IDENTIFICATION OF DISCONTINUOUS PARAMETERS IN DOUBLE PHASE OBSTACLE PROBLEMS

. In this paper, we investigate the inverse problem of identiﬁcation of a discontinuous parameter and a discontinuous boundary datum to an elliptic inclusion problem involving a double phase diﬀerential operator, a multivalued convection term (a multivalued reaction term depending on the gradient), a multivalued boundary condition and an obstacle constraint. First, we apply a surjectivity theorem for multivalued mappings which is formulated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone mapping to examine the existence of a nontrivial solution to the double phase obstacle problem, which exactly relies on the ﬁrst eigenvalue of the Steklov eigenvalue problem for the p -Laplacian. Then, a nonlinear inverse problem driven by the double phase obstacle equation is considered. Finally, by introducing the parameter-to-solution-map, we establish a continuous result of Kuratowski type and prove the solvability of the inverse problem.


Introduction
The aim of this paper is to study an inverse problem to an elliptic differential inclusion problem involving a double phase differential operator, a multivalued convection term (dependence on the gradient of the solution), a multivalued boundary condition and an obstacle constraint. To this end, let Ω be a bounded domain in R N (N ≥ 2) with Lipschitz boundary Γ := ∂Ω such that Γ is divided into three mutually disjoint parts Γ 1 , Γ 2 , and Γ 3 with Γ 1 having positive Lebesgue measure. We study the problem − div a(x)|∇u| p−2 ∇u + µ(x)|∇u| q−2 ∇u + g(x, u) + µ(x)|u| q−2 u ∈ f (x, u, ∇u) in Ω, in Ω, (1.1) where 1 < p < N , p < q, µ : Ω → [0, ∞) is a bounded function, ∂u ∂ν a := a(x)|∇u| p−2 ∇u + µ(x)|∇u| q−2 ∇u · ν, with ν being the outward unit normal vector on Γ, f : Ω × R × R N → 2 R and U : Γ 3 × R → 2 R are two given multivalued functions, Φ : Ω → R is an obstacle function and a : Ω → (0, +∞), h : Γ 2 → R are two possibly discontinuous parameters. The main contribution of the paper is twofold. The first intention of the paper is to establish the nonemptiness, boundedness and closedness of the solution set to problem (1.1) (in the weak sense), in which our main methods are based on a surjectivity theorem for multivalued mappings which is formulated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone mapping, the theory of nonsmooth analysis and the properties of the Steklov eigenvalue problem for the p-Laplacian. The second contribution of the paper is to develop a general framework for studying the inverse problem under consideration and to establish the solvability for such inverse problems. To the best of our knowledge, this is the first work studying the identification of discontinuous parameters for such general nonlinear elliptic equations. The problem under consideration combines several interesting phenomena such as double phase operators, multivalued right-hand sides, mixed boundary conditions and obstacle constraints.
First we point out that, motivated by several applications, the inverse problem of parameter identification in partial differential equations is an important field in mathematics and even though such problems in form of equations and inequalities have been studied a lot, there are still several open problems to be solved. Our work is motivated by the paper of Migórski-Khan-Zeng [39] who studied the inverse problem of mixed quasi-variational inequalities of the form where K : C → 2 C is a set-valued map, T : B × V → V * is a nonlinear map, ϕ : V → R ∪ {+∞} is a functional and m ∈ V * , while V is a real reflexive Banach space, B is another Banach space and C is a nonempty, closed, convex subset of V . Their abstract result applies to p-Laplacian inequalities, see also [38] for hemivariational inequalities. We also mention the works of Clason-Khan-Sama-Tammer [10] for noncoercive variational problems, Gwinner [27] for variational inequalities of second kind, Gwinner-Jadamba-Khan-Sama [28] for an optimization setting and Migórski-Ochal [40] for nonlinear parabolic problems, see also the references therein. In addition we refer to the recent work of Papageorgiou-Vetro [45] about existence and relaxation theorems for different types of problems which can be applied to variational inequalities and control systems.
