On a highly nonlinear self–obstacle optimal control problem

optimal control problem

Daniela Di Donato Department of Mathematics

University of Trento

via Sommarive 14, 38123 Trento - Italy e-mail: [email protected]

and

Dimitri Mugnai

Dipartimento di Matematica e Informatica Universit`a di Perugia

Via Vanvitelli 1, 06123 Perugia - Italy e-mail: [email protected]

Abstract

We consider a non–quadratic optimal control problem associated to a nonlinear elliptic variational inequality, where the obstacle is the control itself. We show that, fixed a desired profile, there exists an optimal so- lution which is not far from it. Detailed characterizations of the optimal solution are given, also in terms of approximating problems.

Keywords: Obstacle problem, Optimal control, Variational inequality 2010AMS Subject Classification: 49J20, 49J40, 49K20, 93C20

1 Introduction

This paper is a contribution to the quest of an optimal control for variational inequalities, a topic which has been faced by several authors under different settings, and we only quote [5], [13], [17], [23], [24] and their references.

We remark that in the framework we shall see below, except for [7] and the recent [10], the relation

solution of a variational inequality ←→ existence of an optimal control in general was given only for linear variational inequalities and quadratic cost functionals (see also [8]). However, in many situations, the variational inequality one is dealing with is nonlinear ([12], [15], [19]) and the cost functional is not quadratic. In this paper we investigate such a situation, where several difficulties

1

show up and a simple adaptation of the known results is not possible. However, our results cover the ones known in already studied situations. Moreover, as in [2], [3], [4], [6], [7], [9] and [10], the problem under consideration is not a classical one, since the function space where the solution u of the variational inequality must be sought depends on the optimal solution ψ of the cost functional, which, in turn, depends on u. For this reason we call this problem a self–obstacle control problem. We remark that in [2], [3], [4], [6], [9] and [10], only linear problems are considered, while in [7] a semilinear case is treated and in [10]

the authors extend the existence results shown for the Laplace operator in [6]

to the p−Laplacian case. However, in no case their cost functional contains other terms than the energy and the distance from the target function, while we consider the presence of another general nonlinear term (see the definition of J below).

Let us present the problem in details. Ω ⊂ R^N is a bounded domain with sufficiently smooth boundary ∂Ω and

z ∈ L²(Ω) is a given target profile. (1) Denote by H^k(Ω) the usual Sobolev space of order k, k ∈ N, and by H0^k(Ω) the closure of C_c^∞(Ω) in the H^k norm. Setting V = H³(Ω) ∩ H₀²(Ω) and given ψ ∈ V , we define the closed convex set

K(ψ) =n

u ∈ H₀²(Ω) | u ≥ ψ a.e. in Ωo , and we consider the following semilinear variational inequality:











u ∈ K(ψ), Z

Ω

∆u∆(v − u)dx + Z

Ω

g(u)(v − u) dx ≥ Z

Ω

f (v − u) dx ∀ v ∈ K(ψ).

(2)

Here f ∈ L²(Ω) and g is a nondecreasing, C¹ real-valued function for which there exist γ ∈ R and ν ≥ 0 such that

|g(s)| ≤ γ + ν|s| ∀ s ∈ R. (3)

It is easy to see that problem (2) admits a unique solution u = τ (ψ) ∈ H₀²(Ω) which is the minimizer of the functional Iψ : K(ψ) → R defined as

Iψ(v) = 1 2 Z

Ω

|∆v|²dx + Z

Ω

G(v) dx − Z

Ω

f v dx, (4)

where

G(v) = Z v

0

g(s) ds.

We now consider an optimal control problem associated to (2) in which we view ψ as the control and u = τ (ψ) as the corresponding state. For this, we introduce the objective functional J : V → R given by

J (ψ) = 1 2

Z

Ω

|∇∆ψ|²dx + Z

Ω

h(x, ψ) dx + 1 2 Z

Ω

(τ (ψ) − z)²dx,

where h : Ω × R → R is a Carath´eodory function that satisfies one of the following conditions:

H1): there exist α1, α2≥ 0 and p ∈ [1, 2) such that |h(x, s)| ≤ α1+ α2|s|^p for a.e. x ∈ Ω and all s ∈ R,

or

H₂): there exist α₁, α₂ ≥ 0 and p such that 0 ≤ h(x, s) ≤ α1+ α₂|s|^p for a.e.

x ∈ Ω and all s ∈ R, where







p ≥ 1 if N ≤ 6

1 ≤ p ≤ _{N −6}^2N if N ≥ 7.

