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![Page 1: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/1.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Krein spaces applied to Friedrichs’ systems
Kresimir Burazin
Department of Mathematics, University of Osijek
May 2009
Joint work with Nenad Antonic
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 2: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/2.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Overview
Introduction . . . basic notions of Friedrichs’ systems
Abstract formulation . . . a new approach – A. Ern, J.-L. Guermond,G. Caplain
Interdependence of different representations of boundary conditions. . . open problem
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 3: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/3.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Overview
Introduction . . . basic notions of Friedrichs’ systems
Abstract formulation . . . a new approach – A. Ern, J.-L. Guermond,G. Caplain
Interdependence of different representations of boundary conditions. . . open problem
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 4: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/4.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Overview
Introduction . . . basic notions of Friedrichs’ systems
Abstract formulation . . . a new approach – A. Ern, J.-L. Guermond,G. Caplain
Interdependence of different representations of boundary conditions. . . open problem
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 5: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/5.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Motivation
Introduced in:K. O. Friedrichs: Symmetric positive linear differential equations,Communications on Pure and Applied Mathematics 11 (1958), 333–418
Goal:
-treating the equations of mixed type, such as the Tricomi equation:
y∂2u
∂x2+∂2u
∂y2= 0 ;
-unified treatment of equations and systems of different type.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 6: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/6.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Motivation
Introduced in:K. O. Friedrichs: Symmetric positive linear differential equations,Communications on Pure and Applied Mathematics 11 (1958), 333–418
Goal:
-treating the equations of mixed type, such as the Tricomi equation:
y∂2u
∂x2+∂2u
∂y2= 0 ;
-unified treatment of equations and systems of different type.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 7: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/7.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Motivation
Introduced in:K. O. Friedrichs: Symmetric positive linear differential equations,Communications on Pure and Applied Mathematics 11 (1958), 333–418
Goal:
-treating the equations of mixed type, such as the Tricomi equation:
y∂2u
∂x2+∂2u
∂y2= 0 ;
-unified treatment of equations and systems of different type.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 8: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/8.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Friedrichs’ system
Assumptions:d, r ∈ N, Ω ⊆ Rd open and bounded with Lipschitz boundary;
Ak ∈W1,∞(Ω; Mr(R)), k ∈ 1..d, and C ∈ L∞(Ω; Mr(R)) satisfy
(F1) matrix functions Ak are symmetric: Ak = A>k ;
(F2) (∃µ0 > 0) C + C> +d∑
k=1
∂kAk ≥ 2µ0I (ae on Ω) .
The operator L : L2(Ω; Rr) −→ D′(Ω; Rr)
Lu :=d∑
k=1
∂k(Aku) + Cu
is called symmetric positive operator or the Friedrichs operator, and
Lu = f
is called symmetric positive system or the Friedrichs system.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 9: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/9.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Friedrichs’ system
Assumptions:d, r ∈ N, Ω ⊆ Rd open and bounded with Lipschitz boundary;
Ak ∈W1,∞(Ω; Mr(R)), k ∈ 1..d, and C ∈ L∞(Ω; Mr(R))
satisfy
(F1) matrix functions Ak are symmetric: Ak = A>k ;
(F2) (∃µ0 > 0) C + C> +d∑
k=1
∂kAk ≥ 2µ0I (ae on Ω) .
The operator L : L2(Ω; Rr) −→ D′(Ω; Rr)
Lu :=d∑
k=1
∂k(Aku) + Cu
is called symmetric positive operator or the Friedrichs operator, and
Lu = f
is called symmetric positive system or the Friedrichs system.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 10: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/10.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Friedrichs’ system
Assumptions:d, r ∈ N, Ω ⊆ Rd open and bounded with Lipschitz boundary;
Ak ∈W1,∞(Ω; Mr(R)), k ∈ 1..d, and C ∈ L∞(Ω; Mr(R)) satisfy
(F1) matrix functions Ak are symmetric: Ak = A>k ;
(F2) (∃µ0 > 0) C + C> +d∑
k=1
∂kAk ≥ 2µ0I (ae on Ω) .
The operator L : L2(Ω; Rr) −→ D′(Ω; Rr)
Lu :=d∑
k=1
∂k(Aku) + Cu
is called symmetric positive operator or the Friedrichs operator, and
Lu = f
is called symmetric positive system or the Friedrichs system.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 11: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/11.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Friedrichs’ system
Assumptions:d, r ∈ N, Ω ⊆ Rd open and bounded with Lipschitz boundary;
Ak ∈W1,∞(Ω; Mr(R)), k ∈ 1..d, and C ∈ L∞(Ω; Mr(R)) satisfy
(F1) matrix functions Ak are symmetric: Ak = A>k ;
(F2) (∃µ0 > 0) C + C> +d∑
k=1
∂kAk ≥ 2µ0I (ae on Ω) .
The operator L : L2(Ω; Rr) −→ D′(Ω; Rr)
Lu :=d∑
k=1
∂k(Aku) + Cu
is called symmetric positive operator or the Friedrichs operator, and
Lu = f
is called symmetric positive system or the Friedrichs system.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 12: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/12.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Friedrichs’ system
Assumptions:d, r ∈ N, Ω ⊆ Rd open and bounded with Lipschitz boundary;
Ak ∈W1,∞(Ω; Mr(R)), k ∈ 1..d, and C ∈ L∞(Ω; Mr(R)) satisfy
(F1) matrix functions Ak are symmetric: Ak = A>k ;
(F2) (∃µ0 > 0) C + C> +d∑
k=1
∂kAk ≥ 2µ0I (ae on Ω) .
The operator L : L2(Ω; Rr) −→ D′(Ω; Rr)
Lu :=d∑
k=1
∂k(Aku) + Cu
is called symmetric positive operator or the Friedrichs operator,
and
Lu = f
is called symmetric positive system or the Friedrichs system.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 13: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/13.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Friedrichs’ system
Assumptions:d, r ∈ N, Ω ⊆ Rd open and bounded with Lipschitz boundary;
Ak ∈W1,∞(Ω; Mr(R)), k ∈ 1..d, and C ∈ L∞(Ω; Mr(R)) satisfy
(F1) matrix functions Ak are symmetric: Ak = A>k ;
(F2) (∃µ0 > 0) C + C> +d∑
k=1
∂kAk ≥ 2µ0I (ae on Ω) .
The operator L : L2(Ω; Rr) −→ D′(Ω; Rr)
Lu :=d∑
k=1
∂k(Aku) + Cu
is called symmetric positive operator or the Friedrichs operator, and
Lu = f
is called symmetric positive system or the Friedrichs system.Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 14: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/14.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Boundary conditions
Boundary conditions are enforced via matrix valued boundary field:
let
Aν :=d∑
k=1
νkAk ∈ L∞(∂Ω; Mr(R)) ,
where ν = (ν1, ν2, · · · , νd) is the outward unit normal on ∂Ω, and
M ∈ L∞(∂Ω; Mr(R)).