A second interesting phenomenon is the occurrence of the weighted double phase operator, namely, For a ≡ 1, this operator corresponds to the energy functional given by Functionals of the form (1.3) have been initially introduced by Zhikov [56] in 1986 in order to describe models for strongly anisotropic materials and it also turned out its relevance in the study of duality theory as well as in the context of the Lavrentiev phenomenon, see Zhikov [57]. Observe that the energy density in (1.3) changes its ellipticity and growth properties according to the point in the domain. In general, double phase differential operators and corresponding energy functionals interpret various comprehensive natural phenomena, and model several problems in Mechanics, Physics and Engineering Sciences. For example, in the elasticity theory, the modulating coefficient µ(·) dictates the geometry of composites made of two different materials with distinct power hardening exponents p and q, see Zhikov [58]. Functionals given in (1.3) have been intensively studied in the last years concerning regularity of local minimizers. We mention the famous works of Baroni-Colombo-Mingione [2,3], Byun-Oh [7], Colombo-Mingione [12,13], Marcellini [36,37] and Ragusa-Tachikawa [49]. A third interesting phenomenon is not only the multivalued right-hand side which is motivated by several physical applications (see, for example, Panagiotopoulos [42,43], Carl-Le [8] and the references therein) but also its dependence on the gradient of the solutions often called convection term. The main difficulty in the study of gradient dependent right-hand sides is their nonvariational character, that is, the standard variational tools to corresponding energy functionals are not applicable. In the past years several interesting works has been published with convection terms, we refer to the papers of El Manouni-Marino-Winkert [16], Faraci-Motreanu-Puglisi [17], Faraci-Puglisi [18], , Gasiński-Papageorgiou [23], Liu-Motreanu-Zeng [32], Liu-Papageorgiou [33], Marano-Winkert [35], Papageorgiou-Rȃdulescu-Repovš [44] and Zeng-Papageorgiou [55].
The paper is organized as follows. Section 2 recalls preliminary material including Musielak-Orlicz Lebesgue and Musielak-Orlicz Sobolev spaces, the p-Laplacian eigenvalue problem with Steklov boundary condition, pseudomonotone operators and a surjectivity theorem for multivalued mappings. Under very general assumptions on the data, Section 3 proves the nonemptiness and compactness of the solution set to problem (1.1). In Section 4, we present a new existence result to the inverse problem of (1.1).

Preliminaries
The section is devoted to recall some basic definitions and preliminaries which will be used in the next sections to derive the main results of the paper. To this end, let Ω ⊂ R N be a bounded domain with Lipschitz boundary Γ := ∂Ω such that Γ is decomposed into three mutually disjoint parts Γ 1 , Γ 2 and Γ 3 with Γ 1 having positive Lebesgue measure. In what follows, we denote by M (Ω) the space of all measurable functions u : Ω → R. As usual, we identify two functions which differ on a Lebesgue-null set. Let r ∈ [1, ∞) and D be a nonempty subset of Ω. We denote the usual Lebesgue spaces by L r (D) := L r (D; R) and L r (D; R N ) equipped with the standard r-norm · r,D and L r (Γ) stands for the boundary Lebesgue spaces with norm · r,Γ .
For any fixed s > 1, the conjugate of s is defined by s > 1 such that 1 s + 1 s = 1. Moreover, we use the symbols s * and s * to represent the critical exponents to s in the domain and on the boundary, respectively, given by Let us comment on the r-Laplacian eigenvalue problem with Steklov boundary condition given by for 1 < r < ∞. From Lê [30] we know that (2.2) has a smallest eigenvalue λ S 1,r > 0 which is isolated and simple. Besides, we know that λ S 1,r > 0 can be characterized by 3) The following assumptions are supposed in the entire paper: 1 < p < N, p < q < p * and 0 ≤ µ(·) ∈ L ∞ (Ω).
