Optimal potentials for Schr odinger operators Giuseppe Buttazzo - - PowerPoint PPT Presentation

Optimal potentials for Schr¨

• dinger operators

Giuseppe Buttazzo Dipartimento di Matematica Universit` a di Pisa buttazzo@dm.unipi.it http://cvgmt.sns.it

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

Paper appeared on JEP (2014). Work in collaboration with: Augusto Gerolin, Ph. D. student at

• Dipartim. di Matematica - Universit`

a di Pisa, gerolin@mail.dm.unipi.it Berardo Ruffini, Post doc fellow at Universit´ e de Grenoble, berardo.ruffini@sns.it Bozhidar Velichkov, Post doc fellow at

• Dipartim. di Matematica - Universit`

a di Pisa, b.velichkov@sns.it

1

We consider the Schr¨

• dinger operator −∆+

V (x) in a given bounded set Ω. The opti- mization problems we deal with are of the form min

• F(V ) : V ∈ V
• ,

where F is a suitable cost functional and V is a suitable admissible class. We limit our- selves to the case V ≥ 0. The cost functionals we want to include in

• ur framework are of the following types.

2

Integral functionals Given a right-hand side f ∈ L2(Ω) we consider the solution uV of the elliptic PDE −∆u + V (x)u = f(x) in Ω, u ∈ H1

0(Ω).

The integral cost functionals we consider are

• f the form

F(V ) =

• Ω j
• x, uV (x), ∇uV (x)
• dx

where j is a suitable integrand that we as- sume convex in the gradient variable and bounded from below as j(x, s, z) ≥ −a(x) − c|s|2

3

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

0(Ω).

In particular, the energy Ef(V ) defined by Ef(V ) = inf

u∈H1

0(Ω)

• Ω

1

2|∇u|2+1 2V (x)u2−f(x)u

• dx

belongs to this class since, integrating by parts its Euler-Lagrange equation, we have Ef(V ) = −1 2

• Ω f(x)uV dx

which corresponds to the integral functional above 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 Schr¨

• dinger operator −∆+V (x)
• n H1

0(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

• f 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 λn

k → λk ∀k =

⇒ Φ(λn) → Φ(λ)

• resp. Φ(λ) ≤ lim inf

n

Φ(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

Rd V

p+d

2

−

dx

1

p.

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

• ptimization, where one has to minimize a

shape cost F(Ω) in a suitable admissible class A of domains. Again, two interesting classes

• f 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 ∈ H1

0(Ω),

8

and the one of spectral costs F(Ω) = Φ

• λ(Ω)
• being λ(Ω) = (λ1(Ω), . . . ) the spectrum of

the Dirichlet Laplacian in Ω. It is known since the ’80 that, unless adding severe geometrical constraints as convexity

• r uniform exterior cone condition on the

competing domains, the class of domains is not compact. More precisely, sequences Ωn can be constructed such that uΩn converges to some function u which is not of the form uΩ, for any domain Ω.

9

The example, found by Cioranescu-Murat, is illustrated below

Ωn is the complement of the union of small holes. 10

Tuning carefully the radius of the holes we have that uΩn converges weakly H1 to the function u which solves −∆u + cu = f for a suitable constant c. Later Dal Maso-Mosco have characterized all possible 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 denote by M+

0 (D) the class of capacitary measures

• n D, and by H1

µ the Sobolev space

H1

µ =

• u ∈ H1(Rd) :
• Rd |u|2 dµ < +∞
• ,

with norm u2

1,µ =

• Rd |∇u|2 dx +
• Rd u2 dx +
• Rd u2 dµ.

It is a Hilbert space and the existence and uniqueness 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) =

  

if E ⊂ Ω up to cap zero +∞

• therwise.
• 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) of capacitary 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 ∈ H1

0(Ω)

converge in L2(Ω) to the solution of −∆u + µu = f, u ∈ H1

0(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 Rµn converges to the spectrum of Rµ; then the eigenvalues of the Schr¨

• dinger operator −∆+

µn defined on H1

0(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 min{F(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 max

• Ef(V ) : V ≥ 0,
• Ω V 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 H1

0(Ω) of

1 2

• Ω |∇u|2 dx+1

2

• Ω |u|2p/(p−1) dx

p−1

p −

• Ω fu dx

corresponding to the nonlinear PDE −∆u + C|u|2/(p−1)u = f.

18

Similar results for λ1(V ) (see also [Henrot Birkh¨ auser 2006]). If p < 1 the problem max

• Ef(V ) : V ≥ 0,
• Ω V p dx ≤ 1
• has no solution.

Indeed, take for instance f = 1; it is not difficult to construct a se- quence Vn such that

• Ω V p

n dx ≤ 1

and Ef(Vn) → 0. The conclusion follows since no potential V can provide zero energy.

19

An interesting case is when p = 1. The solution of max

• Ef(V ) : V ≥ 0,
• Ω V 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 min

u∈H1

0(Ω)

1

2

• Ω |∇u|2 dx+ 1

2u2

L∞(Ω)−

• Ω uf dx
• ,

and setting M = wL∞(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 min

• F(V ) : V ≥ 0,
• Ω Ψ(V ) dx ≤ 1
• with admissible classes of potentials unbounded

in 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 strictly decreasing with Ψ−1(sp) convex for some p > 1. Then there exists a solution.

22

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

• ptimal 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 min

u∈H1

0(Ω)

• Ω |∇u|2dx+

Ω |u|2p/(1+p)dx

(1+p)/p

−

• Ω 2fudx

which corresponds to the nonlinear PDE −∆u + Cp|u|−2/(p+1)u = f, u ∈ H1

0(Ω)

where Cp is a constant depending on p.

23

Similarly, if Ψ(s) = e−αs, we have Vopt = 1 α

• log
• Ω u2 dx
• − log
• u2

where u solves the auxiliary problem min

u∈H1

0(Ω)

• Ω |∇u|2dx+

1 α

• Ω u2

Ω log(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 every xn → ∞, the sequence of translated poten- tials Vn(x) = V (x + xn) γ-converges to the capacitary measure I∅(E) =

  

if cap(E) = 0 +∞ if cap(E) > 0. Thus increasing and translation invariant func- tionals are never γ-lower semicontinuous.

25

• for the problem

max

• F(V ) : V ≥ 0,
• Rd V 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 = 1B1 the 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

min

• F(V ) : V ≥ 0,
• Rd V −p dx ≤ 1
• we do not have a general existence theorem

but only proofs in some special cases, as the Dirichlet Energy Ef (or the first eigenvalue

• f 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 out

• f a compact set to the related PDE).

28

y

• 3
• 1

1 3 up

The solution up and f = χB(0,1) has a compact support. 29

If we limit ourselves to the spectral optimiza- tion problems min

• λk(V ) : V ≥ 0,
• Rd V −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