Improved energy bounds for Schrödinger operators

HAL Id: hal-00955139

https://hal.archives-ouvertes.fr/hal-00955139v2

Submitted on 28 Jul 2014

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Copyright

Improved energy bounds for Schrödinger operators

Lorenzo Brasco, Giuseppe Buttazzo

To cite this version:

Lorenzo Brasco, Giuseppe Buttazzo. Improved energy bounds for Schrödinger operators. Calculus of

Variations and Partial Differential Equations, Springer Verlag, 2015, 53, pp.977-1014. �10.1007/s00526-

014-0774-1�. �hal-00955139v2�

LORENZO BRASCO AND GIUSEPPE BUTTAZZO

Abstract. Given a potentialV and the associated Schr¨odinger operator−∆+V, we consider the problem of providing sharp upper and lower bound on the energy of the operator. It is known that if for exampleV orV⁻¹enjoys suitable summability properties, the problem has a positive answer. In this paper we show that the corresponding isoperimetric-like inequalities can be improved by means of quantitative stability estimates.

Contents

1. Introduction 1

1.1. Aim of the paper 1

1.2. Main results 3

1.3. Plan of the paper 4

2. Preliminaries 4

3. Maximization problems 8

4. Stability for maximization problems 9

4.1. Stability of the potentials 10

4.2. Stability of the state functions 12

5. Minimization problems 13

6. Stability for minimization problems 15

6.1. Preliminary results 15

6.2. Stability of the potentials 17

6.3. Stability of the state functions 24

Appendix A. Sharp decay estimates for non autonomous Schr¨ odinger equations 24

References 30

1. Introduction

1.1. Aim of the paper. Let Ω ⊂ R

^N

be an open set not necessarily with finite measure (it could be Ω = R

^N

) and V : Ω → R be a potential. We consider the associated Schr¨ odinger operator −∆ + V defined on the homogeneous Sobolev space W

₀^1,2

(Ω). The latter is the closure of C

₀^∞

(Ω) with respect to the norm

kuk

_W1,2 0 (Ω)

:=

Z

Ω

|∇u|

²

dx

1/2

.

We also denote by W

^−1,2

(Ω) its dual space and by h·, ·i the duality pairing between W

₀^1,2

(Ω) and W

^−1,2

(Ω). In this paper we are concerned with the following problem: given a source term f ∈ W

^−1,2

(Ω), find lower and upper bounds on the energy of the relevant Schr¨ odinger operator, i.e.

E

_f

(V ) = − 1 2

Z

Ω

|∇u

_V

|

²

dx − 1 2

Z

Ω

V u

²_V

dx.

Date: 14th July 2014, Marseille et Pise.

2010Mathematics Subject Classification. 49K20, 49Q10, 49Q20.

Key words and phrases. Schr¨odinger operators; optimal potentials; stability inequalities; decay estimates.

1

Here the state function u

V

is a W

₀^1,2

(Ω) solution of

−∆u + V u = f, in Ω.

In the recent paper [10], this problem has been solved for summable potentials or for confining potentials, i.e. for potentials blowing-up at infinity such that 1/V enjoys some summability properties. For example, the harmonic–like potential

V = δ

²

+ |x|

²

γ/2

, δ > 0,

belongs to this class, for suitable γ > 0. In order to provide a deeper insight into the scopes of this work, it is useful to briefly recall some of the results in [10]. In that paper it has been shown that E

_f

(V ) can be universally bounded from above in the class (see [10, Proposition 5.1])

V

₁

=

V : Z

Ω

|V |

^p

≤ 1

, and from below in the class (see [10, Proposition 5.4])

V

₂

=

V ≥ 0 : Z

Ω

V

^−p

≤ 1

.

The value 1 above plays no special role and can be replaced by any constant c > 0. Indeed, one can show that there exist two potentials V

₀

and U

₀

such that

Z

Ω

|V

₀

|

^p

dx = 1 = Z

Ω

|U

₀

|

^−p

dx, and

(1.1) E

_f

(V ) ≤ E

_f

(V

₀

), for every V ∈ V

₁

, (1.2) E

_f

(V ) ≥ E

_f

(U

0

), for every V ∈ V

₂

.

In both the estimates (1.1) and (1.2), the equality sign holds if and only if V = V

₀

and V = U

₀

, respectively. Moreover, the extremal potentials V

₀

and U

₀

can be characterized in terms of the solutions v

0

and u

0

of the semilinear non-autonomous PDEs

−∆v

₀

+ c v

(p+1)/(p−1)

0

= f and − ∆u

₀

+ c u

(p−1)/(p+1)

0

= f,

through the relations V

₀

=

Z

Ω

|v

₀

|

^2p/(p−1)

dx

−1/p

|v

₀

|

^2/(p−1)

and U

₀

= Z

Ω

|u

₀

|

^2p/(p+1)

dx

1/p

|u

₀

|

^−2/(p+1)

. For completeness, we point out that a special class of potentials from the sets V

₁

and V

₂

are given respectively by

(1.3) V (x) =

( |E|

^−1/p

, x ∈ E,

0, otherwise and V (x) =

( |E|

^1/p

, x ∈ E, +∞, otherwise,

where E ⊂ Ω is an open set and |E| denotes its N −dimensional Lebesgue measure. If we suppose for simplicity that |E| = 1, the corresponding operators are given by

W

₀^1,2

(Ω) 3 u 7→ −∆u + u · 1

_E

and W

₀^1,2

(E) 3 u 7→ −∆u + u.