Here, the modular function is given by We know that L H (Ω) is uniformly convex and so a reflexive Banach space. Moreover, we introduce the seminormed space L q µ (Ω) The Musielak-Orlicz Sobolev space W 1,H (Ω) is given by equipped with the norm where ∇u H = |∇u| H . As before, it is known that W 1,H (Ω) is a reflexive Banach space. Next, we introduce a closed subspace V of W 1,H (Ω) given by Of course, V is also a reflexive Banach space. In the following we denote the norm of the dual space V * of V by · V * . Let us recall some embedding results for the spaces L H (Ω) and W 1,H (Ω), see Gasiński-Winkert [26] or Crespo-Blanco-Gasiński-Harjulehto-Winkert [14].
We suppose that a ∈ L ∞ (Ω) such that inf x∈Ω a(x) > 0. (2.5) Next, we introduce the nonlinear operator F : V → V * given by for u, v ∈ V with ·, · being the duality pairing between V and its dual space V * . The following proposition states the main properties of F : V → V * . We refer to Liu We now recall some notions and results concerning nonsmooth analysis and multivalued analysis. Throughout the paper the symbols " w −→ " and "→" stand for the weak and the strong convergence, respectively, in various spaces. Moreover, let us recall the notions of pseudomonotonicity and generalized pseudomonotonicity in the sense of Brézis for multivalued operators (see, e.g., Migórski-Ochal-Sofonea [41, Definition 3.57]) which will be useful in the sequel.
Definition 2.4. Let X be a reflexive real Banach space. The operator A : X → 2 X * is called (a) pseudomonotone (in the sense of Brézis) if the following conditions hold: (i) the set A(u) is nonempty, bounded, closed and convex for all u ∈ X; (ii) A is upper semicontinuous from each finite-dimensional subspace of X to the weak topology on X * ; (b) generalized pseudomonotone (in the sense of Brézis) if the following holds: Let {u n } ⊂ X and then the element u * lies in A(u) and u * n , u n X * ×X → u * , u X * ×X . It is not difficult to see that every pseudomonotone operator is generalized pseudomonotone, see, e.g., Carl-Le-Motreanu [9, Proposition 2.122]. Also, under an additional assumption of boundedness, we obtain the converse statement, see, e.g., Carl-Le-Motreanu [9, Proposition 2.123].
Proposition 2.5. Let X be a reflexive real Banach space and assume that A : X → 2 X * satisfies the following conditions: (i) for each u ∈ X we have that A(u) is a nonempty, closed and convex subset of X * .
then u * ∈ A(u) and u * n , u n X * ×X → u * , u X * ×X . Then the operator A : X → 2 X * is pseudomonotone.
Definition 2.6. Let (X, τ ) be a Hausdorff topological space and let {A n } n∈N ⊂ 2 X be a sequence of sets. We define the τ -Kuratowski lower limit of the sets A n by then A is called τ -Kuratowski limit of the sets A n .
Finally, we recall the following surjectivity theorem for multivalued mappings which is formulated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping, see Le [29,Theorem 2.2].
for all u ∈ D(G) with u X = R, for all ξ ∈ G(u) and for all η ∈ F(u). Then the inclusion is satisfied, then the estimate in (2.7) holds automatically for some R large enough.

Double phase elliptic obstacle inclusion problem
In this section, we are interested in the study of the existence of a solution to the double phase elliptic obstacle inclusion problem (1.1) and in deriving some relevant properties of the solution set to problem (1.1). More precisely, we are going to apply a surjectivity theorem for multivalued mappings, which is formulated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone operator, to examine the solvability of problem (1.1).
First, we formulate the hypotheses on the data of problem (1.1). H(f ): The multivalued convection mapping f : Ω × R × R N → 2 R has nonempty, bounded, closed and convex values and for all η ∈ f (x, s, ξ), for all s ∈ R, for all ξ ∈ R N and for a. a. x ∈ Ω, where 1 < r < p * with the critical exponent p * in the domain Ω given in (2.1) for s = p; for all η ∈ f (x, s, ξ), for all s ∈ R, for all ξ ∈ R N and for a. a. x ∈ Ω.
for all s ∈ R and for a. a. x ∈ Ω, where p < ς < p * ; for a. a. x ∈ Γ 3 and for all s ∈ R, where 1 < δ < p * with the critical exponent p * on the boundary Γ given in (2.1); (v) there exist β U ∈ L 1 (Γ 3 ) + and b U ≥ 0 such that for all ξ ∈ U (x, s), for all s ∈ R and for a. a. x ∈ Γ 3 . H(0): a ∈ L ∞ (Ω) is such that inf x∈Ω a(x) ≥ c Λ > 0 and h ∈ L p (Γ 2 ).