We remark that H2) covers the case of superlinear functions having critical growth at infinity in the sense of the Sobolev embedding.

The purpose of this paper is to investigate the optimal control problem (P ) find ψ ∈ V s.t. J (ψ) = inf

φ∈V J (φ).

In particular, we have

Theorem 1. Problem (P ) has a solution ψ^∗∈ V .

Theorem 1 will be proved in Section 3 in a direct way. However, since the problem appears to be unusual and it may look hard to describe completely the solution, in the next sections we will try to describe solutions of (P ) via an approximation method. In particular, for any δ > 0 and ψ ∈ V , we introduce the problems

(Q_δ)_ψ











∆²uδ,ψ+ g(uδ,ψ) +1

δβ(uδ,ψ− ψ) = f in Ω uδ,ψ=∂uδ,ψ

∂ν = 0 on ∂Ω,

where ν is the outer unit normal to ∂Ω and β : R → R is a suitably chosen function which ensures existence and uniqueness of a solution uδ,ψ(see Section 4 for the precise assumptions). Setting τδ(ψ) = uδ,ψ, we consider the approximate objective functional

J_δ(ψ) :=1 2

Z

Ω

|∇∆ψ|²dx + Z

Ω

h(x, ψ) dx + 1 2

Z

Ω

(τ^δ(ψ) − z)²dx.

and the related optimal control problem (Pδ) find ψδ ∈ V such that Jδ(ψδ) = min

ψ∈V Jδ(ψ).

First, we shall show that there exists an optimal solution ˆψδ for problem (Pδ) with associated a certain ˆu_{δ, ˆψ}

δ; due to the “double” nature of the minimization

problem above, we will also say that (ˆu_{δ, ˆψ}

δ, ˆψδ) = (ˆuδ, ˆψδ) is an optimal pair for problem (P_δ). In the spirit of [25] and the subsequent [20] and [21], we will eventually prove that problem (P_δ) is a good approximation for problem (P ), in the sense that optimal solutions of (P_δ) converge to optimal solutions of (P ) as δ → 0, according to the following result:

Theorem 2. For all δ > 0, let (ˆu_δ, ˆψ_δ) be an optimal pair for problem (P_δ).

Then there exist ˆu ∈ H₀²(Ω) and ˆψ ∈ V such that ψˆδ * ˆψ in V and

ˆ

uδ → ˆu in H₀²(Ω) (5)

as δ → 0. Moreover, ˆu = τ ( ˆψ) and (ˆu, ˆψ) is an optimal pair for problem (P ).

Finally, in the last section we give further descriptions of the optimal solution in terms of adjoint functions, both for the approximating problems and for the original one, and exhibiting a differential equation solved by the optimal obstacle.

Remark 1. The function h being quite general and in particular not convex, problem (P ) may have multiple solutions. Via the approximation procedure sketched above, which gives a unique solution for any fixed ψ, we can prove the convergence of a solution of (Pδ) (possibly not unique) to a solution of (P ).

Of course, if uniqueness is granted (for example adding a convexity assumption on h), we have that the solution of problem (P ) can be approximated by the solution of problem (Pδ).

We conclude this part with some notations. First, we endow H₀²(Ω) with the scalar product

hu, vi = Z

Ω

∆u∆v dx,

which induces the norm kuk := k∆uk_L2(Ω), equivalent to the usual one (see [25, Remark 2.1]), and in particular, there exists S > 0 such that

kukL²(Ω)≤ Skuk ∀ u ∈ H₀²(Ω). (6) In view of this setting, we endow V with the norm

kψkV = k∇∆ψkL²(Ω) ∀ ψ ∈ V,

which is equivalent to the usual one, and the Poincar´e inequality implies the existence of S1> 0 such that

kψk ≤ S1kψkV ∀ ψ ∈ V.

Finally, from now on, we will denote by k · k2the norm in L²(Ω) and by |G| the Lebesgue measure of the set G.