Thus from (1.1) and (1.2) we get min

u∈W₀^1,2(Ω)

1 2

Z

Ω

|∇u|

²

dx + 1 2

Z

E

|u|

²

dx − hf, ui

< E

_f

(V

₀

), and

min

u∈W₀^1,2(E)

1 2

Z

E

|∇u|

²

dx + 1 2

Z

E

|u|

²

dx − hf, ui

> E

_f

(U

0

),

for every E ⊂ Ω with |E| = 1. Observe that for the second problem the set Ω simply acts as a

design region where the admissible domains E have to be contained.

The problem of finding sharp bounds on energetical quantities linked to a Schr¨ odinger oper- ator is quite classical, with many studies devoted to the ground state energy or first eigenvalue

λ

1

(V ) = min

u∈W₀^1,2(Ω)

Z

Ω

|∇u|

²

dx + Z

Ω

V u

²

dx : kuk

_L2(Ω)

= 1

.

For example, the pioneering paper [22] by Keller considers the problem of finding sharp lower bounds on λ

₁

(V ) in the class V

₁

, in the case of space dimension N = 1 and Ω = R . Related problems have been considered by Ashbaugh and Harrell in [2] in higher dimensions. There is a vast literature on the subject, considering optimal bounds for other spectral quantities, like the first excited state or second eigenvalue λ

2

(V ) and the fundamental gap λ

2

(V ) − λ

1

(V ). We also mention the recent paper [8], where the case of successive eigenvalues λ

_k

(V ) for k ≥ 2 is considered, for the non-compact case of Ω = R

^N

. Actually this kind of problems is even older:

indeed, we observe that for potentials of the second form in (1.3), we have λ

₁

(V ) = min

u∈W₀^1,2(E)

Z

E

|∇u|

²

dx + Z

E

|u|

²

dx : kuk

_L2(E)

= 1

= λ

₁

(E) + 1,

where λ

1

(E) now stands for the first eigenvalue of −∆ with Dirichlet boundary conditions on

∂E. Thus the celebrated Faber-Krahn inequality (see [21, Chapter 3])

λ

₁

(E) ≥ λ

₁

(B), with B any ball such that |B| = |E| = 1,

can be seen as a particular instance of these problems. A more general overview on optimization problems of spectral type can be found in [9] and [21], to which we refer the interested reader.

We wish to point out that on the contrary the case of the energy E

_f

(V ) appears to be less investigated.

1.2. Main results. In this paper, we improve the previous sharp bounds (1.1) and (1.2) on the energy of a Schr¨ odinger operator, by means of a quantitative stability result. In other words, we will prove that the energy gap E

_f

(V

0

) − E

_f

(V ) or E

_f

(V ) − E

_f

(U

0

) controls the deviation from optimality of a potential V . Thus it is possible to add a reminder term in the right-hand side of (1.1) and (1.2), which measures the distance of a generic potential V from V

0

or U

0

. The relevant results are summarized in the following couple of theorems, which represent the main results of the paper. We refer the reader to Sections 4 and 6 for the precise statements.

Theorem A (Stability of the maximizer). Let 1 < p < ∞. There exists a constant σ

1

> 0 such that for every V ∈ V

₁

we have



 



 



E

_f

(V ) ≤ E

_f

(V

0

) − σ

1

kV − V

0

k

²_Lp(Ω)

, if p ≥ 2, E

_f

(V ) ≤ E

_f

(V

₀

) − σ

₁

|V |

^p−2

V − |V

₀

|

^p−2

V

₀

2

L^p0(Ω)

, if 1 < p < 2.

In the case of inequality (1.2) we need to distinguish between the case |Ω| < +∞ and

|Ω| = +∞.

Theorem B (Stability of the minimizer). Let Ω ⊂ R

^N

be an open set such that |Ω| = +∞.

Let 1 < p < ∞, r > N/2, and let f ∈ L

^r

(Ω) be a function decaying to 0 at infinity as O(|x|

^−α

) with α > 1 + N/2. Then there exist a constant σ

₂

> 0 and an exponent β = β (p) > 2 such that for every V ∈ V

₂

we have

E

_f

(V ) ≥ E

_f

(U

0

) + σ

2

1 V − 1

U

0

β L^p(Ω)

.

If |Ω| < +∞, the same result holds for every f ∈ W

^−1,2

(Ω) without any additional hypothesis.

Stability results of this type have attracted an increasing interest in recent years. As a non-

exhaustive list of works on the subject, we point out for example [13] and [17] dealing with

the classical isoperimetric inequality, the papers [5, 6, 7, 20] and [23] concerning sharp bounds

for eigenvalues of the Laplacian and [4, 11, 15] about quantitative versions of the Sobolev and Gagliardo-Nirenberg inequalities with sharp constant.

