Optimal potentials for Schr¨ odinger 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¨ odinger 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 our framework are of the following types. 2
Integral functionals Given a right-hand side f ∈ L 2 (Ω) we consider the solution u V of the elliptic PDE u ∈ H 1 − ∆ u + V ( x ) u = f ( x ) in Ω , 0 (Ω) . The integral cost functionals we consider are of the form � � � F ( V ) = Ω j x, u V ( x ) , ∇ u V ( 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 ∈ L 1 (Ω) and c smaller than the first eigenvalue of − ∆ on H 1 0 (Ω). In particular, the energy E f ( V ) defined by � 1 2 |∇ u | 2 +1 � � 2 V ( x ) u 2 − f ( x ) u E f ( V ) = inf dx u ∈ H 1 Ω 0 (Ω) belongs to this class since, integrating by parts its Euler-Lagrange equation, we have E f ( V ) = − 1 � Ω f ( x ) u V dx 2 which corresponds to the integral functional above with j ( x, s, z ) = − 1 2 f ( x ) s. 4
Spectral functionals For every admissible potential V ≥ 0 we consider the spectrum λ ( V ) of the Schr¨ odinger operator − ∆+ V ( x ) on H 1 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 ) = λ 1 ( V ) , λ 2 ( V ) , . . . , where λ k ( 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 : R N → R is a given function. For instance, taking Φ( λ ) = λ k we obtain F ( V ) = λ k ( V ) . We say that Φ is continuous (resp. lsc) if λ n k → λ k ∀ k = ⇒ Φ( λ n ) → Φ( λ ) � � resp. Φ( λ ) ≤ lim inf Φ( λ n ) . n 6
Optimization problems for changing sign po- tentials have been recently considered by Carlen- Frank-Lieb for the cost F ( V ) = λ 1 ( V ). They prove the inequality: � 1 p + d � � p . 2 λ 1 ( V ) ≥ − c p,d R d V dx − 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) = x, u Ω ( x ) , ∇ u Ω ( x ) j dx, where u Ω solves the elliptic PDE (with f given) u ∈ H 1 − ∆ u = f in Ω , 0 (Ω) , 8
and the one of spectral costs � � F (Ω) = Φ λ (Ω) being λ (Ω) = ( λ 1 (Ω) , . . . ) the spectrum of the Dirichlet Laplacian in Ω. It is known since the ’80 that, unless adding severe geometrical constraints as convexity or uniform exterior cone condition on the competing domains, the class of domains is not compact. More precisely, sequences Ω n can be constructed such that u Ω n converges to some function u which is not of the form u Ω , for any domain Ω. 9
The example, found by Cioranescu-Murat, is illustrated below Ω n is the complement of the union of small holes. 10
Tuning carefully the radius of the holes we have that u Ω n converges weakly H 1 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 u Ω n ; they are the functions u µ 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 on D , and by H 1 µ the Sobolev space � � � R d | u | 2 dµ < + ∞ H 1 u ∈ H 1 ( R d ) : µ = , with norm � � � R d |∇ u | 2 dx + R d u 2 dx + R d u 2 dµ. � u � 2 1 ,µ = It is a Hilbert space and the existence and uniqueness of a solution u µ to − ∆ u + µu = f follows by the usual Lax-Milgram method. 12
• Every domain Ω is a capacitary measure, given by if E ⊂ Ω 0 up to cap zero ∞ Ω c ( E ) = + ∞ 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 L 1 ( S ), then the measure µ = V d H d − 1 is of capacitary type. 13
Definition We say that a sequence ( µ n ) of capacitary measures γ -converges to the ca- pacitary measure µ if the sequence of resol- vent operators R µ n : L 2 (Ω) → L 2 (Ω) converges strongly to R µ . In other words, for every f the solutions u n of u ∈ H 1 − ∆ u + µ n u = f, 0 (Ω) converge in L 2 (Ω) to the solution of u ∈ H 1 − ∆ u + µu = f, 0 (Ω) . 14
Properties of the γ -convergence • The γ -convergence is equivalent to: R µ n (1) → R µ (1) . In this way, the distance d γ ( µ 1 , µ 2 ) = � R µ 1 (1) − R µ 2 (1) � L 2 (Ω) metrizes the γ -convergence. • The space M 0 (Ω) endowed with the dis- tance d γ is a compact metric space. • Identifying a domain A with the measure ∞ Ω \ A , the class of all smooth domains A ⊂ Ω is d γ -dense in M 0 (Ω). 15
• The measures of the form V ( x ) dx , with V smooth, are d γ -dense in M 0 (Ω). • If µ n → µ for the γ -convergence, the spec- trum of the compact resolvent operator R µ n converges to the spectrum of R µ ; then the eigenvalues of the Schr¨ odinger operator − ∆+ µ n defined on H 1 0 (Ω) converge to the corre- sponding eigenvalues of the operator − ∆+ µ . 16
The case of bounded constraints Proposition If V n → V weakly in L 1 (Ω) the capacitary measures V n 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 L p (Ω) with p > 1, admit a solution. 17
Example If p > 1 the problem � � � Ω V p dx ≤ 1 max E f ( V ) : V ≥ 0 , has the unique solution � − 1 /p �� Ω | u p | 2 p/ ( p − 1) dx | u p | 2 / ( p − 1) , V p = where u p is the minimizer on H 1 0 (Ω) of � p − 1 1 Ω |∇ u | 2 dx +1 p − � �� � Ω | u | 2 p/ ( p − 1) dx Ω fu dx 2 2 corresponding to the nonlinear PDE − ∆ u + C | u | 2 / ( p − 1) u = f. 18
Similar results for λ 1 ( V ) (see also [Henrot Birkh¨ auser 2006]). If p < 1 the problem � � � Ω V p dx ≤ 1 max E f ( V ) : V ≥ 0 , has no solution. Indeed, take for instance f = 1; it is not difficult to construct a se- quence V n such that � Ω V p n dx ≤ 1 and E f ( V n ) → 0 . The conclusion follows since no potential V can provide zero energy. 19
An interesting case is when p = 1. The solution of � � � max E f ( V ) : V ≥ 0 , Ω V dx ≤ 1 is in principle a measure. However, it is pos- sible to prove that for every f ∈ L 2 (Ω), de- noting by w the solution of the auxiliary prob- lem � 1 Ω |∇ u | 2 dx + 1 � � � 2 � u � 2 min L ∞ (Ω) − Ω uf dx , 2 u ∈ H 1 0 (Ω) and setting M = � w � L ∞ (Ω) , ω + = { w = M } , ω − = { w = − M } , 20
we have V opt = f � � 1 ω + − 1 ω − . M Note that in particular, we deduce the con- ditions of optimality • f ≥ 0 on ω + , • f ≤ 0 on ω − , � � • f dx − f dx = M. ω + ω − 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 L p . 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 ( s p ) convex for some p > 1. Then there exists a solution. 22
If Ψ( s ) = s − p with p > 0, the Examples optimal potential for the energy E f is � 1 /p �� Ω | u | 2 p/ ( p +1) dx | u | − 2 / ( p +1) V opt = where u solves the auxiliary problem � (1+ p ) /p � � � � Ω |∇ u | 2 dx + Ω | u | 2 p/ (1+ p ) dx min − Ω 2 fudx u ∈ H 1 0 (Ω) which corresponds to the nonlinear PDE − ∆ u + C p | u | − 2 / ( p +1) u = f, u ∈ H 1 0 (Ω) where C p is a constant depending on p . 23
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