Boundary condition(Aν −M)u|∂Ω
= 0
allows treatment of different types of boundary conditions.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 15: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/15.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Boundary conditions
Boundary conditions are enforced via matrix valued boundary field:
let
Aν :=d∑
k=1
νkAk ∈ L∞(∂Ω; Mr(R)) ,
where ν = (ν1, ν2, · · · , νd) is the outward unit normal on ∂Ω,
and
M ∈ L∞(∂Ω; Mr(R)).
Boundary condition(Aν −M)u|∂Ω
= 0
allows treatment of different types of boundary conditions.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 16: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/16.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Boundary conditions
Boundary conditions are enforced via matrix valued boundary field:
let
Aν :=d∑
k=1
νkAk ∈ L∞(∂Ω; Mr(R)) ,
where ν = (ν1, ν2, · · · , νd) is the outward unit normal on ∂Ω, and
M ∈ L∞(∂Ω; Mr(R)).
Boundary condition(Aν −M)u|∂Ω
= 0
allows treatment of different types of boundary conditions.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 17: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/17.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Boundary conditions
Boundary conditions are enforced via matrix valued boundary field:
let
Aν :=d∑
k=1
νkAk ∈ L∞(∂Ω; Mr(R)) ,
where ν = (ν1, ν2, · · · , νd) is the outward unit normal on ∂Ω, and
M ∈ L∞(∂Ω; Mr(R)).
Boundary condition(Aν −M)u|∂Ω
= 0
allows treatment of different types of boundary conditions.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 18: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/18.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Boundary conditions
Boundary conditions are enforced via matrix valued boundary field:
let
Aν :=d∑
k=1
νkAk ∈ L∞(∂Ω; Mr(R)) ,
where ν = (ν1, ν2, · · · , νd) is the outward unit normal on ∂Ω, and
M ∈ L∞(∂Ω; Mr(R)).
Boundary condition(Aν −M)u|∂Ω
= 0
allows treatment of different types of boundary conditions.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 19: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/19.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Assumptions on the boundary matrix M
We assume (for ae x ∈ ∂Ω)
(FM1) (∀ ξ ∈ Rr) M(x)ξ · ξ ≥ 0 ,
(FM2) Rr = Ker(Aν(x)−M(x)
)+ Ker
(Aν(x) + M(x)
).
such M is called the admissible boundary condition.
Boundary problem: for given f ∈ L2(Ω; Rr) find u such thatLu = f
(Aν −M)u|∂Ω= 0 .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 20: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/20.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Assumptions on the boundary matrix M
We assume (for ae x ∈ ∂Ω)
(FM1) (∀ ξ ∈ Rr) M(x)ξ · ξ ≥ 0 ,
(FM2) Rr = Ker(Aν(x)−M(x)
)+ Ker
(Aν(x) + M(x)
).
such M is called the admissible boundary condition.
Boundary problem: for given f ∈ L2(Ω; Rr) find u such thatLu = f
(Aν −M)u|∂Ω= 0 .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 21: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/21.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Assumptions on the boundary matrix M
We assume (for ae x ∈ ∂Ω)
(FM1) (∀ ξ ∈ Rr) M(x)ξ · ξ ≥ 0 ,
(FM2) Rr = Ker(Aν(x)−M(x)
)+ Ker
(Aν(x) + M(x)
).
such M is called the admissible boundary condition.
Boundary problem: for given f ∈ L2(Ω; Rr) find u such thatLu = f
(Aν −M)u|∂Ω= 0 .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 22: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/22.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Assumptions on the boundary matrix M
We assume (for ae x ∈ ∂Ω)
(FM1) (∀ ξ ∈ Rr) M(x)ξ · ξ ≥ 0 ,
(FM2) Rr = Ker(Aν(x)−M(x)
)+ Ker
(Aν(x) + M(x)
).
such M is called the admissible boundary condition.
Boundary problem: for given f ∈ L2(Ω; Rr) find u such thatLu = f
(Aν −M)u|∂Ω= 0 .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 23: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/23.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Different ways to enforce boundary conditions
Instead of(Aν −M)u = 0 on ∂Ω ,
we propose boundary conditions with
u(x) ∈ N(x) , x ∈ ∂Ω ,
where N = N(x) : x ∈ ∂Ω is a family of subspaces of Rr.
Boundary problem: Lu = f
u(x) ∈ N(x) , x ∈ ∂Ω.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 24: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/24.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Different ways to enforce boundary conditions
Instead of(Aν −M)u = 0 on ∂Ω ,
we propose boundary conditions with
u(x) ∈ N(x) , x ∈ ∂Ω ,
where N = N(x) : x ∈ ∂Ω is a family of subspaces of Rr.
Boundary problem: Lu = f
u(x) ∈ N(x) , x ∈ ∂Ω.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 25: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/25.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Assumptions on N
maximal boundary conditions: (for ae x ∈ ∂Ω)
(FX1)N(x) is non-negative with respect to Aν(x):
(∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≥ 0 ;
(FX2)there is no non-negative subspace with respect to
Aν(x), which contains N(x) ;
or
Let N(x) and N(x) := (Aν(x)N(x))⊥ satisfy (for ae x ∈ ∂Ω)
(FV1)(∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≥ 0
(∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≤ 0
(FV2) N(x) = (Aν(x)N(x))⊥ and N(x) = (Aν(x)N(x))⊥ .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 26: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/26.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Assumptions on N
maximal boundary conditions: (for ae x ∈ ∂Ω)
(FX1)N(x) is non-negative with respect to Aν(x):
(∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≥ 0 ;
(FX2)there is no non-negative subspace with respect to
Aν(x), which contains N(x) ;
or
Let N(x) and N(x) := (Aν(x)N(x))⊥ satisfy (for ae x ∈ ∂Ω)
(FV1)(∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≥ 0
(∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≤ 0
(FV2) N(x) = (Aν(x)N(x))⊥ and N(x) = (Aν(x)N(x))⊥ .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 27: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/27.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Assumptions on N
maximal boundary conditions: (for ae x ∈ ∂Ω)
(FX1)N(x) is non-negative with respect to Aν(x):
(∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≥ 0 ;
(FX2)there is no non-negative subspace with respect to
Aν(x), which contains N(x) ;
or
Let N(x) and N(x) := (Aν(x)N(x))⊥ satisfy (for ae x ∈ ∂Ω)
(FV1)(∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≥ 0
(∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≤ 0
(FV2) N(x) = (Aν(x)N(x))⊥ and N(x) = (Aν(x)N(x))⊥ .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 28: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/28.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Equivalence of different descriptions of boundary conditions
Theorem
It holds
(FM1)–(FM2) ⇐⇒ (FX1)–(FX2) ⇐⇒ (FV1)–(FV2) ,
withN(x) := Ker
(Aν(x)−M(x)
).