Among these papers, the recent one [12] is very much related with the subject here consid- ered. In [12] a quantitative stability estimate for λ

₁

(V ) is proved, for potentials belonging to the class (here r > N/2)

V

⁰

=

V : Z

R^N

|V

₋

|

^r

≤ 1

,

where V

−

is the negative part of V . In this case λ

₁

(V ) admits a sharp lower bound, corre- sponding to the negative potential

W

₀

= − Z

R^N

|w

₀

|

^2r/(r−1)

−1/r

|w

₀

|

^2r/(r−1)

, where w

₀

is an extremal of the Gagliardo-Nirenberg inequality

Z

R^N

|u|

^2r/(r−1)

dx

(r−1)/r

≤ C Z

R^N

|∇u|

²

dx

1−ϑ

Z

R^N

|u|

²

dx

ϑ

. The parameter 0 < ϑ = ϑ(N, r) < 1 above is uniquely determined by scaling invariance.

1.3. Plan of the paper. Along all the paper, for the sake of simplicity, we assume N ≥ 3. In this way the Sobolev exponent 2

^∗

is finite; all the results also apply, with minor modifications, to the cases N = 1 and N = 2 by using the corresponding Sobolev embeddings.

We start with Section 2 where we fix the main notations and prove some basic results which will be used throughout the whole paper. Then Section 3 is concerned with maximization problems for E

_f

, under a constraint on the L

^p

norm of the admissible potentials. The relevant quantitative stability result (Theorem 4.3) is then considered in Section 4. Minimization prob- lems are adressed in Section 5, while the last section of the paper contains the corresponding stability result (Theorem 6.6). Finally, a self-contained Appendix on sharp decay estimates for finite energy solutions of

−∆u + c u

^q−1

= f, for c > 0, 1 < q < 2, concludes the paper (Theorem A.1).

2. Preliminaries

In the paper we mainly focus on the following three model cases:

• Ω = R

^N

;

• Ω = ω × R , with ω ⊂ R

^N−1

open set with finite Lebesgue measure (waveguide);

• Ω ⊂ R

^N

with finite Lebesgue measure (compact case).

For N ≥ 3, we define

2

^∗

= 2 N N − 2 .

The following embedding properties of W

₀^1,2

(Ω) are well known.

Proposition 2.1. Let N ≥ 3; then

i) if Ω = R

^N

, we have the continuous embedding W

₀^1,2

(Ω) , → L

^2∗

(Ω), but W

₀^1,2

(Ω) 6⊂

L

^s

(Ω) for s 6= 2

^∗

;

ii) if Ω = ω × R is a waveguide, we have the continuous embedding W

₀^1,2

(Ω) , → L

^s

(Ω) for every 2 ≤ s ≤ 2

^∗

;

iii) if |Ω| < +∞, we have the continuous embedding W

₀^1,2

(Ω) , → L

^s

(Ω) for every 0 < s ≤ 2

^∗

.

Moreover, this is compact for 0 < s < 2

^∗

.

In what follows, for N ≥ 3 we set

(2.1) T

N

:= inf

u∈W₀^1,2(R^N)

Z

R^N

|∇v|

²

dx : kvk

_L2∗

(R^N)

= 1

< +∞.

This infimum is finite by Proposition 2.1 and attained on W

₀^1,2

( R

^N

), see for example [26].

Let f ∈ W

^−1,2

(Ω); for every potential V belonging to the admissible class V = n

V : Ω → (−∞, +∞] : V Borel measurable, kV

₋

k

_LN/2(Ω)

< T

_N

o , we define its energy by

(2.2) E

_f

(V ) = min

u∈W₀^1,2(Ω)

1 2

Z

Ω

|∇u|

²

dx + 1 2

Z

Ω

V u

²

dx − hf, ui.

Proposition 2.2. For every V ∈ V the minimization problem (2.2) admits a solution u

_V

∈ W

₀^1,2

(Ω). Moreover the energy inequality

(2.3) ku

_V

k

_W1,2

0 (Ω)

≤ T

_N

T

_N

− kV

₋

k

_LN/2(Ω)

kf k

_W−1,2(Ω)

. holds.

Proof. At first we observe that, taking u = 0 gives E

_f

(V ) ≤ 0. Moreover, since V ∈ V, the energy functional is bounded from below, because

(2.4) Z

Ω

V u

²

dx ≥ − Z

Ω

V

−

u

²

dx ≥ −kV

₋

k

_LN/2(Ω)

kuk

²_L2^∗_(Ω)

≥ − kV

₋

k

_LN/2(Ω)

T

_N

Z

Ω

|∇u|

²

dx, and thus

1 2

Z

Ω

|∇u|

²

dx+ 1 2

Z

Ω

V u

²

dx − hf, ui

≥ 1

2 1 − kV

−

k

_LN/2(Ω)

T

N

− δ

! Z

Ω

|∇u|

²

dx − 1

2 δ kfk

²_W−1,2(Ω)

, (2.5)

for every 0 < δ ≤ 1. Thus E

_f

(V ) is finite. Let now {u

_n

}

n∈N

⊂ W

₀^1,2

(Ω) be a minimizing sequence; we can assume that

1 2

Z

Ω

|∇u

_n

|

²

dx + 1 2

Z

Ω

V u

²_n

dx − hf, u

_n

i ≤ E

_f

(V ) + 1.