2 Preliminaries

We recall the following notion of capacity (see [14, Definition 4.1.1]). Let E be a set which is compactly contained Ω and set

WE =n

v ∈D(Ω) : v ≥ 1 in a neighbourhood of Eo ,

where, as usual, D(Ω) denotes the set of smooth functions having compact support in Ω; then the (Sobolev)C2,2capacity of E relative to Ω is defined by

C2,2(E, Ω) := inf

v∈WK

nkvk²_H2(Ω)

o ,

Due to the equivalence between k · k and the complete norm k · k_H2(Ω)in H₀²(Ω), it is readily seen thatC2,2 is equivalent to C_2,2, defined as

C_2,2(E, Ω) := inf

v∈W_ωk∆vk²₂, and, in particular, there exists c > 0 such that

C2,2(E, Ω) ≤C2,2(E, Ω) ≤ cC2,2(E, Ω) for every E ⊂⊂ Ω. (7) We note that, adding the condition “v ≥ 0 in Ω” in the definition of W_E, we obtain the related capacities C_2,2⁺ andC2,2⁺, with the obvious inequalities

C2,2≤ C_2,2⁺ andC2,2≤C2,2⁺. (8) Moreover, by trivial extension of test functions outside Ω, we also get that

Cap(E) := Cap(E, R^N) ≤ Cap(E, Ω) for every E ⊂ Ω, (9) where we have denoted by Cap any of the capacities introduced above.

With such a definition, a standard technique (for example, see the proof of [14, Property 2, p. 211]), gives that

Lemma 1. Cap is an outer measure in Ω.

We will also need the following result, for which we obviously exclude the easy case N ≤ 4:

Lemma 2. Let u, v ∈ H²(Ω) and N ≥ 5. If u ≥ v a.e. (with respect to the Lebesgue measure), then ˆu ≥ ˆv C_2,2⁺ (·, Ω)−a.e., where ˆu denotes the precise representative for u defined a.e. by

ˆ

u(x) = lim

R→0

1

|B(x, R)|

Z

B(x,R)

u(y) dy,

where the limit exists for C_2,2⁺ (·, Ω)−a.e. x ∈ Ω.

Proof. By [14, Property 2, p. 210] (or adapting the proof of [22, Lemma 2.1] to our setting), we immediately have that have that there exists c1> 0 such that

C_2,2⁺ (E, Ω) ≤ c₁C_2,2⁺ (E, R^N) (10) for every E ⊂ Ω.

By using (9), (7) and (10), we immediately find

C2,2⁺(E) ≤C2,2⁺(E, Ω) ≤ cC_2,2⁺ (E, Ω) ≤ cc₁C_2,2⁺ (E) ≤ cc₁C2,2⁺(E), (11) so that all involved capacities are equivalent.

The existence of the limit in the statement of the theorem is guaranteed C2,2−a.e. by [1, Theorem 6.2.1 and Proposition 2.3.13] (actually the space functions used therein to defineC2,2 is the Schwarz class, but the two definitions are equivalent by an obvious density argument). Moreover, by [1, Theorem 2.2.7 and Proposition 2.3.13], for any compact set K ⊂ Ω the extremal function for C2,2(K) is nonnegative. Hence, C2,2⁺(K) ≤C2,2(K), and, as a consequence of (8) and (11), the claim follows.

Moreover, by Lemma 1, C_2,2⁺ (·, Ω) is a capacity in the sense of [16], i.e. a countably subadditive, monotone set function with values on the [0, ∞] such that C_2,2⁺ (∅, Ω) = 0. Since ˆu ≥ ˆv a.e. in Ω, then (ˆv − ˆu)⁺= 0 a.e. in Ω. Moreover, by definition, if G is open and |E| = 0, then C_2,2⁺ (G, Ω) = C_2,2⁺ (G \ E, Ω). Hence, by [16], we get that (ˆv − ˆu)⁺ = 0 C_2,2⁺ (·, Ω)−a.e. in Ω, and the theorem is completely proved.

Remark 2. Lemma 2 was already stated in [2], but the use of nonnegative functions in W was not required therein. However, this condition is used, for example, in proving Lemma 4, and for this we had to prove Lemma 2 again.

In what follows, we will always consider the precise representatives of any function, but for the sake of simplicity we drop the “hats”.

Lemma 3. Let ψ ∈ V and u = τ (ψ) solve (2) with obstacle ψ. Then exists a nonnegative Radon measure µ such that

∆²u + g(u) − f = µ in D⁰(Ω), (12) and moreover µ ∈ H⁻²(Ω), i.e.

Z

Ω

∆u∆ξdx + Z

Ω

(g(u) − f )ξ dx = Z

Ω

ξ dµ, ∀ ξ ∈ H₀²(Ω). (13) Here u denotes the precise representatives of u and byR ξdµ we mean the duality hξ, µi_H2

0(Ω),H⁻²(Ω).