H(1): The inequality holds
for all η ∈ f (x, s, ξ), for all s ∈ R, for all ξ ∈ R N and for a. a. x ∈ Ω, where 1 < 1 < p; H(U )(v)': there exist β U ∈ L 1 (Γ 3 ) + and b U ≥ 0 such that for all ξ ∈ U (x, s), for all s ∈ R and for a. a. x ∈ Γ 3 , where 1 < 2 < p, then hypothesis H(1) can be removed. Indeed, it follows from Young's inequality with ε > 0 that for all η ∈ f (y, s, ξ), for all ξ ∈ U (x, s), for all s ∈ R, for all ξ ∈ R N , for a. a. y ∈ Ω and for a. a. x ∈ Γ 3 with some c 1 (ε), , then the inequality in H(1) holds automatically.
Let K be a subset of V given by Under H(Φ) we see that the set K is a nonempty, closed and convex subset of V . In fact, from H(Φ) (that is, Φ(x) ≥ 0 for a. a. x ∈ Ω), we know that 0 ∈ K, i. e., K = ∅. Furthermore, it is clear that K is convex. For the closedness, let {u n } n∈N ⊂ K be a sequence such that u n → u in V for some u ∈ V . The continuity of V into L p (Ω) implies that u n → u in L p (Ω). Passing to a subsequence if necessary, we may suppose that u n (x) → u(x) for a. a. x ∈ Ω. Therefore, Hence, u ∈ K and so K is closed. Next, we state the definition of a weak solution to problem (1.1).
is satisfied for all v ∈ K, where the set K is defined by (3.1).
The following theorem which is the main result in this section shows that for each pair (a, h) ∈ L ∞ (Ω) + × L p (Γ 2 ) satisfying H(0), the solution set to problem (1.1), denoted by S(a, h), is nonempty, bounded, and weakly closed. Proof. We divide the proof into three parts.
I Existence: First, we consider the following nonlinear functions F : Ly, z L p (Ω)×L p (Ω) := Ω |y| p−2 yz dx for all u, v ∈ V , for all w ∈ L ς (Ω) and for all y, z ∈ L p (Ω).
Let u ∈ V be fixed. By the Yankov-von Neumann-Aumann selection theorem (see e. g. Papageorgiou-Winkert [47, Theorem 2.7.25]) and assumptions H(f )(i) and (ii), we know that the multivalued function x → f (x, u(x), ∇u(x)) admits a measurable selection. Let η : Ω → R be a measurable selection of x → f (x, u(x), ∇u(x)), that is, η(x) ∈ f (x, u(x), ∇u(x)) for a. a. x ∈ Ω. From H(f )(iii) and the inequality (|r 1 | + |r 2 |) s ≤ 2 s−1 (|r 1 | s + |r 2 | s ) for all r 1 , r 2 ∈ R with s ≥ 1, it follows that there exist constants M 1 , M 2 > 0 satisfying where we have used the fact that the embeddings of V into W 1,p (Ω) and of V into L r (Ω) are continuous. Hence, η ∈ L r (Ω). This permits us to consider the Nemytskij operator N f : V ⊂ L r (Ω) → 2 L r (Ω) associated to the multivalued mapping f defined by Similarly, because of hypotheses H(U )(i), (ii) and (iii), for each u ∈ L δ (Γ 3 ) fixed, we are able to find a measurable function ξ : for some M 3 > 0. Therefore, in what follows, we denote by N U : L δ (Γ 3 ) → 2 L δ (Γ3) the Nemytskij operator corresponding to the multivalued mapping U defined by , ω : V → L ς (Ω) and β : V → L p (Ω) be the embedding operators of V to L r (Ω), V to L ς (Ω) and V to L p (Ω), respectively, with its adjoint operators ι * : L r (Ω) → V * , ω * : L ς (Ω) → V * and β * : L p (Ω) → V * , respectively. Also, we denote by γ : V → L δ (Γ 3 ) the trace operator of V into L δ (Γ 3 ) with its adjoint operator γ * : L δ (Γ 3 ) → V * . Consider the indicator function of the set K formulated as Under the definitions above, we could use a standard procedure for variation calculus to obtain that u ∈ K is a weak solution of problem (1.1) if and only if it solves the following nonlinear inclusion problem: where ∂ c I K is the convex subdifferential operator of I K .
