Optimal potentials for

Schrodinger operators

Giuseppe Buttazzo

Dipartimento di Matematica

Universita di Pisa

buttazzo@dm.unipi.it

http://cvgmt.sns.it

“New trends in modeling, control and inverse problems”Enrique Zuazua’s CIMI ChairToulouse, June 16–19, 2014

Paper appeared on JEP (2014). Work incollaboration with:

Augusto Gerolin, Ph. D. student atDipartim. di Matematica - Universita di Pisa,gerolin@mail.dm.unipi.it

Berardo Ruffini, Post doc fellow atUniversite de Grenoble,berardo.ruffini@sns.it

Bozhidar Velichkov, Post doc fellow atDipartim. di Matematica - Universita di Pisa,b.velichkov@sns.it

1

We consider the Schrodinger operator −∆+

V (x) in a given bounded set Ω. The opti-

mization problems we deal with are of the

form

minF (V ) : V ∈ V

,

where F is a suitable cost functional and Vis a suitable admissible class. We limit our-

selves to the case V ≥ 0.

The cost functionals we want to include in

our framework are of the following types.

2

Integral functionals Given a right-hand sidef ∈ L2(Ω) we consider the solution uV of theelliptic PDE

−∆u+ V (x)u = f(x) in Ω, u ∈ H10(Ω).

The integral cost functionals we consider areof the form

F (V ) =∫

Ωj(x, uV (x),∇uV (x)

)dx

where j is a suitable integrand that we as-sume convex in the gradient variable andbounded from below as

j(x, s, z) ≥ −a(x)− c|s|2

3

with a ∈ L1(Ω) and c smaller than the firsteigenvalue of −∆ on H1

0(Ω). In particular,the energy Ef(V ) defined by

Ef(V ) = infu∈H1

0(Ω)

∫Ω

(1

2|∇u|2+

1

2V (x)u2−f(x)u

)dx

belongs to this class since, integrating byparts its Euler-Lagrange equation, we have

Ef(V ) = −1

2

∫Ωf(x)uV dx

which corresponds to the integral functionalabove with

j(x, s, z) = −1

2f(x)s.

4

Spectral functionals For every admissible

potential V ≥ 0 we consider the spectrum

λ(V ) of the Schrodinger operator −∆+V (x)

on H10(Ω).

If Ω is bounded or has finite measure, or if

the potential V satisfies some suitable inte-

gral properties, the operator −∆+V (x) has a

compact resolvent and so its spectrum λ(V )

is discrete:

λ(V ) =(λ1(V ), λ2(V ), . . .

),

where λk(V ) are the eigenvalues counted with

their multiplicity.

5

The spectral cost functionals we consider are

of the form

F (V ) = Φ(λ(V )

)where Φ : RN → R is a given function. For

instance, taking Φ(λ) = λk we obtain

F (V ) = λk(V ).

We say that Φ is continuous (resp. lsc) if

λnk → λk ∀k =⇒ Φ(λn)→ Φ(λ)(resp. Φ(λ) ≤ lim inf

nΦ(λn)

).

6

Optimization problems for changing sign po-

tentials have been recently considered by Carlen-

Frank-Lieb for the cost F (V ) = λ1(V ). They

prove the inequality:

λ1(V ) ≥ −cp,d( ∫

RdVp+d

2− dx

)1p.

Our goal is to obtain similar inequalities for

more general cost functionals and integral

constraints on the potential; on the other

hand, we limit ourselves to the case of non-

negative potentials.

7

The motivation

Problems of the same kind arise in shape

optimization, where one has to minimize a

shape cost F (Ω) in a suitable admissible class

A of domains. Again, two interesting classes

of problems are the one of integral costs

F (Ω) =∫j(x, uΩ(x),∇uΩ(x)

)dx,

where uΩ solves the elliptic PDE (with f

given)

−∆u = f in Ω, u ∈ H10(Ω),

8

and the one of spectral costs

F (Ω) = Φ(λ(Ω)

)being λ(Ω) = (λ1(Ω), . . . ) the spectrum ofthe Dirichlet Laplacian in Ω.

It is known since the ’80 that, unless addingsevere geometrical constraints as convexityor uniform exterior cone condition on thecompeting domains, the class of domains isnot compact. More precisely, sequences Ωn

can be constructed such that uΩn convergesto some function u which is not of the formuΩ, for any domain Ω.