Proof. Though standard, we give the proof of this Lemma for completeness.

Take an arbitrary ξ ∈D(Ω) with ξ ≥ 0. Clearly, v = u + ξ ∈ K(ψ), and putting v into (2), we get

Z

Ω

∆u∆ξ dx + Z

Ω

(g(u) − f )ξ dx ≥ 0.

By the Riesz Theorem for positive distributions, there exists a nonnegative Radon measure µ such that (12) holds.

Furthermore, since the left hand side defines a linear bounded operator in H₀²(Ω), by density of D(Ω) in H0²(Ω), we have that (13) also holds for any ξ ∈ H₀²(Ω).

Lemma 4. Let ψ ∈ V , and let u = τ (ψ) and µ ∈ H⁻²(Ω) as in Lemma 3.

Then:

1. there exists a positive constant M such that

µ(ω) ≤ M C_2,2⁺ (ω, Ω) (14)

for every µ-measurable set ω ⊂ Ω whose distance from ∂Ω is positive;

2. u = ψ µ−a.e., where u and ψ are the precise representatives defined C_2,2⁺ −a.e. in Ω.

Proof. 1. Let ξ ∈ Wω. Using Lemma 3, (3) and the continuous embedding H₀²(Ω) ,→ L²(Ω), we have

µ(ω) = Z

Ω

χωdµ ≤ Z

Ω

ξ dµ

= Z

Ω

∆u∆ξ dx + Z

Ω

(g(u) − f )ξ dx

≤ k∆uk2k∆ξk2+ Z

Ω

(γ + ν|u| + |f |)|ξ| dx

≤ k∆uk2k∆ξk2+ (γ|Ω|¹² + νkuk2+ kf k2)kξk2

≤ M k∆ξk2,

where M = M (u, |Ω|, kf k2). The claim follows at once.

2. By Lemma 2, we have u ≥ ψ C_2,2⁺ −a.e., that is C_2,2⁺ n

x ∈ Ω : ψ(x) >

u(x) a.e. in Ωo , Ω

= 0. Thus, by part 1, we obtain

µn

x ∈ Ω : ψ > u a.e.o

= 0. (15)

Taking v = ψ ∈ K(ψ) in (2) and using (13), we can write Z

Ω

(ψ − u) dµ ≥ 0.

Thus µn

x ∈ Ω : u > ψ a, e.o

= 0, that is ψ ≥ u µ−a.e.; by (15) we get the claim.

Lemma 5. Let ψ, ψ1∈ V , and set u = τ (ψ) and u1= τ (ψ1). Then there exists C = C(kuk2, ku1k2) > 0 such that

ku − u1k ≤ 1

2kψ − ψ1k +1

2pkψ − ψ1k²+ 4SCkψ − ψ1k, (16) where S is given in (6), and C = 2γ|Ω|¹² + ν(kuk2+ ku1k2).

Proof. By Theorem 3, it follows that there exist two nonnegative measures µ ∈ H⁻²(Ω) and µ₁∈ H⁻²(Ω) such that

∆²u + g(u) − f = µ inD⁰(Ω)

∆²u1+ g(u1) − f = µ1 inD⁰(Ω) (17) and

u = ψ µ − a.e.

u₁= ψ₁ µ₁− a.e. (18)

By Lemma 2, we obtain u ≥ ψ C_2,2⁺ −a.e. in Ω. Thus C_2,2⁺ 

x ∈ Ω : u(x) < ψ(x) a.e. , Ω

= 0.

Hence, by part 1 of Lemma 4, we obtain µ1{x ∈ Ω : u(x) < ψ(x) a.e.} = 0.

Therefore,

− Z

Ω

u dµ1≤ − Z

Ω

ψ dµ1. (19)

Similarly, we have

− Z

Ω

u1dµ ≤ − Z

Ω

ψ1dµ. (20)

Using (13) first for u and ξ = u − u₁, then for u₁ and ξ = u₁− u, and summing up, by the monotonicity of g, we get

ku − u1k²= − Z

Ω

(g(u) − g(u1))(u − u1) dx +

Z

Ω

u dµ + Z

Ω

u1dµ1− Z

Ω

u1dµ − Z

Ω

u dµ1

≤ Z

Ω

u dµ + Z

Ω

u1dµ1− Z

Ω

u1dµ − Z

Ω

u dµ1. Taking into account (18), (19) and (20) we have

ku − u1k²≤ Z

Ω

ψ dµ + Z

Ω

ψ1dµ1− Z

Ω

ψ1dµ − Z

Ω

ψ dµ1

= Z

Ω

(ψ − ψ₁) d(µ − µ₁).