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Classical results
Friedrichs:- uniqueness of classical solution- existence of weak solution (under an additional assumptions)
Contributions:
C. Morawetz, P. Lax, L. Sarason, R. S. Phillips, J. Rauch, . . .
- the meaning of traces for functions in the graph space
- well-posedness results under additional assumptions (on Aν)
- regularity of solution
- numerical treatment
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 30: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/30.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Classical results
Friedrichs:- uniqueness of classical solution- existence of weak solution (under an additional assumptions)
Contributions:
C. Morawetz, P. Lax, L. Sarason, R. S. Phillips, J. Rauch, . . .
- the meaning of traces for functions in the graph space
- well-posedness results under additional assumptions (on Aν)
- regularity of solution
- numerical treatment
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 31: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/31.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Classical results
Friedrichs:- uniqueness of classical solution- existence of weak solution (under an additional assumptions)
Contributions:
C. Morawetz, P. Lax, L. Sarason, R. S. Phillips, J. Rauch, . . .
- the meaning of traces for functions in the graph space
- well-posedness results under additional assumptions (on Aν)
- regularity of solution
- numerical treatment
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 32: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/32.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Classical results
Friedrichs:- uniqueness of classical solution- existence of weak solution (under an additional assumptions)
Contributions:
C. Morawetz, P. Lax, L. Sarason, R. S. Phillips, J. Rauch, . . .
- the meaning of traces for functions in the graph space
- well-posedness results under additional assumptions (on Aν)
- regularity of solution
- numerical treatment
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 33: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/33.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Classical results
Friedrichs:- uniqueness of classical solution- existence of weak solution (under an additional assumptions)
Contributions:
C. Morawetz, P. Lax, L. Sarason, R. S. Phillips, J. Rauch, . . .
- the meaning of traces for functions in the graph space
- well-posedness results under additional assumptions (on Aν)
- regularity of solution
- numerical treatment
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 34: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/34.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Classical results
Friedrichs:- uniqueness of classical solution- existence of weak solution (under an additional assumptions)
Contributions:
C. Morawetz, P. Lax, L. Sarason, R. S. Phillips, J. Rauch, . . .
- the meaning of traces for functions in the graph space
- well-posedness results under additional assumptions (on Aν)
- regularity of solution
- numerical treatment
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 35: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/35.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
New approach...
A. Ern, J.-L. Guermond, G. Caplain: An Intrinsic Criterion for theBijectivity Of Hilbert Operators Related to Friedrichs’ Systems,Communications in Partial Differential Equations, 32 (2007), 317–341
-abstract setting (operators on Hilbert spaces)
-intrinsic criterion for bijectivity of Friedrichs’ operator
-avoiding the question of traces for functions in the graph space
-investigation of different formulations of boundary conditions
. . . and new open questions
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 36: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/36.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
New approach...
A. Ern, J.-L. Guermond, G. Caplain: An Intrinsic Criterion for theBijectivity Of Hilbert Operators Related to Friedrichs’ Systems,Communications in Partial Differential Equations, 32 (2007), 317–341
-abstract setting (operators on Hilbert spaces)
-intrinsic criterion for bijectivity of Friedrichs’ operator
-avoiding the question of traces for functions in the graph space
-investigation of different formulations of boundary conditions
. . . and new open questions
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 37: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/37.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
New approach...
A. Ern, J.-L. Guermond, G. Caplain: An Intrinsic Criterion for theBijectivity Of Hilbert Operators Related to Friedrichs’ Systems,Communications in Partial Differential Equations, 32 (2007), 317–341
-abstract setting (operators on Hilbert spaces)
-intrinsic criterion for bijectivity of Friedrichs’ operator
-avoiding the question of traces for functions in the graph space
-investigation of different formulations of boundary conditions
. . . and new open questions
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 38: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/38.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
New approach...
A. Ern, J.-L. Guermond, G. Caplain: An Intrinsic Criterion for theBijectivity Of Hilbert Operators Related to Friedrichs’ Systems,Communications in Partial Differential Equations, 32 (2007), 317–341
-abstract setting (operators on Hilbert spaces)
-intrinsic criterion for bijectivity of Friedrichs’ operator
-avoiding the question of traces for functions in the graph space
-investigation of different formulations of boundary conditions
. . . and new open questions
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 39: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/39.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
New approach...
A. Ern, J.-L. Guermond, G. Caplain: An Intrinsic Criterion for theBijectivity Of Hilbert Operators Related to Friedrichs’ Systems,Communications in Partial Differential Equations, 32 (2007), 317–341
-abstract setting (operators on Hilbert spaces)
-intrinsic criterion for bijectivity of Friedrichs’ operator
-avoiding the question of traces for functions in the graph space
-investigation of different formulations of boundary conditions
. . . and new open questions
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 40: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/40.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
New approach...
A. Ern, J.-L. Guermond, G. Caplain: An Intrinsic Criterion for theBijectivity Of Hilbert Operators Related to Friedrichs’ Systems,Communications in Partial Differential Equations, 32 (2007), 317–341
-abstract setting (operators on Hilbert spaces)
-intrinsic criterion for bijectivity of Friedrichs’ operator
-avoiding the question of traces for functions in the graph space
-investigation of different formulations of boundary conditions
. . . and new open questions
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 41: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/41.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Assumptions
L - a real Hilbert space (L′ ≡ L), D ⊆ L a dense subspace, andT, T : D −→ L linear unbounded operators satisfying
(T1) (∀ϕ,ψ ∈ D) 〈Tϕ | ψ 〉L = 〈ϕ | Tψ 〉L ;
(T2) (∃ c > 0)(∀ϕ ∈ D) ‖(T + T )ϕ‖L ≤ c‖ϕ‖L ;
(T3) (∃µ0 > 0)(∀ϕ ∈ D) 〈 (T + T )ϕ | ϕ 〉L ≥ 2µ0‖ϕ‖2L .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
An example: Friedrichs operator
Let D := D(Ω; Rr), L = L2(Ω; Rr) and T, T : D −→ L defined with
Tu :=d∑
k=1
∂k(Aku) + Cu ,
Tu :=−d∑
k=1
∂k(A>k u) + (C> +d∑
k=1
∂kA>k )u ,
where Ak and C are as before (they satisfy (F1)–(F2)).
Then T i T satisfy (T1)–(T3)
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Extensions
(D, 〈 · | · 〉T ) is an inner product space, where
〈 · | · 〉T := 〈 · | · 〉L + 〈T · | T · 〉L .
‖ · ‖T is called graph norm.
W0 - the completion of DT, T : D −→ L are continuous with respect to (‖ · ‖T , ‖ · ‖L). . . extension by density to L(W0, L)The following imbeddings are continuous:
W0 → L ≡ L′ →W ′0 .
Let T ∗ ∈ L(L,W ′0) be the adjoint operator of T : W0 −→ L
(∀u ∈ L)(∀ v ∈W0) W ′0〈 T ∗u, v 〉W0 = 〈u | T v 〉L .
Therefore T = T ∗|W0
Analogously T = T ∗|W0
Abusing notation: T, T ∈ L(L,W ′0) . . . (T1)–(T3)
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Extensions
(D, 〈 · | · 〉T ) is an inner product space, where
〈 · | · 〉T := 〈 · | · 〉L + 〈T · | T · 〉L .
‖ · ‖T is called graph norm.W0 - the completion of D
T, T : D −→ L are continuous with respect to (‖ · ‖T , ‖ · ‖L). . . extension by density to L(W0, L)The following imbeddings are continuous:
W0 → L ≡ L′ →W ′0 .
