 2 Ωε.

i.e. if τ(g) = 1, then

u(gx) = u(x) 8x 2 Ωε,

and if τ(g) = 1, then

u(gx) = u(x) 8x 2 Ωε.

Therefore, if τ is surjective, u changes sign.

Page 117: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeA trick to produce sign changing solutions

Let τ : G ! Z/2 := f1,1g be a group homomorphism. We look for solutions u which satisfy

u(gx) = τ(g)u(x) 8g 2 G , 8x 2 Ωε.

i.e. if τ(g) = 1, then

u(gx) = u(x) 8x 2 Ωε,

and if τ(g) = 1, then

u(gx) = u(x) 8x 2 Ωε.

Therefore, if τ is surjective, u changes sign.

Page 118: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeA trick to produce sign changing solutions

Let τ : G ! Z/2 := f1,1g be a group homomorphism. We look for solutions u which satisfy

u(gx) = τ(g)u(x) 8g 2 G , 8x 2 Ωε.

i.e. if τ(g) = 1, then

u(gx) = u(x) 8x 2 Ωε,

and if τ(g) = 1, then

u(gx) = u(x) 8x 2 Ωε.

Therefore, if τ is surjective, u changes sign.

Page 119: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeSymmetries & sign changing solutions

ExampleG2n := group generated by the rotation $2n by

πn about the

z-axis in R3,

τ($k2n) := (1)k , k = 0, 1, . . . , 2n 1.

A function satisfying u(gx) = τ(g)u(x) for G4 is:

Page 120: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeSymmetries & sign changing solutions

ExampleG2n := group generated by the rotation $2n by

πn about the

z-axis in R3,

τ($k2n) := (1)k , k = 0, 1, . . . , 2n 1.

A function satisfying u(gx) = τ(g)u(x) for G4 is:

Page 121: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeSymmetries & sign changing solutions

ExampleG2n := group generated by the rotation $2n by

πn about the

z-axis in R3,

τ($k2n) := (1)k , k = 0, 1, . . . , 2n 1.

A function satisfying u(gx) = τ(g)u(x) for G4 is:

Page 122: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeSymmetries & sign changing solutions

We look for solutions to

(2,ε)

∆u = juj2

2 u in Ωε,u = 0 ∂Ωε,

where Ωε := fx 2 Ω : dist(x ,M) > εg,

which satisfy

u(gx) = τ(g)u(x) 8g 2 G , x 2 Ωε.

Page 123: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeSymmetries & sign changing solutions

We look for solutions to

(2,ε)

∆u = juj2

2 u in Ωε,u = 0 ∂Ωε,

where Ωε := fx 2 Ω : dist(x ,M) > εg, which satisfy

u(gx) = τ(g)u(x) 8g 2 G , x 2 Ωε.

Page 124: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeExistence

Theorem (C.-Grossi-Pistoia 2010)For each ε > 0 suciently small problem (2,ε) has at leastone nontrivial solution u which satises

u(gx) = τ(g)u(x) 8g 2 G , x 2 Ωε.

u is positive if τ is the trivial homomorphism,

u changes sign if τ is surjective.

Page 125: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeExistence

Theorem (C.-Grossi-Pistoia 2010)For each ε > 0 suciently small problem (2,ε) has at leastone nontrivial solution u which satises

u(gx) = τ(g)u(x) 8g 2 G , x 2 Ωε.

u is positive if τ is the trivial homomorphism,

u changes sign if τ is surjective.

Page 126: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeMultiplicity in highly symmetric domains

ExampleΩ R3 a solid of revolution about the z-axis, M := S1 f0gsuch that

M Ω and Ω \ (z-axis) = ∅.

M

Ωε is invariant under rotations about the z-axis, hence (ε) has innitely many rotationaly invariant solutions.

Page 127: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeMultiplicity in highly symmetric domains

ExampleΩ R3 a solid of revolution about the z-axis, M := S1 f0gsuch that

M Ω and Ω \ (z-axis) = ∅.