By using again (13), we get ku − u₁k²≤

Z

Ω

∆(u − u₁)∆(ψ − ψ₁) dx + Z

Ω

(g(u) − g(u₁))(ψ − ψ₁) dx

≤ k∆ψ − ∆ψ1k2k∆u − ∆u1k2+ Ckψ − ψ₁k2

≤ kψ − ψ1kku − u1k + SCkψ − ψ1k,

with C = 2γ|Ω|¹² + ν(kuk2+ ku1k2). The proof is complete.

3 Existence of an Optimal Control

Now we are ready to prove Theorem 1.

Proof of Theorem 1. First, we observe that by H1) or H2) we have in any case inf

ψ∈V J (ψ) > −∞.

Now, take a minimizing sequence (ψn)n in V . By the very definition of the objective functional J , there exist M1, M2≥ 0 such that

J (ψn) ≥ M1kψnk²− M2, (21) and, more precisely, if H₂) is true, we get J (ψ_n) ≥ kψ_nk². Thus, the sequence (ψ_n)_n∈N is bounded in V . Up to a subsequence, there exists ψ^∗∈ V such that

ψ_n* ψ^∗ in V (22)

and

ψn→ ψ^∗ in H₀²(Ω). (23)

Now, we prove that the sequence (un)_n∈Nis bounded in H₀²(Ω), where un = τ (ψn). From (2) for un with v = ψn ∈ K(ψn) and using the properties of function g, we get

kunk²≤ hun, ψni − Z

Ω

(g(ψn) − f )(un− ψn) dx

≤ kunkkψnk + Z

Ω

(γ + ν|ψ_n| + |f |)|un− ψn| dx

≤ kunkkψnk + (γ|Ω|¹² + νkψnk2+ kf k2)(kunk2+ kψnk2).

Since the sequence (ψn)_n∈N is bounded in V , we immediately get that there exists C1> 0 such that

kunk ≤ C1 ∀ n ∈ N. (24)

Hence, we can suppose that un converges to a certain u weakly in H₀²(Ω), strongly in L²(Ω) and a.e. in Ω.

Moreover, u is the (unique) solution of (2) with obstacle ψ^∗, that is u = u^∗ = τ (ψ^∗). Indeed, from Lemma 5 applied to un and u^∗ and using (23), we obtain

un→ u^∗ in H₀²(Ω), and by uniqueness of the weak limit, we have u = u^∗.

Finally, we prove that ψ^∗is a minimum for the objective functional J . First, by (22), we have

lim inf

n→+∞

Z

Ω

|∇∆ψn|²dx ≥ Z

Ω

|∇∆ψ^∗|²dx.

and, by Fatou’s Lemma, if H1) or H2) hold, we have lim inf

n→+∞

Z

Ω

h(x, ψn) dx ≥ Z

Ω

h(x, ψ^∗) dx.

Note that, unless H2) is in force with p = 2N/(N − 6) and N ≥ 7, the previous inequality is actually an equality.

As a consequence, we obtain inf

ψ∈V J (ψ) = lim

n→+∞J (ψn)

≥ lim inf

n→+∞

1 2

Z

Ω

(∇∆ψ_n)²dx + lim inf

n→+∞

Z

Ω

h(x, ψ_n)+

+ lim

n→+∞

1 2

Z

Ω

(τ (ψ_n) − z)²dx

≥ 1 2

Z

Ω

(∇∆ψ^∗)²dx + Z

Ω

h(x, ψ^∗) + Z

Ω

(τ (ψ^∗) − z)²dx

= J (ψ^∗) ≥ inf

ψ∈VJ (ψ), and the theorem is proved.

4 Approximating Problems

We introduce the nonnegative function

B(r) =







0, if r > 0,

−¹₃r³, if r ∈ [−¹₂, 0],

1

2r²+¹₄r +₂₄¹, if r < −¹₂,

and its derivative β ∈ C¹(R) (which is a nonpositive and nondecreasing func- tion):

β(r) =







0, if r > 0,

−r², if r ∈ [−¹₂, 0]

r + ¹₄, if r < −¹₂.