Let T ∗ ∈ L(L,W ′0) be the adjoint operator of T : W0 −→ L
(∀u ∈ L)(∀ v ∈W0) W ′0〈 T ∗u, v 〉W0 = 〈u | T v 〉L .
Therefore T = T ∗|W0
Analogously T = T ∗|W0
Abusing notation: T, T ∈ L(L,W ′0) . . . (T1)–(T3)
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 45: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/45.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Extensions
(D, 〈 · | · 〉T ) is an inner product space, where
〈 · | · 〉T := 〈 · | · 〉L + 〈T · | T · 〉L .
‖ · ‖T is called graph norm.W0 - the completion of DT, T : D −→ L are continuous with respect to (‖ · ‖T , ‖ · ‖L). . . extension by density to L(W0, L)
The following imbeddings are continuous:
W0 → L ≡ L′ →W ′0 .
Let T ∗ ∈ L(L,W ′0) be the adjoint operator of T : W0 −→ L
(∀u ∈ L)(∀ v ∈W0) W ′0〈 T ∗u, v 〉W0 = 〈u | T v 〉L .
Therefore T = T ∗|W0
Analogously T = T ∗|W0
Abusing notation: T, T ∈ L(L,W ′0) . . . (T1)–(T3)
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 46: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/46.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Extensions
(D, 〈 · | · 〉T ) is an inner product space, where
〈 · | · 〉T := 〈 · | · 〉L + 〈T · | T · 〉L .
‖ · ‖T is called graph norm.W0 - the completion of DT, T : D −→ L are continuous with respect to (‖ · ‖T , ‖ · ‖L). . . extension by density to L(W0, L)The following imbeddings are continuous:
W0 → L ≡ L′ →W ′0 .
Let T ∗ ∈ L(L,W ′0) be the adjoint operator of T : W0 −→ L
(∀u ∈ L)(∀ v ∈W0) W ′0〈 T ∗u, v 〉W0 = 〈u | T v 〉L .
Therefore T = T ∗|W0
Analogously T = T ∗|W0
Abusing notation: T, T ∈ L(L,W ′0) . . . (T1)–(T3)
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 47: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/47.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Extensions
(D, 〈 · | · 〉T ) is an inner product space, where
〈 · | · 〉T := 〈 · | · 〉L + 〈T · | T · 〉L .
‖ · ‖T is called graph norm.W0 - the completion of DT, T : D −→ L are continuous with respect to (‖ · ‖T , ‖ · ‖L). . . extension by density to L(W0, L)The following imbeddings are continuous:
W0 → L ≡ L′ →W ′0 .
Let T ∗ ∈ L(L,W ′0) be the adjoint operator of T : W0 −→ L
(∀u ∈ L)(∀ v ∈W0) W ′0〈 T ∗u, v 〉W0 = 〈u | T v 〉L .
Therefore T = T ∗|W0
Analogously T = T ∗|W0
Abusing notation: T, T ∈ L(L,W ′0) . . . (T1)–(T3)
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 48: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/48.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Extensions
(D, 〈 · | · 〉T ) is an inner product space, where
〈 · | · 〉T := 〈 · | · 〉L + 〈T · | T · 〉L .
‖ · ‖T is called graph norm.W0 - the completion of DT, T : D −→ L are continuous with respect to (‖ · ‖T , ‖ · ‖L). . . extension by density to L(W0, L)The following imbeddings are continuous:
W0 → L ≡ L′ →W ′0 .
Let T ∗ ∈ L(L,W ′0) be the adjoint operator of T : W0 −→ L
(∀u ∈ L)(∀ v ∈W0) W ′0〈 T ∗u, v 〉W0 = 〈u | T v 〉L .
Therefore T = T ∗|W0
Analogously T = T ∗|W0
Abusing notation: T, T ∈ L(L,W ′0) . . . (T1)–(T3)
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 49: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/49.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Extensions
(D, 〈 · | · 〉T ) is an inner product space, where
〈 · | · 〉T := 〈 · | · 〉L + 〈T · | T · 〉L .
‖ · ‖T is called graph norm.W0 - the completion of DT, T : D −→ L are continuous with respect to (‖ · ‖T , ‖ · ‖L). . . extension by density to L(W0, L)The following imbeddings are continuous:
W0 → L ≡ L′ →W ′0 .
Let T ∗ ∈ L(L,W ′0) be the adjoint operator of T : W0 −→ L
(∀u ∈ L)(∀ v ∈W0) W ′0〈 T ∗u, v 〉W0 = 〈u | T v 〉L .
Therefore T = T ∗|W0
Analogously T = T ∗|W0
Abusing notation: T, T ∈ L(L,W ′0) . . . (T1)–(T3)
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 50: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/50.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Extensions
(D, 〈 · | · 〉T ) is an inner product space, where
〈 · | · 〉T := 〈 · | · 〉L + 〈T · | T · 〉L .
‖ · ‖T is called graph norm.W0 - the completion of DT, T : D −→ L are continuous with respect to (‖ · ‖T , ‖ · ‖L). . . extension by density to L(W0, L)The following imbeddings are continuous:
W0 → L ≡ L′ →W ′0 .
Let T ∗ ∈ L(L,W ′0) be the adjoint operator of T : W0 −→ L
(∀u ∈ L)(∀ v ∈W0) W ′0〈 T ∗u, v 〉W0 = 〈u | T v 〉L .
Therefore T = T ∗|W0
Analogously T = T ∗|W0
Abusing notation: T, T ∈ L(L,W ′0) . . . (T1)–(T3)
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 51: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/51.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Extensions
(D, 〈 · | · 〉T ) is an inner product space, where
〈 · | · 〉T := 〈 · | · 〉L + 〈T · | T · 〉L .
‖ · ‖T is called graph norm.W0 - the completion of DT, T : D −→ L are continuous with respect to (‖ · ‖T , ‖ · ‖L). . . extension by density to L(W0, L)The following imbeddings are continuous:
W0 → L ≡ L′ →W ′0 .
Let T ∗ ∈ L(L,W ′0) be the adjoint operator of T : W0 −→ L
(∀u ∈ L)(∀ v ∈W0) W ′0〈 T ∗u, v 〉W0 = 〈u | T v 〉L .
Therefore T = T ∗|W0
Analogously T = T ∗|W0
Abusing notation: T, T ∈ L(L,W ′0) . . . (T1)–(T3)Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Formulation of the problem
Lemma
The graph space
W := u ∈ L : Tu ∈ L = u ∈ L : T u ∈ L ,
is a Hilbert space with respect to 〈 · | · 〉T .
Problem: for given f ∈ L find u ∈W such that Tu = f .
Find sufficient conditions on V 6 W such that T|V : V −→ L is an
isomorphism.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Formulation of the problem
Lemma
The graph space
W := u ∈ L : Tu ∈ L = u ∈ L : T u ∈ L ,
is a Hilbert space with respect to 〈 · | · 〉T .
Problem: for given f ∈ L find u ∈W such that Tu = f .
Find sufficient conditions on V 6 W such that T|V : V −→ L is an
isomorphism.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Formulation of the problem
Lemma
The graph space
W := u ∈ L : Tu ∈ L = u ∈ L : T u ∈ L ,
is a Hilbert space with respect to 〈 · | · 〉T .