M

Ωε is invariant under rotations about the z-axis, hence

(ε) has innitely many rotationaly invariant solutions.

Page 128: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeMultiplicity in highly symmetric domains

ExampleΩ R3 a solid of revolution about the z-axis, M := S1 f0gsuch that

M Ω and Ω \ (z-axis) = ∅.

M

Ωε is invariant under rotations about the z-axis, hence (ε) has innitely many rotationaly invariant solutions.

Page 129: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeMultiplicity in highly symmetric domains

CorollaryLet m 2 N, Ω and M as above Then, for ε small enough, (ε)has m pairs of solutions u1, . . . ,um such that

un($k2nx) = (1)k un(x), k = 0, 1, . . . , 2n 1.

Proof.We apply the previous theorem to the group G2n :

+ + + ++

+

+

_

_ _

_ _

_

_

G2 G4 G8

Page 130: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeMultiplicity in highly symmetric domains

CorollaryLet m 2 N, Ω and M as above Then, for ε small enough, (ε)has m pairs of solutions u1, . . . ,um such that

un($k2nx) = (1)k un(x), k = 0, 1, . . . , 2n 1.

Proof.We apply the previous theorem to the group G2n :

+ + + ++

+

+

_

_ _

_ _

_

_

G2 G4 G8

Page 131: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeOpen problems

ProblemIn general domains of the form (without symmetries)

Ωε := fx 2 Ω : dist(x ,M) > εg,

work in progress by Juan Carlos Fernández shows there areat least two solutions.

Is it true that, as for punctured domains, the number ofsolutions increases arbitrarily as ε ! 0?

Are there bubble towers? Are there multibump solutions? Are there solutions with layers concentrating along M as

ε ! 0?

Page 132: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeOpen problems

ProblemIn general domains of the form (without symmetries)

Ωε := fx 2 Ω : dist(x ,M) > εg,

work in progress by Juan Carlos Fernández shows there areat least two solutions.

Is it true that, as for punctured domains, the number ofsolutions increases arbitrarily as ε ! 0?

Are there bubble towers? Are there multibump solutions? Are there solutions with layers concentrating along M as

ε ! 0?

Page 133: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeOpen problems

ProblemIn general domains of the form (without symmetries)

Ωε := fx 2 Ω : dist(x ,M) > εg,

work in progress by Juan Carlos Fernández shows there areat least two solutions.

Is it true that, as for punctured domains, the number ofsolutions increases arbitrarily as ε ! 0?

Are there bubble towers?

Are there multibump solutions? Are there solutions with layers concentrating along M as

ε ! 0?

Page 134: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeOpen problems

ProblemIn general domains of the form (without symmetries)

Ωε := fx 2 Ω : dist(x ,M) > εg,

work in progress by Juan Carlos Fernández shows there areat least two solutions.

Is it true that, as for punctured domains, the number ofsolutions increases arbitrarily as ε ! 0?

Are there bubble towers? Are there multibump solutions?

Are there solutions with layers concentrating along M asε ! 0?

Page 135: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Domains with a thin holeOpen problems

ProblemIn general domains of the form (without symmetries)

Ωε := fx 2 Ω : dist(x ,M) > εg,

work in progress by Juan Carlos Fernández shows there areat least two solutions.

Is it true that, as for punctured domains, the number ofsolutions increases arbitrarily as ε ! 0?

Are there bubble towers? Are there multibump solutions? Are there solutions with layers concentrating along M as

ε ! 0?

Page 136: Mónica Clapp - NAU jan.ucc.nau.edu web serverjan.ucc.nau.edu/~jmn3/var12/talks/Clapp(1).pdf · Mónica Clapp Introduction Yamabe™s problem The variational problem The classical

critical &supercritical

Mónica Clapp

Introduction

Yamabesproblem

Thevariationalproblem

The classicalresults

The geometricstructure

Multiplesolutions

Punctureddomains

Symmetries

Thin holes

The end

Thanks

Thank you very much for your attention !