By the Weierstrass Theorem, it is easy to see that for any δ > 0 and ψ ∈ V , the problem

min

v∈H₀²(Ω)

 1 2

Z

Ω

|∆v|²dx +1 δ

Z

Ω

B(v − ψ) dx + Z

Ω

G(v) dx − Z

Ω

f v dx

 (25) has a unique solution uδ,ψ, which is also a weak solution of

(Qδ)ψ











∆²u_δ,ψ+ g(u_δ,ψ) +1

δβ(u_δ,ψ− ψ) = f in Ω uδ,ψ=∂uδ,ψ

∂ν = 0 on ∂Ω,

where ν is the outer unit normal to ∂Ω.

Let τ_δ : V → H₀²(Ω) be defined as τ_δ(ψ) = u_δ,ψ, and consider the following approximate objective functional:

Jδ(ψ) :=1 2

Z

Ω

|∇∆ψ|²dx + Z

Ω

h(x, ψ) dx + 1 2

Z

Ω

(τ^δ(ψ) − z)²dx.

Now we consider the optimal control problem defined as follows:

(Pδ) find ψδ ∈ V such that Jδ(ψδ) = min

ψ∈V Jδ(ψ)

As stated in the Introduction, first, we show that there exists an optimal solution ˆψδ for problem (Pδ) with associated ˆu_{δ, ˆψ}

δ, and then we prove that opti- mal solutions of (Pδ) converge to optimal solutions of (P ) as δ → 0. Moreover, we recall that we say that (ˆu_{δ, ˆψ}

δ, ˆψ_δ) = (ˆu_δ, ˆψ_δ) is an optimal pair for problem (Pδ).

Now we prove the following

Proposition 1. The operator τδ : V → H₀²(Ω) is Lipschitz continuous.

Proof. Let ϕ, ψ ∈ V and set u = τ_δ(ϕ), v = τ_δ(ψ). We recall that, for each ψ ∈ V , u_δ is weak solution of (Q_δ)_ϕ if

Z

Ω

∆uδ∆w dx + Z

Ω

g(uδ)w dx +1 δ

Z

Ω

β(uδ− ϕ)w dx

= Z

Ω

f w dx ∀ w ∈ H₀²(Ω).

(26)

Hence, writing the analogous equation for v and subtracting, we immediately have

hu − v, wi + Z

Ω

(g(u) − g(v))w dx +1 δ Z

Ω

[β(u − ϕ) − β(v − ψ)]w dx = 0.

Choosing w = u − v, gives hu − v, u − vi +

Z

Ω

(g(u) − g(v))(u − v) dx +1

δ Z

Ω

β(u − ϕ) − β(v − ψ)(u − ϕ) − (v − ψ) − (ψ − ϕ) dx = 0.

By the monotonicity of g and β we obtain ku − vk²+1

δ Z

Ω

β(u − ϕ) − β(v − ψ)(ϕ − ψ) dx ≤ 0.

Since β is of class C¹ and |β⁰| ≤ 1, by the Lagrange Theorem, the H¨older and the Minkowski inequalities and by (6), we obtain

ku − vk²≤1

δk(u − v) − (ϕ − ψ)k2kϕ − ψk2

≤S²

δ (ku − vk + kϕ − ψk)kϕ − ψk.

By the Schwartz inequality we get that there exist two positive constant Cδ and Dδ such that

kτδ(ϕ) − τδ(ψ)k = ku − vk ≤ Cδkϕ − ψk ≤ Dδkϕ − ψkV.

As usual, from now on we will simply write δ → 0 in place of any sequence (δn)_n∈Nsuch that δn→ 0 as n → ∞, so that we shall write uδ instead of of uδ_n, and so on.

Lemma 6. Let ψ ∈ V . Then exists u ∈ H₀²(Ω) such that u_δ,ψ= τ_δ(ψ) * u in H₀²(Ω) as δ → 0,

where u = τ (ψ) ∈ K(ψ) is the solution of the variational inequalities (2) with obstacle ψ. Moreover, there exist C1 = C1(ν) > 0 and C2 = C2(|Ω|, kf k2) > 0 such that

kuδk ≤ C1kψk2+ C2 for all δ > 0. (27) In particular, we can choose C1= 2ν + 2 and C2= 2γ|Ω|^1/2+ 2kf k2.