By (2.5), if we take δ 1 the sequence {u

_n

}

_n∈N

is bounded in W

₀^1,2

(Ω), so u

_n

weakly converges (up to a subsequence) in W

₀^1,2

(Ω) to a function u ∈ W

₀^1,2

(Ω). Moreover, by the compact Sobolev embedding of Proposition 2.1 iii), we have strong convergence in L

^s

(Ω

⁰

), for every (smooth) Ω

⁰

b Ω and every 1 ≤ s < 2

^∗

. In particular, u

_n

converges almost everywhere (up to a subsequence) in Ω to u. Also observe that still by Proposition 2.1 we have weak convergence of u

²_n

to u

²

in L

^2∗/2

(Ω). By using this and the Fatou Lemma, we get

lim inf

n→∞

Z

Ω

V u

²_n

dx = lim inf

n→∞

Z

Ω

V

+

u

²_n

dx − Z

Ω

V

−

u

²_n

dx

≥ Z

Ω

V u

²

dx.

Finally, the weak lower semicontinuity of the norm implies 1

2 Z

Ω

|∇u|

²

dx + 1 2

Z

Ω

V u

²

dx − hf, ui

≤ lim inf

n→∞

1 2

Z

Ω

|∇u

_n

|

²

dx + 1 2

Z

Ω

V u

²_n

dx − hf, u

_n

i

= E

_f

(V ),

which gives the existence of a minimizer u

_V

.

This function u

_V

satisfies the Euler-Lagrange equation Z

Ω

h∇u, ∇ϕi dx + Z

Ω

V u ϕ dx = hf, ϕi, for every ϕ ∈ C

₀^∞

(Ω) ∩ L

²

(Ω; V ), where for V ∈ V we set

L

²

(Ω; V ) =

ϕ : Z

Ω

|V | ϕ

²

dx < +∞

.

By density, the previous equation holds for every ϕ ∈ W

₀^1,2

(Ω) ∩ L

²

(Ω; V ). By taking u

V

as a test function and then appealing to

|hf, ui| ≤ 1

2 δ kf k

²_W−1,2(Ω)

+ δ 2 kuk

²

W₀^1,2(Ω)

, we have the estimate

1 − δ

2 Z

Ω

|∇u

_V

|

²

dx + Z

Ω

V u

²_V

dx ≤ 1

2 δ kf k

²_W−1,2(Ω)

. By using (2.4) and choosing

δ = 1 − kV

−

k

_LN/2(Ω)

T

N

,

we get (2.3).

Remark 2.3. If kV

₋

k

_LN/2

≥ T

N

, in general problem (2.2) is not well-posed. Indeed, take Ω = R

^N

and indicate by U ∈ W

₀^1,2

(R

^N

) a positive function such that

Z

R^N

|∇U |

²

dx = T

N

and Z

R^N

|U |

^2∗

dx = 1.

We then take

V = −T

_N

U

^4/(N−2)

, and f ∈ L

^(2∗)0

( R

^N

) such that

Z

R^N

f U dx > 0.

By evaluating the functional in (2.2) on the sequence u

_n

= n U , we get 1

2 Z

Ω

|∇u

_n

|

²

dx + 1 2

Z

Ω

V u

²_n

dx − hf, u

_n

i = −n hf, U i, thus the functional is unbounded from below.

Lemma 2.4. Let V

1

, V

2

∈ V be two admissible potentials and let u

1

, u

2

∈ W

₀^1,2

(Ω) be solutions of (2.2). Then

(2.6)

Z

Ω

V

₁

u

²₁

dx − Z

Ω

V

₂

u

²₂

dx

≤ C kfk

_W−1,2(Ω)

ku

₁

− u

₂

k

_W1,2 0 (Ω)

, where

C = 1 + T

_N

T

N

− k(V

₁

)

−

k

_LN/2(Ω)

+ T

_N

T

N

− k(V

₂

)

−

k

_LN/2(Ω)

. In particular C = 3 if V

1

, V

2

are nonnegative.

Proof. From the respective PDEs, we obtain Z

Ω

|∇u

_i

|

²

dx + Z

Ω

V

i

u

²_i

dx = hf, u

_i

i, i = 1, 2, so that

Z

Ω

V

₁

u

²₁

dx − Z

Ω

V

₂

u

²₂

dx

≤ Z

Ω

|∇u

₁

|

²

dx − Z

Ω

|∇u

₂

|

²

dx

+ kf k

_W−1,2(Ω)

ku

₁

− u

₂

k

_W1,2 0 (Ω)

.

Finally, we observe that

k∇u

₁

k

²_L2(Ω)

− k∇u

₂

k

²_L2(Ω)

=

k∇u

₁

k

_L2(Ω)

+ k∇u

₂

k

_L2(Ω)

k∇u

₁

k

_L2(Ω)

− k∇u

₂

k

_L2(Ω)

≤

k∇u

₁

k

_L2(Ω)

+ k∇u

₂

k

_L2(Ω)

ku

₁

− u

₂

k

_W1,2 0 (Ω)

,

then we can conclude by using (2.3).