Problem: for given f ∈ L find u ∈W such that Tu = f .
Find sufficient conditions on V 6 W such that T|V : V −→ L is an
isomorphism.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Boundary operator
Boundary operator D ∈ L(W,W ′):
W ′〈Du, v 〉W := 〈Tu | v 〉L − 〈u | T v 〉L , u, v ∈W .
Lemma
D is symmetric and satisfies
KerD = W0
ImD = W 00 = g ∈W ′ : (∀u ∈W0) W ′〈 g, u 〉W = 0 .
In particular, ImD is closed in W ′.
If T is the Friedrichs operator L, then for u, v ∈ D(Rd; Rr) we have
W ′〈Du, v 〉W =∫
∂Ω
Aν(x)u|∂Ω(x) · v|∂Ω
(x)dS(x) .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Boundary operator
Boundary operator D ∈ L(W,W ′):
W ′〈Du, v 〉W := 〈Tu | v 〉L − 〈u | T v 〉L , u, v ∈W .
Lemma
D is symmetric and satisfies
KerD = W0
ImD = W 00 = g ∈W ′ : (∀u ∈W0) W ′〈 g, u 〉W = 0 .
In particular, ImD is closed in W ′.
If T is the Friedrichs operator L, then for u, v ∈ D(Rd; Rr) we have
W ′〈Du, v 〉W =∫
∂Ω
Aν(x)u|∂Ω(x) · v|∂Ω
(x)dS(x) .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Boundary operator
Boundary operator D ∈ L(W,W ′):
W ′〈Du, v 〉W := 〈Tu | v 〉L − 〈u | T v 〉L , u, v ∈W .
Lemma
D is symmetric and satisfies
KerD = W0
ImD = W 00 = g ∈W ′ : (∀u ∈W0) W ′〈 g, u 〉W = 0 .
In particular, ImD is closed in W ′.
If T is the Friedrichs operator L, then for u, v ∈ D(Rd; Rr) we have
W ′〈Du, v 〉W =∫
∂Ω
Aν(x)u|∂Ω(x) · v|∂Ω
(x)dS(x) .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Well-posedness theorem
Let V and V be subspaces of W that satisfy
(V1)(∀u ∈ V ) W ′〈Du, u 〉W ≥ 0
(∀ v ∈ V ) W ′〈Dv, v 〉W ≤ 0
(V2) V = D(V )0 , V = D(V )0 .
Theorem
Under assumptions (T1)− (T3) and (V 1)− (V 2), the operatorsT|V : V −→ L and T|V
: V −→ L are isomorphisms.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Well-posedness theorem
Let V and V be subspaces of W that satisfy
(V1)(∀u ∈ V ) W ′〈Du, u 〉W ≥ 0
(∀ v ∈ V ) W ′〈Dv, v 〉W ≤ 0
(V2) V = D(V )0 , V = D(V )0 .
Theorem
Under assumptions (T1)− (T3) and (V 1)− (V 2), the operatorsT|V : V −→ L and T|V
: V −→ L are isomorphisms.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Correlation with old assumptions
(V1)(∀u ∈ V ) W ′〈Du, u 〉W ≥ 0 ,
(∀ v ∈ V ) W ′〈Dv, v 〉W ≤ 0 ,
(V2) V = D(V )0 , V = D(V )0 ,
(FV1)(∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≥ 0 ,
(∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≤ 0 ,
(FV2) N(x) = (Aν(x)N(x))⊥ and N(x) = (Aν(x)N(x))⊥ ,
(for ae x ∈ ∂Ω)
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Correlation with old assumptions
(V1)(∀u ∈ V ) W ′〈Du, u 〉W ≥ 0 ,
(∀ v ∈ V ) W ′〈Dv, v 〉W ≤ 0 ,
(V2) V = D(V )0 , V = D(V )0 ,
(FV1)(∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≥ 0 ,
(∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≤ 0 ,
(FV2) N(x) = (Aν(x)N(x))⊥ and N(x) = (Aν(x)N(x))⊥ ,
(for ae x ∈ ∂Ω)
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Assumptions in classical setting
maximal boundary conditions: (for ae x ∈ ∂Ω)
(FX1) (∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≥ 0 ,
(FX2)there is no non-negative subspace with respect to
Aν(x), which contains N(x) ,
admissible boundary condition: there exists a matrix functionM : ∂Ω −→ Mr(R) such that (for ae x ∈ ∂Ω)
(FM1) (∀ ξ ∈ Rr) M(x)ξ · ξ ≥ 0 ,
(FM2) Rr = Ker(Aν(x)−M(x)
)+ Ker
(Aν(x) + M(x)
).
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Assumptions in classical setting
maximal boundary conditions: (for ae x ∈ ∂Ω)
(FX1) (∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≥ 0 ,
(FX2)there is no non-negative subspace with respect to
Aν(x), which contains N(x) ,
admissible boundary condition: there exists a matrix functionM : ∂Ω −→ Mr(R) such that (for ae x ∈ ∂Ω)
(FM1) (∀ ξ ∈ Rr) M(x)ξ · ξ ≥ 0 ,
(FM2) Rr = Ker(Aν(x)−M(x)
)+ Ker
(Aν(x) + M(x)
).
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Correlation with old assumptions – maximal b.c.
maximal boundary conditions: (for ae x ∈ ∂Ω)
(FX1) (∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≥ 0 ,
(FX2)there is no non-negative subspace with respect to
Aν(x), which contains N(x) ,
subspace V is maximal non-negative with respect to D:(X1)V is non-negative with respect to D: (∀ v ∈ V ) W ′〈Dv, v 〉W ≥ 0 ,
(X2)there is no non-negative subspace with respect to D that contains V .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Correlation with old assumptions – maximal b.c.
maximal boundary conditions: (for ae x ∈ ∂Ω)
(FX1) (∀ ξ ∈ N(x)) Aν(x)ξ · ξ ≥ 0 ,
(FX2)there is no non-negative subspace with respect to
Aν(x), which contains N(x) ,
subspace V is maximal non-negative with respect to D:(X1)V is non-negative with respect to D: (∀ v ∈ V ) W ′〈Dv, v 〉W ≥ 0 ,
(X2)there is no non-negative subspace with respect to D that contains V .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Correlation with old assumptions – admissible b.c.
admissible boundary condition: there exist a matrix functionM : ∂Ω −→ Mr(R) such that (for ae x ∈ ∂Ω)
(FM1) (∀ ξ ∈ Rr) M(x)ξ · ξ ≥ 0 ,
(FM2) Rr = Ker(Aν(x)−M(x)
)+ Ker
(Aν(x) + M(x)
).
admissible boundary condition: there exist M ∈ L(W,W ′) that satisfy
(M1) (∀u ∈W ) W ′〈Mu, u 〉W ≥ 0 ,
(M2) W = Ker (D −M) + Ker (D +M) .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Correlation with old assumptions – admissible b.c.
admissible boundary condition: there exist a matrix functionM : ∂Ω −→ Mr(R) such that (for ae x ∈ ∂Ω)
(FM1) (∀ ξ ∈ Rr) M(x)ξ · ξ ≥ 0 ,
(FM2) Rr = Ker(Aν(x)−M(x)
)+ Ker
(Aν(x) + M(x)
).
admissible boundary condition: there exist M ∈ L(W,W ′) that satisfy
(M1) (∀u ∈W ) W ′〈Mu, u 〉W ≥ 0 ,
(M2) W = Ker (D −M) + Ker (D +M) .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Equivalence of different descriptions of b.c.