Proof. Since ψ is fixed and no ambiguity occurs, we simply write uδ = uδ,ψ. First we show (27). Since uδ solves (Qδ)ψ, choosing w = ψ − uδ as test function, we have

Z

Ω

|∆uδ|²dx + Z

Ω

(g(u_δ) − g(ψ))(u_δ− ψ) dx = Z

Ω

∆u_δ∆ψ dx+

− Z

Ω

(g(ψ) − f )(u_δ− ψ) dx +1 δ Z

Ω

β(u_δ− ψ)(ψ − uδ)dx.

Using the monotony of g and β, we get kuδk²≤

Z

Ω

∆u_δ∆ψ dx − Z

Ω

(g(ψ) − f )(u_δ− ψ) dx.

Hence, by (3), we have kuδk²≤ kuδkkψk +

Z

Ω

(γ + ν|ψ| + |f |)|uδ− ψ| dx

≤ kuδkkψk + (γ|Ω|¹² + νkψk2+ kf k2)(kuδk2+ kψk2),

and (27) holds. In particular, (u_δ)_δ is bounded in H₀²(Ω) and thus there exists u ∈ H₀²(Ω) such that, up to a subsequence,

uδ* u in H₀²(Ω) as δ → 0. (28) Now we prove that u ∈ K(ψ). By (26), we have

δ

Z

Ω

∆uδ∆w dx + Z

Ω

(g(uδ) − f )w dx



= − Z

Ω

β(uδ− ψ)w dx ∀ w ∈ H₀²(Ω).

The convergence of u_δ guarantees that [R

Ω∆u_δ∆w dx +R

Ω(g(u_δ) − f )w dx] is uniformly bounded with respect to δ, and thus

lim

δ→0

Z

Ω

β(uδ− ψ)w dx = 0.

On the other hand, by (28), this gives Z

Ω

β(u − ψ)w dx = 0 ∀ w ∈ H₀²(Ω), and thus u ∈ K(ψ).

Finally, we are able to show that u solves (2). We consider an arbitrary v ∈ K(ψ) and using the non-negativity of the function B, we have

1 2

Z

Ω

|∆v|²dx + Z

Ω

G(v) dx − Z

Ω

f v dx = 1 2

Z

Ω

|∆v|²dx + Z

Ω

G(v) dx

− Z

Ω

f v dx +1 δ

Z

Ω

B(v − ψ) dx

≥1 2

Z

Ω

|∆uδ|²dx + Z

Ω

G(uδ) dx+

− Z

Ω

f uδdx + 1 δ Z

Ω

B(uδ− ψ) dx

≥1 2

Z

Ω

|∆uδ|²dx + Z

Ω

G(uδ) dx+

− Z

Ω

f uδdx.

Using (28), we can write 1

2 Z

Ω

|∆v|²dx + Z

Ω

G(v) dx − Z

Ω

f v dx ≥ 1 2lim inf

δ→0

Z

Ω

|∆uδ|²dx + lim

δ→0

Z

Ω

G(uδ) dx − lim

δ→0

Z

Ω

f uδdx

≥ 1 2 Z

Ω

|∆u|²dx + Z

Ω

G(u) dx − Z

Ω

f u dx,

Thus u is the minimizer of (25) in K(ψ), i.e. u = τ (ψ).

Proposition 2. Problem (P_δ) has an optimal pair (ˆu_δ, ˆψ_δ), where ˆu_δ = τ_δ( ˆψ_δ).

Moreover, there exist positive constants a1(ν, |Ω|, kf k2), a2(ν, |Ω|, kf k2), a3 = a3(ν, |Ω|, kf k2) > 0 such that

kˆu_δk ≤ a₁k ˆψ_δk_V + a₂≤ 2a₁kzk₂+ a₃ for every δ > 0. (29) Proof. Take δ > 0 and consider a minimizing sequence (ψ_k^δ)_k∈N ⊆ V for the objective functional Jδ. It is readily seen that (ψ^δ_k)_k∈N is bounded in V , thus there exists C > 0 such that kψ_k^δkV ≤ C. Hence we can suppose that there exists ˆψδ∈ V such that

ψ^δ_k* ˆψδ in V as k → +∞. (30) Now consider the associated (sub)subsequence (u^δ_k)_k∈N, where u^δ_k = τδ(ψ^δ_k).