Lemma 2.5. Let V

1

, V

2

∈ V be two admissible potentials. Let u

1

, u

2

∈ W

₀^1,2

(Ω) be solutions of (2.2) such that

u

_i

∈ L

²

(Ω; V

₁

) ∩ L

²

(Ω; V

₂

), i = 1, 2.

Then we have

(2.7) E

_f

(V

₁

) − E

_f

(V

₂

) = 1 2

Z

Ω

(V

₁

− V

₂

) u

₁

u

₂

dx.

In particular there holds

(2.8) |E

_f

(V

1

) − E

_f

(V

2

)| ≤ 1 2

Z

Ω

|V

₁

− V

2

| u

²₁

dx

1/2

Z

Ω

|V

₁

− V

2

| u

²₂

dx

1/2

.

Proof. We first observe that, from the hypothesis on the potentials, we can use u

1

− u

2

as a test function for the equations solved by u

₁

and u

₂

, i.e.

(2.9) Z

Ω

h∇u

_i

, ∇ϕi dx+

Z

Ω

V

_i

u

_i

ϕ dx = hf, ϕi, for every ϕ ∈ W

₀^1,2

(Ω)∩L

²

(Ω; V

_i

), i = 1, 2.

We have

E

_f

(V

1

) − E

_f

(V

2

) = 1 2

Z

Ω

|∇u

₁

|

²

dx + 1 2

Z

Ω

V

1

u

²₁

dx − hf, u

₁

i

− 1 2

Z

Ω

|∇u

₂

|

²

dx − 1 2

Z

Ω

V

₂

u

²₂

dx + hf, u

₂

i.

On the other hand 1 2

Z

Ω

|∇u

₁

|

²

dx − 1 2

Z

Ω

|∇u

₂

|

²

dx = 1 2

Z

Ω

h∇u

₁

, ∇(u

₁

− u

2

)i dx

− 1 2

Z

Ω

h∇u

₂

, ∇(u

₂

− u

₁

)i dx, thus by appealing to (2.9), we get

1 2

Z

Ω

|∇u

₁

|

²

dx − 1 2

Z

Ω

|∇u

₂

|

²

dx = − 1 2

Z

Ω

V

1

u

1

(u

1

− u

2

) dx + 1

2 hf, u

₁

− u

2

i + 1

2 Z

Ω

V

₂

u

₂

(u

₂

− u

₁

) dx − 1

2 hf, u

₂

− u

₁

i

= − 1 2

Z

Ω

V

1

u

²₁

dx + 1 2

Z

Ω

V

2

u

²₂

dx + hf, u

₁

− u

2

i + 1

2 Z

Ω

(V

₁

− V

₂

) u

₁

u

₂

dx.

This concludes the proof of (2.7). The estimate (2.8) just follows by applying H¨ older inequality.

Remark 2.6. Observe that if V

₁

, V

₂

∈ L

^p

(Ω) with p > 1 and u

₁

, u

₂

∈ L

^2p/(p−1)

(Ω), then the hypotheses of the previous Lemma are verified and (2.8) gives the Lipschitz estimate

|E

_f

(V

1

) − E

_f

(V

2

)| ≤ 1

2 kV

₁

− V

2

k

_Lp(Ω)

q

ku

₁

k

_L2p/(p−1)(Ω)

q

ku

₂

k

_L2p/(p−1)(Ω)

. By Proposition 2.1, the condition u

₁

, u

₂

∈ L

^2p/(p−1)

(Ω) is verified for example if

• |Ω| = +∞ and p = N/2;

• Ω = ω × R is a waveguide and p ≥ N/2;

• |Ω| < +∞ and p ≥ N/2.

3. Maximization problems

In this section we fix p > 1 and we consider the optimization problem for potentials

(3.1) max

V∈V

E

_f

(V ) : Z

Ω

|V |

^p

dx ≤ 1

. We also introduce the strictly convex functional

(3.2) G

_p,f

(u) = 1 2

Z

Ω

|∇u|

²

dx + 1 2

Z

Ω

|u|

^2p/(p−1)

dx

(p−1)/p

− hf, ui, u ∈ W

₀^1,2

(Ω), where it is intended that G

_p,f

(u) = +∞ if u 6∈ L

^2p/(p−1)

(Ω). We recall the following existence result from [10]. We give the proof for the reader’s convenience.

Proposition 3.1. The problem (3.1) admits a solution and is equivalent to

(3.3) max

V∈V

E

_f

(V ) : V ≥ 0, Z

Ω

V

^p

dx = 1

. The solution V

0

is unique and is of the form

(3.4) V

₀

=

Z

Ω

|v

₀

|

^2p/(p−1)

dx

−1/p

v

^2/(p−1)₀

,

where v

₀

∈ W

₀^1,2

(Ω) ∩ L

^2p/(p−1)

(Ω) is the unique minimizer of G

_p,f

. Moreover, we have

(3.5) E

_f

(V

0

) = G

p,f

(v

0

).