Theorem
It holds
(FM1)–(FM2) ⇐⇒ (FX1)–(FX2) ⇐⇒ (FV1)–(FV2) ,
withN(x) := Ker
(Aν(x)−M(x)
).
Theorem
A. Ern, J.-L. Guermond, G. Caplain: It holds
(M1)–(M2)=⇒←−
(V1)–(V2) =⇒ (X1)–(X2) ,
withV := Ker (D −M) .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Equivalence of different descriptions of b.c.
Theorem
It holds
(FM1)–(FM2) ⇐⇒ (FX1)–(FX2) ⇐⇒ (FV1)–(FV2) ,
withN(x) := Ker
(Aν(x)−M(x)
).
Theorem
A. Ern, J.-L. Guermond, G. Caplain: It holds
(M1)–(M2)=⇒←−
(V1)–(V2) =⇒ (X1)–(X2) ,
withV := Ker (D −M) .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
(M1)–(M2) ←− (V1)–(V2)
Theorem
Let V and V satisfy (V1)–(V2), and suppose that there exist operatorsP ∈ L(W,V ) and Q ∈ L(W, V ) such that
(∀ v ∈ V ) D(v − Pv) = 0 ,
(∀ v ∈ V ) D(v −Qv) = 0 ,DPQ = DQP .
Let us define M ∈ L(W,W ′) (for u, v ∈W ) with
W ′〈Mu, v 〉W = W ′〈DPu,Pv 〉W −W ′〈DQu,Qv 〉W+ W ′〈D(P +Q− PQ)u, v 〉W −W ′〈Du, (P +Q− PQ)v 〉W .
Then V := Ker (D −M), V := Ker (D +M∗), and M satisfies(M1)–(M2).
Lemma
Suppose additionally that V + V is closed. Then the operators P and Qfrom previous theorem do exist.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
(M1)–(M2) ←− (V1)–(V2)
Theorem
Let V and V satisfy (V1)–(V2), and suppose that there exist operatorsP ∈ L(W,V ) and Q ∈ L(W, V ) such that
(∀ v ∈ V ) D(v − Pv) = 0 ,
(∀ v ∈ V ) D(v −Qv) = 0 ,DPQ = DQP .
Let us define M ∈ L(W,W ′) (for u, v ∈W ) with
W ′〈Mu, v 〉W = W ′〈DPu,Pv 〉W −W ′〈DQu,Qv 〉W+ W ′〈D(P +Q− PQ)u, v 〉W −W ′〈Du, (P +Q− PQ)v 〉W .
Then V := Ker (D −M), V := Ker (D +M∗), and M satisfies(M1)–(M2).
Lemma
Suppose additionally that V + V is closed. Then the operators P and Qfrom previous theorem do exist.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Our contribution
Theorem
A. Ern, J.-L. Guermond, G. Caplain: It holds
(M1)–(M2)=⇒←−
(V1)–(V2) =⇒ (X1)–(X2) ,
withV := Ker (D −M) .
Theorem
It holds
(M1)–(M2)=⇒←−/
(V1)–(V2)=⇒⇐=
(X1)–(X2) ,
withV := Ker (D −M) .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Our contribution
Theorem
A. Ern, J.-L. Guermond, G. Caplain: It holds
(M1)–(M2)=⇒←−
(V1)–(V2) =⇒ (X1)–(X2) ,
withV := Ker (D −M) .
Theorem
It holds
(M1)–(M2)=⇒←−/
(V1)–(V2)=⇒⇐=
(X1)–(X2) ,
withV := Ker (D −M) .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
New notation
[u | v ] := W ′〈Du, v 〉W = 〈Tu | v 〉L − 〈u | T v 〉L , u, v ∈W
is an indefinite inner product on W .
(V1)(∀ v ∈ V ) [ v | v ] ≥ 0 ,
(∀ v ∈ V ) [ v | v ] ≤ 0 ;
(V2) V = V [⊥] , V = V [⊥] .
([⊥] stands for [ · | · ]-orthogonal complement).
subspace V is maximal non-negative in (W, [ · | · ]):
(X1) V is non-negative in (W, [ · | · ]): (∀ v ∈ V ) [ v | v ] ≥ 0 ,
(X2) there is no non-negative subspace in (W, [ · | · ]) that contains V .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
New notation
[u | v ] := W ′〈Du, v 〉W = 〈Tu | v 〉L − 〈u | T v 〉L , u, v ∈W
is an indefinite inner product on W .
(V1)(∀ v ∈ V ) [ v | v ] ≥ 0 ,
(∀ v ∈ V ) [ v | v ] ≤ 0 ;
(V2) V = V [⊥] , V = V [⊥] .
([⊥] stands for [ · | · ]-orthogonal complement).
subspace V is maximal non-negative in (W, [ · | · ]):
(X1) V is non-negative in (W, [ · | · ]): (∀ v ∈ V ) [ v | v ] ≥ 0 ,
(X2) there is no non-negative subspace in (W, [ · | · ]) that contains V .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
New notation
[u | v ] := W ′〈Du, v 〉W = 〈Tu | v 〉L − 〈u | T v 〉L , u, v ∈W
is an indefinite inner product on W .
(V1)(∀ v ∈ V ) [ v | v ] ≥ 0 ,
(∀ v ∈ V ) [ v | v ] ≤ 0 ;
(V2) V = V [⊥] , V = V [⊥] .
([⊥] stands for [ · | · ]-orthogonal complement).
subspace V is maximal non-negative in (W, [ · | · ]):
(X1) V is non-negative in (W, [ · | · ]): (∀ v ∈ V ) [ v | v ] ≥ 0 ,
(X2) there is no non-negative subspace in (W, [ · | · ]) that contains V .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
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IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Krein spaces
(W, [ · | · ]) is not a Krein space – it is degenerated because its Grammoperator G := j D (where j : W ′ −→W is canonical isomorphism) haslarge kernel:
KerG = W0 .
Theorem
If G is the Gramm operator of the space W , then the quotient spaceW := W/KerG is a Krein space if and only if ImG is closed.
W := W/W0 is the Krein space, with
[ u | v ] := [u | v ] , u, v ∈W .
Important: ImD is closed and KerD = W0!
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 78: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/78.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Krein spaces
(W, [ · | · ]) is not a Krein space – it is degenerated because its Grammoperator G := j D (where j : W ′ −→W is canonical isomorphism) haslarge kernel:
KerG = W0 .
Theorem
If G is the Gramm operator of the space W , then the quotient spaceW := W/KerG is a Krein space if and only if ImG is closed.