Using (27), we obtain the existence of a constant C3= C3(kf k2) > 0 such that ku^k_δk ≤ C1k ˆψδk + C2≤ C3 for all k ∈ N,

and the existence of a function ˆu_δ∈ H₀²(Ω) such that, up to a subsequence, u^δ_k* ˆuδ in H₀²(Ω) as k → +∞. (31) Moreover, using (30) and (31) we can also suppose that

 ψ_k^δ → ˆψ_δ in H₀²(Ω),

u^δ_k → ˆu_δ in H₀¹(Ω) (32) as k → +∞.

Since u^δ_k solve (Qδ)ψ_k, we have Z

Ω

∆u^δ_k∆w dx + Z

Ω

(g(u^δ_k) − f )w dx + Z

Ω

1

δβ(u^δ_k− ψ^δ_k)w dx = 0 for every w ∈ H₀²(Ω). By (31) and the continuity of g and β, we get

Z

Ω

∆ˆuδ∆w + Z

Ω

(g(ˆuδ) − f )w +1 δ

Z

Ω

β(ˆuδ− ˆψδ)w dx = 0

for every w ∈ H₀²(Ω), i.e. ˆuδ= τδ( ˆψδ).

We now prove that ψδ is a minimum point for the functional Jδ. Indeed, by (30) and (32) we can write

lim inf

k→+∞

Z

Ω

|∇∆ψ^δ_k|²dx ≥ Z

Ω

|∇∆ ˆψδ|²dx and

u^δ_k → ˆuδ in L²(Ω) and a.e. in Ω.

Then we obtain inf

ψ∈VJδ(ψ) = lim

k→+∞Jδ(ψ_k^δ)

≥ 1 2lim inf

k→+∞

Z

Ω

|∇∆ψ_k^δ|²dx +1 2lim inf

k→+∞

Z

Ω

h(x, ψ_k^δ) dx +1

2 lim

k→+∞

Z

Ω

(τ_δ(ψ^δ_k) − z)²dx

≥ Z

Ω

|∇∆ ˆψ_δ|²dx +1 2

Z

Ω

h(x, ˆψ_δ) dx +1 2

Z

Ω

(τ_δ( ˆψ_δ) − z)²dx

= J_δ( ˆψ_δ) ≥ inf

ψ∈V J_δ(ψ)

and then ˆψδ is the optimal solution for Jδ. Here, as already seen above, lim infR h(x, ψ_k^δ) is actually a limit, unless H2) holds with p = 2N/(N − 6) and N ≥ 7.

Finally, we are able to show (29). Putting dµδ = −1

δβ(ˆuδ− ˆψδ) dx, (33) we obviously have

Z

Ω

ˆ uδdµδ ≤

Z

Ω

ψˆδdµδ

(recall that β(r) = 0 if r ≥ 0). Being ˆuδ= τδ( ˆψδ), we have Z

Ω

|∆ˆuδ|²dx = Z

Ω

(f − g(ˆuδ))ˆuδdx − 1 δ Z

Ω

β(ˆuδ− ˆψδ)ˆuδdx

= Z

Ω

(f − g(ˆuδ))ˆuδdx + Z

Ω

ˆ uδdµδ

≤ Z

Ω

(f − g(ˆuδ))ˆuδdx + Z

Ω

ψˆδdµδ

= Z

Ω

(f − g(ˆuδ))ˆuδdx − 1 δ Z

Ω

ψˆδβ(ˆuδ− ˆψδ) dx

= Z

Ω

∆ˆu_δ∆ ˆψ_δdx + Z

Ω

(g(ˆu_δ) − f )( ˆψ_δ− ˆu_δ) dx

= Z

Ω

∆ˆu_δ∆ ˆψ_δdx + Z

Ω

(g(ˆu_δ) − g( ˆψ_δ))( ˆψ_δ− ˆu_δ) dx +

Z

Ω

(g( ˆψ_δ) − f )( ˆψ_δ− ˆu_δ) dx

Integrating by parts, using the properties of g and the Poincar´e inequality, we finally get

Z

Ω

|∆ˆuδ|²dx ≤ − Z

Ω

∇ˆuδ∇∆ ˆψδdx + Z

Ω

(g( ˆψδ) − f )( ˆψδ− ˆuδ) dx

≤ k∇ˆu_δk2k∇∆ ˆψ_δk2+ (γ|Ω|¹² + νk ˆψ_δk2+ kf k₂)(kˆu_δk2+ k ˆψ_δk2)

≤ Ck∆ˆuδk2k ˆψδkV + (γ|Ω|¹² + νk ˆψδk2+ kf k2)(kˆuδk2+ k ˆψδk2)