9

The example, found by Cioranescu-Murat, isillustrated below

Ωn is the complement of the union of small holes.

10

Tuning carefully the radius of the holes wehave that uΩn converges weakly H1 to thefunction u which solves

−∆u+ cu = f

for a suitable constant c.

Later Dal Maso-Mosco have characterized allpossible limits of sequences of the form uΩn;they are the functions uµ solutions of

−∆u+ µu = f

where µ is a capacitary measure (i.e. µ(E) =0 for all sets E with cap(E) = 0).

11

To have a functional framework, we denoteby M+

0 (D) the class of capacitary measureson D, and by H1

µ the Sobolev space

H1µ =

u ∈ H1(Rd) :

∫Rd|u|2 dµ < +∞

,

with norm

‖u‖21,µ =∫Rd|∇u|2 dx+

∫Rdu2 dx+

∫Rdu2 dµ.

It is a Hilbert space and the existence anduniqueness of a solution uµ to

−∆u+ µu = f

follows by the usual Lax-Milgram method.

12

• Every domain Ω is a capacitary measure,

given by

∞Ωc(E) =

0 if E ⊂ Ω up to cap zero

+∞ otherwise.

• Every potential V is a capacitary measure,

given by µ = V dx.

• If S is a smooth d− 1 manifold and V ≥ 0

is in L1(S), then the measure µ = V dHd−1

is of capacitary type.

13

Definition We say that a sequence (µn) ofcapacitary measures γ-converges to the ca-pacitary measure µ if the sequence of resol-vent operators

Rµn : L2(Ω)→ L2(Ω)

converges strongly to Rµ. In other words,for every f the solutions un of

−∆u+ µnu = f, u ∈ H10(Ω)

converge in L2(Ω) to the solution of

−∆u+ µu = f, u ∈ H10(Ω).

14

Properties of the γ-convergence

• The γ-convergence is equivalent to:

Rµn(1)→ Rµ(1).

In this way, the distance

dγ(µ1, µ2) = ‖Rµ1(1)−Rµ2(1)‖L2(Ω)

metrizes the γ-convergence.

• The space M0(Ω) endowed with the dis-tance dγ is a compact metric space.

• Identifying a domain A with the measure∞Ω\A, the class of all smooth domains A ⊂Ω is dγ-dense in M0(Ω).

15

• The measures of the form V (x) dx, with V

smooth, are dγ-dense in M0(Ω).

• If µn → µ for the γ-convergence, the spec-

trum of the compact resolvent operator Rµnconverges to the spectrum of Rµ; then the

eigenvalues of the Schrodinger operator −∆+

µn defined on H10(Ω) converge to the corre-

sponding eigenvalues of the operator −∆+µ.

16

The case of bounded constraints

Proposition If Vn → V weakly in L1(Ω) the

capacitary measures Vn dx γ-converge to V dx.

As a consequence, all the optimization prob-

lems of the form

minF (V ) : V ∈ V

with F γ-l.s.c (very weak assumption) and

V closed convex and bounded in Lp(Ω) with

p > 1, admit a solution.

17

Example If p > 1 the problem

maxEf(V ) : V ≥ 0,

∫ΩV p dx ≤ 1

has the unique solution

Vp =(∫

Ω|up|2p/(p−1) dx

)−1/p|up|2/(p−1),

where up is the minimizer on H10(Ω) of

1

2

∫Ω|∇u|2 dx+

1

2

(∫Ω|u|2p/(p−1) dx

)p−1p −

∫Ωfu dx

corresponding to the nonlinear PDE

−∆u+ C|u|2/(p−1)u = f.

18

Similar results for λ1(V ) (see also [HenrotBirkhauser 2006]).

If p < 1 the problem

maxEf(V ) : V ≥ 0,

∫ΩV p dx ≤ 1

has no solution. Indeed, take for instancef = 1; it is not difficult to construct a se-quence Vn such that∫

ΩV pn dx ≤ 1 and Ef(Vn)→ 0.

The conclusion follows since no potential Vcan provide zero energy.

19

An interesting case is when p = 1.