Proof. We start by proving that we can restrict the optimization to positive potentials that saturate the constraint on the L

^p

norm. We have

sup

V∈V

E

_f

(V ) : V ≥ 0, Z

Ω

|V |

^p

dx = 1

≤ sup

V∈V

E

_f

(V ) : Z

Ω

|V |

^p

dx ≤ 1

. On the other hand it is immediate to see that

E

_f

(V ) ≤ E

_f

|V | kV k

_Lp(Ω)

, for every V ∈ L

^p

(Ω) \ {0} with Z

Ω

|V |

^p

dx ≤ 1, thus the two suprema coincide.

In order to characterize the optimal potential V

₀

, we observe that for every u ∈ L

^2p/(p−1)

(Ω) and every admissible potential, we get

(3.6)

Z

Ω

V u

²

dx ≤ Z

Ω

|u|

^2p/(p−1)

dx

(p−1)/p

,

thanks to H¨ older inequality. By appealing to the definition of the energy E

_f

(V ), we then get E

_f

(V ) ≤ 1

2 Z

Ω

|∇u|

²

dx + 1 2

Z

Ω

|u|

^2p/(p−1)

dx

(p−1)/p

− hf, ui, ∀u ∈ W

₀^1,2

(Ω).

By taking the infimum on u, we obtain E

_f

(V ) ≤ min

u∈W₀^1,2(Ω)

G

p,f

(u), for every V admissible.

On the other hand, we see that if v

0

is a minimizer of G

_p,f

and (V

0

, v

0

) achieves equality in (3.6), we have equality in the last inequality. By appealing to the equality cases in H¨ older

inequality, we get the characterization (3.4).

Remark 3.2. For the sake of completeness we observe that by a standard homogeneity argu- ment

E

_f

(V ) = − 1

2 sup

u∈W₀^1,2(Ω)\{0}

hf, ui

²

Z

Ω

|∇u|

²

dx + Z

Ω

V u

²

dx .

By using (3.6) we can infer E

_f

(V ) ≤ − 1

2 sup

u∈W₀^1,2(Ω)\{0}

hf, ui

²

Z

Ω

|∇u|

²

dx + Z

Ω

|u|

^2p/(p−1)

dx

(p−1)/p

,

for every V ∈ L

^p

(Ω) with unit norm. This implies that sup

E

_f

(V ) : Z

Ω

|V |

^p

dx = 1

= − 1

2 sup

u∈W₀^1,2(Ω)\{0}

hf, ui

²

Z

Ω

|∇u|

²

dx + Z

Ω

|u|

^2p/(p−1)

(p−1)/p

,

so that E

_f

(V

₀

) is related to the best constant in a Poincar´ e-Sobolev type inequality, i.e.

Z

Ω

|∇u|

²

dx + Z

Ω

|u|

^2p/(p−1)

dx

(p−1)/p

≥ 1

2 |E

_f

(V

0

)| hu, f i

²

, ∀u ∈ W

₀^1,2

(Ω), with equality holding if and only if u is proportional to v

0

.

4. Stability for maximization problems In what follows c

1

will denote the constant

c

1

:=

Z

Ω

|v

₀

|

^2p/(p−1)

dx

(p−1)/2p

,

where v

₀

is the unique minimizer of G

_p,f

. In this section we prove a quantitative improvement of the inequality

E

_f

(V

0

) ≥ E

_f

(V ), for every V ∈ V such that Z

Ω

|V |

^p

dx ≤ 1.

At this aim, we need the following result, see [12, Theorem 3.1] for a proof. An earlier related result could be found in [14, Proposition 2.6].

Lemma 4.1 (Quantitative H¨ older inequality). Let 2 ≤ q < ∞ and q

⁰

= q/(q − 1). For every f ∈ L

^q

(Ω) and g ∈ L

^q0

(Ω) such that kf k

_Lq(Ω)

= kgk

_Lq0

(Ω)

= 1, we have (4.1)

Z

Ω

f g dx

≤ 1 − q

⁰

− 1 4

|f|

^q−2

f − g

2 L^q0(Ω)

, and

(4.2)

Z

Ω

f g dx

≤ 1 − 1 q 2

^q−1

f − |g|

^q0−2

g

q L^q(Ω)

.

Remark 4.2. In the case q = ∞ no quantitative inequality of the previous kind may hold. In fact, by taking Ω = (0, 1) and the functions

f

_n

= 1

_[0,1−1/n]

and g = 1 2 √

x , we obtain

n→∞

lim

Z

Ω

f

n

g dx

− 1

= 0 while

f

n

− g

|g|

L^∞(Ω)

= kf

_n

− 1k

_L^∞_(Ω)

= 1

4.1. Stability of the potentials. This is the main result of this section.

Theorem 4.3 (Stability of maximal potentials). Let V

₀

be the optimal potential achieving the maximum in (3.1). Then for every V ∈ V such that kV k

_Lp(Ω)

≤ 1 we have

(4.3) E

_f

(V

₀

) − E

_f

(V ) ≥ σ

⁰_M

|V |

^p−2

V − V

₀^p−1

2

L^p0(Ω)

, for p ≥ 2, and

(4.4) E

_f

(V

₀

) − E

_f

(V ) ≥ σ

_M⁰⁰

kV − V

₀

k

²_Lp(Ω)

, for 1 < p < 2,

where σ

_M⁰

> 0 and σ

⁰⁰_M

> 0 are two constants depending only on p and c

₁

(see Remark 4.4 below).