W := W/W0 is the Krein space, with
[ u | v ] := [u | v ] , u, v ∈W .
Important: ImD is closed and KerD = W0!
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 79: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/79.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Krein spaces
(W, [ · | · ]) is not a Krein space – it is degenerated because its Grammoperator G := j D (where j : W ′ −→W is canonical isomorphism) haslarge kernel:
KerG = W0 .
Theorem
If G is the Gramm operator of the space W , then the quotient spaceW := W/KerG is a Krein space if and only if ImG is closed.
W := W/W0 is the Krein space
, with
[ u | v ] := [u | v ] , u, v ∈W .
Important: ImD is closed and KerD = W0!
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 80: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/80.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Krein spaces
(W, [ · | · ]) is not a Krein space – it is degenerated because its Grammoperator G := j D (where j : W ′ −→W is canonical isomorphism) haslarge kernel:
KerG = W0 .
Theorem
If G is the Gramm operator of the space W , then the quotient spaceW := W/KerG is a Krein space if and only if ImG is closed.
W := W/W0 is the Krein space, with
[ u | v ] := [u | v ] , u, v ∈W .
Important: ImD is closed and KerD = W0!
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 81: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/81.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Krein spaces
(W, [ · | · ]) is not a Krein space – it is degenerated because its Grammoperator G := j D (where j : W ′ −→W is canonical isomorphism) haslarge kernel:
KerG = W0 .
Theorem
If G is the Gramm operator of the space W , then the quotient spaceW := W/KerG is a Krein space if and only if ImG is closed.
W := W/W0 is the Krein space, with
[ u | v ] := [u | v ] , u, v ∈W .
Important: ImD is closed and KerD = W0!
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 82: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/82.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Quotient Krein space
Lemma
Let U ⊇W0 and Y be subspaces of W . Then
a) U is closed if and only if U := v : v ∈ U is closed in W ;
b)(U + Y ) = u+ v +W0 : u ∈ U, v ∈ Y = U + Y ;
c) U + Y is closed if and only if U + Y is closed;
d) (Y )[⊥] = Y [⊥].
e) if Y is maximal non-negative (non-positive) in W , than Y is maximalnon-negative (non-positive) in W ;
f) if U is maximal non-negative (non-positive) in W , then U is maximalnon-negative (non-positive) in W .
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 83: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/83.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
(V1)–(V2) ⇐⇒ (X1)–(X2)
Theorem
a) If subspaces V and V satisfy (V1)–(V2), then V is maximalnon-negative in W (satisfies (X1)–(X2)) and V is maximal non-positivein W .
b) If V is maximal non-negative in W , then V and V := V [⊥] satisfy(V1)–(V2).
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 84: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/84.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
(M1)–(M2) ←− (V1)–(V2)
Theorem
Let V and V satisfy (V1)–(V2), and suppose that there exist operatorsP ∈ L(W,V ) and Q ∈ L(W, V ) such that
(∀ v ∈ V ) D(v − Pv) = 0 ,
(∀ v ∈ V ) D(v −Qv) = 0 ,DPQ = DQP .
Let us define M ∈ L(W,W ′) (for u, v ∈W ) with
W ′〈Mu, v 〉W = W ′〈DPu,Pv 〉W −W ′〈DQu,Qv 〉W+ W ′〈D(P +Q− PQ)u, v 〉W −W ′〈Du, (P +Q− PQ)v 〉W .
Then V := Ker (D −M), V := Ker (D +M∗), and M satisfies(M1)–(M2).
Lemma
Suppose additionally that V + V is closed. Then the operators P and Qfrom previous theorem do exist.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 85: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/85.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
(M1)–(M2) ←− (V1)–(V2)
Lemma
If codimW0(= dimW/W0) is finite, then the set V + V is closedwhenever V and V satisfy (V1)–(V2).
-this corresponds to d = 1.
Sufficient conditions for a counter example:
Theorem
Let subspaces V and V of space W satisfy (V1)–(V2), V ∩ V = W0, andW 6= V + V .
Then V + V is not closed in W .
Moreover, there exists no operators P and Q with desired properties.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 86: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/86.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
(M1)–(M2) ←− (V1)–(V2)
Lemma
If codimW0(= dimW/W0) is finite, then the set V + V is closedwhenever V and V satisfy (V1)–(V2).
-this corresponds to d = 1.
Sufficient conditions for a counter example:
Theorem
Let subspaces V and V of space W satisfy (V1)–(V2), V ∩ V = W0, andW 6= V + V .
Then V + V is not closed in W .
Moreover, there exists no operators P and Q with desired properties.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 87: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/87.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
(M1)–(M2) ←− (V1)–(V2)
Lemma
If codimW0(= dimW/W0) is finite, then the set V + V is closedwhenever V and V satisfy (V1)–(V2).
-this corresponds to d = 1.
Sufficient conditions for a counter example:
Theorem
Let subspaces V and V of space W satisfy (V1)–(V2), V ∩ V = W0, andW 6= V + V .
Then V + V is not closed in W .
Moreover, there exists no operators P and Q with desired properties.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 88: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/88.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
(M1)–(M2) ←− (V1)–(V2)
Lemma
If codimW0(= dimW/W0) is finite, then the set V + V is closedwhenever V and V satisfy (V1)–(V2).
-this corresponds to d = 1.
Sufficient conditions for a counter example:
Theorem
Let subspaces V and V of space W satisfy (V1)–(V2), V ∩ V = W0, andW 6= V + V .
Then V + V is not closed in W .
Moreover, there exists no operators P and Q with desired properties.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 89: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/89.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
(M1)–(M2) ←− (V1)–(V2)
Lemma
If codimW0(= dimW/W0) is finite, then the set V + V is closedwhenever V and V satisfy (V1)–(V2).
-this corresponds to d = 1.
Sufficient conditions for a counter example:
Theorem
Let subspaces V and V of space W satisfy (V1)–(V2), V ∩ V = W0, andW 6= V + V .
Then V + V is not closed in W .
Moreover, there exists no operators P and Q with desired properties.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 90: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/90.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
(M1)–(M2) ←− (V1)–(V2)
Lemma
If codimW0(= dimW/W0) is finite, then the set V + V is closedwhenever V and V satisfy (V1)–(V2).
-this corresponds to d = 1.
Sufficient conditions for a counter example:
Theorem
Let subspaces V and V of space W satisfy (V1)–(V2), V ∩ V = W0, andW 6= V + V .
Then V + V is not closed in W .
Moreover, there exists no operators P and Q with desired properties.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 91: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/91.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Counter example
Let Ω ⊆ R2, µ > 0 and f ∈ L2(Ω) be given.
Scalar elliptic equation
−4u+ µu = f
can be written as Friedrichs’ system:
p +∇u = 0µu+ divp = f
.
Then W = L2div(Ω)×H1(Ω) . For α > 0 we define (Robin b. c.)
V := (p, u)> ∈W : Tdivp = αTH1u ,V := (r, v)> ∈W : Tdivr = −αTH1v .
Lemma
The above V and V satisfy (V1)-(V2), V ∩ V = W0 and V + V 6= W .There exists an operator M ∈ L(W,W ′), that satisfies (M1)–(M2) andV = Ker (D −M).
