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

“Modelisation with optimal transport” Grenoble, October 3–4, 2013

Work in collaboration with: Augusto Gerolin, Ph. D. student at

• Dipartim. di Matematica - Universit`

a di Pisa, gerolin@mail.dm.unipi.it Berardo Ruffini, Ph. D. student at Scuola Normale Superiore di Pisa, berardo.ruffini@sns.it Bozhidar Velichkov, Ph. D. student at Scuola Normale Superiore 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 γ-convergence We denote by M+

0 (Ω) the class of capaci-

tary measures on Ω, i.e. the Borel (not nec- essarily finite) measures µ on Ω such that µ(E) = 0 for any set E ⊂ Ω of capacity zero. For any capacitary measure µ ∈ M+

0 (Ω), we

define the Sobolev space H1

µ =

• u ∈ H1(Rd) :
• Rd |u|2 dµ < +∞
• ,

8

which is a Hilbert space when endowed with the norm u1,µ, where u2

1,µ =

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

If u / ∈ H1

µ, we set u1,µ = +∞.

In particular the measure ∞K(E) =

  

if cap(E ∩ K) = 0 +∞ if cap(E ∩ K) > 0 is a capacitary measure and, taking K = Ωc, the space H1

µ(Ω) becomes in this case the

usual Sobolev space H1

0(Ω).

9

Definition We say that a sequence (µn) of capacitary measures γ-converges to the ca- pacitary measure µ if the sequence of func- tionals · 1,µn Γ-converges to the functional · 1,µ in L2(Ω), i.e. the following two con- ditions are satisfied:

• for every un → u in L2(Ω) we have

u2

1,µ ≤ lim inf n→∞ un2 1,µn;

• for every u ∈ L2(Ω), there exists un → u in

L2(Ω) such that u2

1,µ = lim n→∞ un2 1,µn.

10

For every µ ∈ M0(Ω) and f ∈ L2(Ω) we may consider the PDE formally written as

  

−∆u + µu = f u ∈ H1

0(Ω)

whose precise meaning has to be given in the weak form

• Ω ∇u∇ϕ dx +
• Ω uϕ dµ =
• Ω fϕ dx

for every ϕ ∈ H1

0(Ω) ∩ L2 µ(Ω). The resolvent

• perator Rµ : L2(Ω) → L2(Ω) associates to

every f ∈ L2(D) the unique solution u of the PDE above.

11

Properties of the γ-convergence

• The γ-convergence is equivalent to:

Rµn(f) → Rµ(f) for every f ∈ L2(D). Actually, it is enough to have Rµn(1) → Rµ(1). In this way, the distance dγ(µ1, µ2) = Rµ1(1) − Rµ2(1)L2(Ω) is equivalent to 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).

12

• 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.

13

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.

14

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.

15

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.

16

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},

17

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.

18

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.

19

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.

20

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.

21

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.

22

• 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.

23

y

• 3
• 1

1 3 up

The solution up and f = χB(0,1) does not have a compact support. 24

• 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).

25

y

• 3
• 1

1 3 up

The solution up and f = χB(0,1) has a compact support. 26

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.

27

In [Bucur-B.-Velichkov] (paper in prepara- tion) we are able to show that the two prob- lems above have a positive answer. For the moment the proof cannot be adapted to other kinds of cost functionals F(V ), as for instance integral functionals or spectral functionals.

28