Proof. We start observing that by hypothesis kV − V

₀

k

_Lp(Ω)

≤ 2, and

|V |

^p−2

V − |V

₀

|

^p−2

V

₀

L^p0(Ω)

≤ 2, thus we can always suppose

(4.5) E

_f

(V

₀

) − E

_f

(V ) ≤ min

c

²₁

4 , 1

, because otherwise (4.3) and (4.4) are trivially true, with constants

σ

_M⁰

= σ

⁰⁰_M

= 1 4 min

c

²₁

4 , 1

.

By using v

0

as a test function in the variational problem defining E

_f

(V ) and recalling the definition (3.2) of G

_p,f

, we get

E

_f

(V ) ≤ G

_p,f

(v

₀

) + 1 2

"

Z

Ω

V v

²₀

dx − Z

Ω

|v

₀

|

^2p/(p−1)

dx

(p−1)/p

#

= E

_f

(V

0

) + 1 2

"

Z

Ω

V v

₀²

dx − Z

Ω

|v

₀

|

^2p/(p−1)

dx

(p−1)/p

# .

The optimal potential V

₀

and v

₀

are linked through (3.4), thus by substituting above we get E

_f

(V ) − E

_f

(V

0

) ≤ c

²₁

2 Z

Ω

(V − V

0

) V

₀^p−1

dx = c

²₁

2

Z

Ω

V V

₀^p−1

dx − 1

. From the previous inequality we obtain

(4.6) E

_f

(V

0

) − E

_f

(V ) ≥ c

²₁

2

"

1 − Z

Ω

|V |

^p

dx

1/p

#

+ c

²₁

2

"

Z

Ω

|V |

^p

dx

1/p

− Z

Ω

V V

₀^p−1

dx

# .

The two terms inside the square brackets are both positive, since kV k

_Lp(Ω)

≤ kV

₀^p−1

k

_Lp0

(Ω)

= 1.

The previous estimate in particular implies that (4.7)

Z

Ω

|V |

^p

dx

1/p

≥ 1 2 ,

since otherwise we would contradict (4.5). We now distinguish two cases.

Case p ≥ 2. By applying (4.1) with the choices

f = V

kV k

_Lp(Ω)

, g = V

₀^p−1

, q = p and q

⁰

= p

⁰

, we get

Z

Ω

|V |

^p

dx

1/p

− Z

Ω

V V

₀^p−1

dx ≥ p

⁰

− 1 8

|V |

^p−2

V

kV k

^p−1_Lp(Ω)

− V

₀^p−1

2

L^p0(Ω)

(4.8) ,

where we used (4.7) to estimate the L

^p

norm of V from below. We now observe that by the triangle inequality and convexity of t 7→ t

²

, we get

|V |

^p−2

V

kV k

^p−1_Lp(Ω)

− V

₀^p−1

2

L^p0(Ω)

≥ 1 2

|V |

^p−2

V − V

₀^p−1

2 L^p0(Ω)

−

|V |

^p−2

V

kV k

^p−1_Lp(Ω)

− |V |

^p−2

V

2

L^p0(Ω)

= 1 2

|V |

^p−2

V − V

₀^p−1

2 L^p0(Ω)

−

1 − kV k

^p−1_Lp(Ω)

2

≥ 1 2

|V |

^p−2

V − V

₀^p−1

2

L^p0(Ω)

− (p − 1)

²

2

c

²₁

2

E

_f

(V

0

) − E

_f

(V

0

)

2

, where we used that for p ≥ 2

1 − t

^p−1

≤ (p − 1) (1 − t), for every 0 ≤ t ≤ 1,

and (4.6) in the last inequality. By inserting this information in (4.8), combining with (4.6) and using that E

_f

(V

0

) − E

_f

(V ) ≤ 1, we end up with (4.3).

Case 1 < p < 2. By applying (4.1) this time with the choices f = V

₀^p−1

, g = V

kV k

_Lp(Ω)

, q = p

⁰

and q

⁰

= p, we get

Z

Ω

|V |

^p

dx

1/p

− Z

Ω

V V

₀^p−1

dx ≥ p − 1 8

V kV k

_Lp(Ω)

− V

0

2

L^p(Ω)

(4.9) ,

where we used again (4.7). We can estimate the remainder term as before

V

kV k

_L^p_(Ω)

− V

₀

2

L^p(Ω)

≥ 1

2 kV − V

₀

k

²_Lp(Ω)

−

V

kV k

_L^p_(Ω)

− V

2

L^p(Ω)

= 1

2 kV − V

₀

k

²_Lp(Ω)

−

1 − kV k

_Lp(Ω)

2

≥ 1

2 kV − V

0

k

²_Lp(Ω)

− 2

c

²₁

2

E

_f

(V

0

) − E

_f

(V )

2

,

where we used again (4.6) in the last inequality. We can obtain the desired result by combining

(4.6), (4.9) and the previous estimate.