Observe that the functions F , G and L are bounded. The latter combined with (3.

2), (3.3) and hypotheses H(f ) and H(U ) implies that for each u ∈ V the set
is nonempty, bounded, closed and convex. We show that H is a pseudomonotone operator. Let {u n } n∈N ⊂ V , {ζ n } n∈N ⊂ V * be sequences and let (u, ζ) ∈ V × V * be such that ζ n ∈ H(u n ) for each n ∈ N, ζ n w −→ ζ and lim sup n→∞ ζ n , u n − u ≤ 0. (3.4) Then, for every n ∈ N, there are η n ∈ N f (u n ) and ξ n ∈ N U (u n ) such that ζ n = F u n + ω * Gu n − β * Lu n − ι * η n − γ * ξ n for all n ∈ N.
Taking (3.2) and (3.3) into account, we can see that the sequences {η n } n∈N ⊂ L r (Ω) and {ξ n } n∈N ⊂ L δ (Γ 3 ) are both bounded. Without any loss of generality, we may assume that there exist functions Recall that V is embedded compactly into L ς (Ω), L r (Ω) and L p (Ω), respectively, and γ : From Proposition 2.3 we know that F is of type (S + ). Therefore, Passing to a subsequence if necessary, we may assume that u n (x) → u(x) and ∇u n (x) → ∇u(x) for a. a. x ∈ Ω. (3.6) Applying Mazur's Theorem there exists a sequence {χ n } n∈N of convex combinations to {η n } n∈N satisfying Therefore, we can suppose that χ n (x) → η(x) for a. a. x ∈ Ω. Due to the convexity of f we see that Recall that f is u.s.c. and has nonempty, bounded, closed and convex values (see hypotheses H(f )(i) and (ii)). So, we can use Proposition 4.1.9 of Denkowski-Migórski-Papageorgiou [15] to infer that the graph of (s, ξ) → f (x, s, ξ) is closed for a. a. x ∈ Ω. Taking the convergence properties in (3.6) and χ n (x) → η(x) for a. a. x ∈ Ω into account, we obtain η(x) ∈ f (x, u(x), ∇u(x)) for a. a.
x ∈ Ω. This shows that η ∈ N f (u). Applying the same arguments as we did before, we conclude that ξ ∈ N U (u). Recall that F , G and L are continuous. So we can use the the convergence (3.4) in order to get This implies that ζ ∈ H(u). Hence, we have This shows that H is a generalized pseudomonotone operator. Employing Proposition 2.5, we conclude that H is pseudomonotone.
Next, we show the coercivity of H . To this end, we introduce a subspace W of W 1,p (Ω) defined by Because Γ 1 has positive measure, it is not difficult to prove that W endowed with the norm u W := ∇u p,Ω for all u ∈ W is a reflexive and separable Banach space. Moreover, since the embedding of V into W is continuous, there exists a constant C V W > 0 such that Let u ∈ V and ζ ∈ H(u) be arbitrary. Then, we can find functions η ∈ N f (u) and ξ ∈ N U (u) such that ζ = F u + ω * Gu − β * Lu − ι * η − γ * ξ and Keeping in mind that ς > p, it follows from Young's Inequality and the eigenvalue problem of the p-Laplacian with Steklov boundary condition (see (2.2) and (2.3)) that the following inequalities hold with some c(ε) > 0. Using (3.9) and (3.10) in (3.8), we get whereM 0 > 0 is defined byM It is well-known that I K is a proper, convex and l.s.c. function. Note that (see e.g., Proposition 1.10 of Brézis [6]) where we have used the fact that 0 ∈ K. Combining the inequality above and (3.11) gives for all ζ ∈ H(u) and for all κ ∈ ∂ c I K (u) with some M 4 > 0. Therefore, we infer that (2.8) is satisfied with u 0 = 0, G = ∂ c I K and F = H. Thus, all conditions of Theorem 2.7 are verified. Using this theorem, we conclude that problem (1.1) has at least one weak solution u ∈ K. Recalling that f (x, 0, 0) = {0}, u turns out to be a nontrivial weak solution of problem (1.1).