The solution of

maxEf(V ) : V ≥ 0,

∫ΩV dx ≤ 1

is in principle a measure. However, it is pos-sible to prove that for every f ∈ L2(Ω), de-noting by w the solution of the auxiliary prob-lem

minu∈H1

0(Ω)

1

2

∫Ω|∇u|2 dx+

1

2‖u‖2L∞(Ω)−

∫Ωuf dx

,

and setting M = ‖w‖L∞(Ω), ω+ = w = M,ω− = w = −M,

20

we have

Vopt =f

M

(1ω+ − 1ω−

).

Note that in particular, we deduce the con-

ditions of optimality

• f ≥ 0 on ω+,

• f ≤ 0 on ω−,

•∫ω+

f dx−∫ω−f dx = M.

21

The case of unbounded constraints

We consider now problems of the form

minF (V ) : V ≥ 0,

∫Ω

Ψ(V ) dx ≤ 1

with admissible classes of potentials unboundedin every Lp. For example:• Ψ(s) = s−p, for any p > 0;• Ψ(s) = e−αs, for any α > 0.

Theorem Let Ω be bounded, F increas-ing and γ-lower semicontinuous, Ψ strictlydecreasing with Ψ−1(sp) convex for somep > 1. Then there exists a solution.

22

Examples If Ψ(s) = s−p with p > 0, the

optimal potential for the energy Ef is

Vopt =(∫

Ω|u|2p/(p+1) dx

)1/p|u|−2/(p+1)

where u solves the auxiliary problem

minu∈H1

0(Ω)

∫Ω|∇u|2dx+

( ∫Ω|u|2p/(1+p)dx

)(1+p)/p−∫

Ω2fudx

which corresponds to the nonlinear PDE

−∆u+ Cp|u|−2/(p+1)u = f, u ∈ H10(Ω)

where Cp is a constant depending on p.

23

Similarly, if Ψ(s) = e−αs, we have

Vopt =1

α

(log

(∫Ωu2 dx

)− log

(u2))

where u solves the auxiliary problem

minu∈H1

0(Ω)

∫Ω|∇u|2dx+

1

α

∫Ωu2( ∫

Ωlog(u2)dx− log(u2)

)dx−

∫Ω

2fudx.

which corresponds to the nonlinear PDE

−∆u+ Cαu−1

αu log(u2) = f, u ∈ H1

0(Ω)

where Cα is a constant depending on α.

24

PROBLEMS WITH Ω = Rd

When Ω = Rd most of the cost function-als are not γ-lower semicontinuous; for ex-ample, if V (x) is any potential, with V =+∞ outside a compact set, then, for everyxn → ∞, the sequence of translated poten-tials Vn(x) = V (x + xn) γ-converges to thecapacitary measure

I∅(E) =

0 if cap(E) = 0

+∞ if cap(E) > 0.

Thus increasing and translation invariant func-tionals are never γ-lower semicontinuous.

25

• for the problem

maxF (V ) : V ≥ 0,

∫RdV p dx ≤ 1

most of the results obtained in the case Ω

bounded can be repeated.

In the cases F = Ef and F = λ1 in gen-

eral the optimal potentials are not compactly

supported, even if f is compactly supported.

For instance, taking f = 1B1the optimal

potential Vopt is radially decreasing and sup-

ported in the whole Rd.

26

y

-3 -1 1 3

up

The solution up and f = χB(0,1) does not have a compact support.

27

• for the problem

minF (V ) : V ≥ 0,

∫RdV −p dx ≤ 1

we do not have a general existence theorembut only proofs in some special cases, as theDirichlet Energy Ef (or the first eigenvalueof the Dirichlet Laplacian).

In these cases, if f is compactly supported,we have that 1/Vopt is compactly supported,that is Vopt = +∞ out of a compact set(hence the Dirichlet condition is imposed outof a compact set to the related PDE).

28

y

-3 -1 1 3up

The solution up and f = χB(0,1) has a compact support.

29

If we limit ourselves to the spectral optimiza-

tion problems

minλk(V ) : V ≥ 0,

∫RdV −p dx ≤ 1

the problems are:

Problem 1. for every k an optimal potential

Vk exists;

Problem 2. for every k the optimal poten-

tial Vk above is such that 1/Vk is compactly

supported.

30

In [Bucur-B.-Velichkov] (SIAM J. Math. Anal.

(to appear), http://cvgmt.sns.it) we showed

the two problems above have a positive an-

swer.

For the moment the proof cannot be adapted

to other kinds of cost functionals F (V ), as

for instance integral functionals or spectral

functionals.

31