The question whether (V1)–(V2) implies (M1)–(M2) is still open.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 92: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/92.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Counter example
Let Ω ⊆ R2, µ > 0 and f ∈ L2(Ω) be given. Scalar elliptic equation
−4u+ µu = f
can be written as Friedrichs’ system:
p +∇u = 0µu+ divp = f
.
Then W = L2div(Ω)×H1(Ω) . For α > 0 we define (Robin b. c.)
V := (p, u)> ∈W : Tdivp = αTH1u ,V := (r, v)> ∈W : Tdivr = −αTH1v .
Lemma
The above V and V satisfy (V1)-(V2), V ∩ V = W0 and V + V 6= W .There exists an operator M ∈ L(W,W ′), that satisfies (M1)–(M2) andV = Ker (D −M).
The question whether (V1)–(V2) implies (M1)–(M2) is still open.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 93: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/93.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Counter example
Let Ω ⊆ R2, µ > 0 and f ∈ L2(Ω) be given. Scalar elliptic equation
−4u+ µu = f
can be written as Friedrichs’ system:
p +∇u = 0µu+ divp = f
.
Then W = L2div(Ω)×H1(Ω) . For α > 0 we define (Robin b. c.)
V := (p, u)> ∈W : Tdivp = αTH1u ,V := (r, v)> ∈W : Tdivr = −αTH1v .
Lemma
The above V and V satisfy (V1)-(V2), V ∩ V = W0 and V + V 6= W .There exists an operator M ∈ L(W,W ′), that satisfies (M1)–(M2) andV = Ker (D −M).
The question whether (V1)–(V2) implies (M1)–(M2) is still open.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 94: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/94.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Counter example
Let Ω ⊆ R2, µ > 0 and f ∈ L2(Ω) be given. Scalar elliptic equation
−4u+ µu = f
can be written as Friedrichs’ system:
p +∇u = 0µu+ divp = f
.
Then W = L2div(Ω)×H1(Ω) .
For α > 0 we define (Robin b. c.)
V := (p, u)> ∈W : Tdivp = αTH1u ,V := (r, v)> ∈W : Tdivr = −αTH1v .
Lemma
The above V and V satisfy (V1)-(V2), V ∩ V = W0 and V + V 6= W .There exists an operator M ∈ L(W,W ′), that satisfies (M1)–(M2) andV = Ker (D −M).
The question whether (V1)–(V2) implies (M1)–(M2) is still open.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 95: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/95.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Counter example
Let Ω ⊆ R2, µ > 0 and f ∈ L2(Ω) be given. Scalar elliptic equation
−4u+ µu = f
can be written as Friedrichs’ system:
p +∇u = 0µu+ divp = f
.
Then W = L2div(Ω)×H1(Ω) . For α > 0 we define (Robin b. c.)
V := (p, u)> ∈W : Tdivp = αTH1u ,V := (r, v)> ∈W : Tdivr = −αTH1v .
Lemma
The above V and V satisfy (V1)-(V2), V ∩ V = W0 and V + V 6= W .There exists an operator M ∈ L(W,W ′), that satisfies (M1)–(M2) andV = Ker (D −M).
The question whether (V1)–(V2) implies (M1)–(M2) is still open.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 96: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/96.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Counter example
Let Ω ⊆ R2, µ > 0 and f ∈ L2(Ω) be given. Scalar elliptic equation
−4u+ µu = f
can be written as Friedrichs’ system:
p +∇u = 0µu+ divp = f
.
Then W = L2div(Ω)×H1(Ω) . For α > 0 we define (Robin b. c.)
V := (p, u)> ∈W : Tdivp = αTH1u ,V := (r, v)> ∈W : Tdivr = −αTH1v .
Lemma
The above V and V satisfy (V1)-(V2), V ∩ V = W0 and V + V 6= W .
There exists an operator M ∈ L(W,W ′), that satisfies (M1)–(M2) andV = Ker (D −M).
The question whether (V1)–(V2) implies (M1)–(M2) is still open.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 97: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/97.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Counter example
Let Ω ⊆ R2, µ > 0 and f ∈ L2(Ω) be given. Scalar elliptic equation
−4u+ µu = f
can be written as Friedrichs’ system:
p +∇u = 0µu+ divp = f
.
Then W = L2div(Ω)×H1(Ω) . For α > 0 we define (Robin b. c.)
V := (p, u)> ∈W : Tdivp = αTH1u ,V := (r, v)> ∈W : Tdivr = −αTH1v .
Lemma
The above V and V satisfy (V1)-(V2), V ∩ V = W0 and V + V 6= W .There exists an operator M ∈ L(W,W ′), that satisfies (M1)–(M2) andV = Ker (D −M).
The question whether (V1)–(V2) implies (M1)–(M2) is still open.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 98: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/98.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Counter example
Let Ω ⊆ R2, µ > 0 and f ∈ L2(Ω) be given. Scalar elliptic equation
−4u+ µu = f
can be written as Friedrichs’ system:
p +∇u = 0µu+ divp = f
.
Then W = L2div(Ω)×H1(Ω) . For α > 0 we define (Robin b. c.)
V := (p, u)> ∈W : Tdivp = αTH1u ,V := (r, v)> ∈W : Tdivr = −αTH1v .
Lemma
The above V and V satisfy (V1)-(V2), V ∩ V = W0 and V + V 6= W .There exists an operator M ∈ L(W,W ′), that satisfies (M1)–(M2) andV = Ker (D −M).
The question whether (V1)–(V2) implies (M1)–(M2) is still open.
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 99: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/99.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Open questions
(V1)–(V2) =⇒ (M1)–(M2)?
What is relationship between classical results and the new ones (matrixfield on boundary M and boundary operator M)?
New examples. . .
Can theory of Krein spaces give further results?
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 100: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/100.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Open questions
(V1)–(V2) =⇒ (M1)–(M2)?
What is relationship between classical results and the new ones (matrixfield on boundary M and boundary operator M)?
New examples. . .
Can theory of Krein spaces give further results?
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 101: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/101.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Open questions
(V1)–(V2) =⇒ (M1)–(M2)?
What is relationship between classical results and the new ones (matrixfield on boundary M and boundary operator M)?
New examples. . .
Can theory of Krein spaces give further results?
Kresimir Burazin Krein spaces applied to Friedrichs’ systems
![Page 102: Krein spaces applied to Friedrichs' systems - unizg.hrweb.math.pmf.unizg.hr/najman_conference/2nd/slides/Burazin.pdf · Introduction Abstract formulation Interdependence of di erent](https://reader033.fdocuments.us/reader033/viewer/2022042800/5a72ec2b7f8b9aa2538e2d85/html5/thumbnails/102.jpg)
IntroductionAbstract formulation
Interdependence of different representations of boundary conditions
Open questions
(V1)–(V2) =⇒ (M1)–(M2)?
What is relationship between classical results and the new ones (matrixfield on boundary M and boundary operator M)?
New examples. . .
Can theory of Krein spaces give further results?
Kresimir Burazin Krein spaces applied to Friedrichs’ systems