II Boundedness: Suppose that the solution set S(a, h) is unbounded. Then, without loss of generality, there exists a sequence {u n } n∈N ⊂ S(a, h) such that u n V → +∞ as n → ∞.
Employing the same arguments as in the proof of the first part, we get the estimate for all n ∈ N and for some M 5 > 0. Letting n → ∞ in the inequality above, we get a contradiction. Therefore, the solution set S(a, h) is bounded in V .
III Closedness: Let {u n } n∈N ⊂ S(a, h) be a sequence such that u n w −→ u in V for some u ∈ K. Then, there exist functions η n ∈ N f (u n ) and ξ n ∈ N U (u n ) such that for all v ∈ K. Due to the boundedness of the operators N f and N U we may suppose that there are functions η ∈ L r (Ω) and ξ ∈ L δ (Γ 3 ) satisfying η n w −→ η in L r (Ω) and ξ n w −→ ξ in L δ (Γ 3 ).
Taking v = u in (3.13) and passing to the upper limit as n → ∞ in the resulting inequality, we get Applying Proposition 2.3 we obtain that u n → u in V . Using the upper semicontinuity of f and U , one has η ∈ N f (u) and ξ ∈ N U (u). Passing to the upper limit as n → ∞ in equality (3.13), we derive that u ∈ S(a, h) and so, S(a, h) is weakly closed. This completes the proof.

An inverse problem for double phase elliptic obstacle inclusion systems
The section is concerned with the study of an inverse problem to identify a discontinuous parameter in the domain and a discontinuous boundary datum for the double phase elliptic obstacle problem given in (1.1).
For any g ∈ L 1 (Ω) fixed, in what follows, we denote by TV(g) the total variation of the function g given by TV(g) := sup By BV(Ω), we denote the function space of all integrable functions with bounded variation, namely, BV(Ω) := g ∈ L 1 (Ω) : TV(g) < +∞ .
It is well-known that BV(Ω) endowed with the norm g BV(Ω) := g 1,Ω + TV(g) for all g ∈ BV(Ω) is a Banach space. In the sequel, let H be a nonempty, closed and convex subset of L p (Γ 3 ). Given positive constants c Λ and d Λ , we denote by Λ the set of all admissible parameters for the double phase differential operator given in (1.2) defined by Obviously, we see that the admissible set Λ is a closed and convex subset of both BV(Ω) and L ∞ (Ω).
Given two regularization parameters κ > 0 and τ > 0 and the known observed or measured datum z ∈ L p (Ω; R N ), we consider the inverse problem formulated in the following regularized optimal control framework: and S(a, h) stands for the solution set of the double phase elliptic obstacle problem (1.1) with respect to a ∈ L ∞ (Ω) and h ∈ L p (Γ 2 ).
The main result in this section is the following existence result for the regularized optimal control problem given in Problem 4.1. Proof. The proof of this theorem is divided into four steps.
Step 1: The functional C defined in (4.2) is well-defined.
We only need to verify that for (a, h) ∈ Λ × H fixed, the optimal problem min u∈S(a,h) From Theorem 3.3 we know that {u n } n∈N is bounded in V . Passing to a subsequence if necessary, we can assume that u n w −→ u * in V for some u * ∈ V . This fact along with the weak closedness of S(a, h) ensures that u * ∈ S(a, h). On the other hand, the weak lower semicontinuity of the norm This means that for each (a, h) ∈ Λ × H there exists u * ∈ S(a, h) such that inf u∈S(a,h) Hence, C is well-defined. For any (a, h) ∈ Λ × H and u ∈ S(a, h) fixed, it follows from (3.12) that for some M 6 > 0. Therefore, we conclude that S maps bounded sets of Λ × H ⊂ BV(Ω) × L p (Γ 2 ) into bounded sets of K.