Remark 4.4. From the proof, we can see that a possible value for σ

_M⁰

is σ

_M⁰

= 1

4 min

(p

⁰

− 1) c

⁴₁

8 c

²₁

+ 2 (p − 1) , 1, c

²₁

4

, p ≥ 2, while for σ

⁰⁰_M

we could take

σ

_M⁰⁰

= 1 4 min

(p − 1) c

⁴₁

8 c

²₁

+ 2 (p − 1) , 1, c

²₁

4

, 1 < p < 2.

Remark 4.5. We point out that we could have used (4.2) in place of (4.1). In this way, one could obtain stability estimates of the type

(4.10) E

_f

(V

0

) − E

_f

(V ) ≥ e σ

_M⁰

kV − V

0

k

^p_Lp(Ω)

, for p ≥ 2, and

E

_f

(V

0

) − E

_f

(V ) ≥ e σ

_M⁰⁰

|V |

^p−2

V − V

₀^p−1

p⁰

L^p0(Ω)

, for 1 < p < 2.

We also notice that these estimates are asymptotically worse than (4.3) and (4.4). Indeed, when V is of the form V

ε

= V

0

+ ε ψ, for ε 1 and ψ ∈ L

^p

(Ω), it is not difficult to see that

kV

_ε

− V

0

k

_Lp(Ω)

' ε and

|V

_ε

|

^p−2

V

ε

− V

₀^p−1

L^p0(Ω)

' ε,

so that we have

|V

_ε

|

^p−2

V

_ε

− V

₀^p−1

2

L^p0(Ω)

kV

_ε

− V

₀

k

^p_Lp(Ω)

if p ≥ 2, kV

_ε

− V

0

k

²_Lp(Ω)

|V

_ε

|

^p−2

V

ε

− V

₀^p−1

p⁰

L^p0(Ω)

if 1 < p < 2.

4.2. Stability of the state functions. We have the following stability result for the mini- mization of G

p,f

.

Proposition 4.6. Let 1 < p < ∞ and let v

₀

be the unique minimizer of G

_p,f

defined by (3.2), then for every u ∈ W

₀^1,2

(Ω) we have

(4.11) G

_p,f

(u) − G

_p,f

(v

0

) ≥ 1

2 ku − v

0

k

²

W₀^1,2(Ω)

.

Proof. We first observe that if u 6∈ L

^2p/(p−1)

(Ω), then G

p,f

(u) = +∞ and (4.11) trivially holds.

Thus, let us take u ∈ W

₀^1,2

(Ω) ∩ L

^2p/(p−1)

(Ω) with u 6= v

0

and set d = ku − v

₀

k

_W1,2

0 (Ω)

and ϕ = u − v

₀

d .

Then u can be written as u = v

₀

+ d ϕ and ϕ has unitary norm in W

₀^1,2

(Ω). We then get G

_p,f

(u) − G

_p,f

(v

₀

) ≥ d

"

Z

Ω

h∇v

₀

, ∇ϕi dx + Z

Ω

|v

₀

|

^2p/(p−1)

dx

−1/p

Z

Ω

|v

₀

|

^2/(p−1)

v

₀

ϕ dx − hf, ϕi

#

+ d

²

2

Z

Ω

|∇ϕ|

²

dx,

where we used the convexity of the C

¹

map Φ(s) = kv

₀

+ s ϕk

²_L2p/(p−1)(Ω)

=

Z

Ω

|v

₀

+ s ϕ|

^2p/(p−1)

dx

(p−1)/p

, s ∈ R , so that

Φ(d) ≥ Φ(0) + Φ

⁰

(0)d.

It is now sufficient to observe that Z

Ω

h∇v

₀

, ∇ϕi dx + Z

Ω

|v

₀

|

^2p/(p−1)

dx

−1/p

Z

Ω

|v

₀

|

^2/(p−1)

v

0

ϕ dx − hf, ϕi = 0,

by minimality of v

0

, thus we directly get (6.2).

As a consequence of the previous result, we get that if (u, V ) is almost realizing the equality in (3.6), then u is near to the optimizer v

₀

in the W

₀^1,2

norm.

Corollary 4.7. Let V be an admissible potential for (3.1) and u a corresponding energy func- tion. Then

(4.12) ku − v

₀

k

²

W₀^1,2(Ω)

≤

"

Z

Ω

|u|

^2p/(p−1)

dx

(p−1)/p

− Z

Ω

V u

²

dx

# . Proof. We already observed that

E

_f

(V ) ≤ G

p,f

(v), for every v ∈ W

₀^1,2

(Ω).

By taking the energy function v

0

corresponding to V

0

, we get E

_f

(V ) ≤ G

_p,f

(v

0

) ≤ G

_p,f

(u) − 1

2 ku − v

0

k

²

W₀^1,2(Ω)

, where we used (4.11). Thus we have

1

2 ku − v

0

k

²

W₀^1,2(Ω)

≤ G

_p,f

(u) − E

_f

(V ) = 1 2

"

Z

Ω

|u|

^2p/(p−1)

dx

(p−1)/p

− Z

Ω

V u

²

dx

# ,

which concludes the proof.