Step 2: If {(a n , h n )} n∈N ⊂ Λ × H is a sequence such that {a n } n∈N is bounded in BV(Ω), a n → a in L 1 (Ω) and h n w −→ h in H for some (a, h) ∈ L 1 (Ω) × H, then a ∈ Λ and one has ∅ = w-lim sup n→∞ S(a n , h n ) ⊂ S(a, h). Let {(a n , h n )} n∈N ⊂ Λ × H be a sequence such that a n → a in L 1 (Ω) and h n w −→ h in H for some (a, h) ∈ L 1 (Ω) × H. By the properties of Λ (that is, Λ is nonempty, closed and convex in BV(Ω) and L 1 (Ω)), one has (a, h) ∈ Λ × H. Moreover, the boundedness of {a n } n∈N ⊂ BV(Ω) ∩ L ∞ (Ω) and the map S implies that ∪ n≥1 S(a n , h n ) is bounded in K. Also, the reflexivity of V guarantees that the set w-lim sup n→∞ S(a n , h n ) is nonempty.
For any u ∈ w-lim sup n→∞ S(a n , h n ), passing to a subsequence if necessary, there exists a sequence {u n } n∈N ⊂ K such that u n ∈ S(a n , h n ) and u n w −→ u in V.
Hence, for every n ∈ N, we are able to find functions η n ∈ N f (u n ) and ξ n ∈ N U (u n ) such that Ω a n (x)|∇u n | p−2 ∇u n + µ(x)|∇u n | q−2 ∇u n · ∇(v − u n ) dx gives Ω a n (x)|∇u n | p−2 ∇u n + µ(x)|∇u n | q−2 ∇u n · ∇(u − u n ) dx Hypotheses H(f )(iii) and H(U )(iv) imply that the sequences {η n } n∈N and {ξ n } n∈N are bounded in L r (Ω) and L δ (Γ 3 ), respectively. Since the embeddings of V to L ς (Ω) and L r (Ω) are compact, we obtain where we have also used the compactness of V → L p (Γ 2 ) and V → L δ (Γ 3 ).
When 1 < p < 2, we can apply (4.8) and get Since a n → a in L 1 (Ω), without loss of generality, we may assume that a n (x) → a(x) for a. a. x ∈ Ω.
Letting n → ∞ in equality (4.4) and using the convergence properties above we obtain Ω a(x)|∇u| p−2 ∇u + µ(x)|∇u| q−2 ∇u · ∇(v − u) dx for all v ∈ K. Therefore, we can observe that u ∈ K is a solution of problem (1.1) corresponding to (a, h) ∈ Λ × H, that is, u ∈ S(a, h). Hence w-lim sup n→∞ S(a n , h n ) ⊂ S(a, h) and so we have proved (4.3).
Step 3: If {(a n , h n )} n∈N ⊂ Λ × H is such that {a n } n∈N is bounded in BV(Ω), a n → a in L 1 (Ω) and h n w −→ h in L p (Γ 2 ) for some (a, h) ∈ L 1 (Ω) × H, then the inequality C(a, h) ≤ lim inf n→∞ C(a n , h n ) (4.12) holds. Let {(a n , h n )} n∈N ⊂ Λ × H be such that a n → a in L 1 (Ω) and h n w −→ h in L p (Γ 2 ) for some (a, h) ∈ L 1 (Ω) × H. From Step 2 one has a ∈ Λ. Let {u n } n∈N ⊂ K be a sequence such that u n ∈ S(a n , h n ) and inf u∈S(an,hn) ∇u − z L p (Ω;R N ) = ∇u n − z L p (Ω;R N ) (4.13) for each n ∈ N.
Recalling that ∪ n≥1 S(a n , h n ) is bounded, passing to a subsequence if necessary, we have u n w −→ u * in V for some u * ∈ K, that is, u * ∈ w-lim sup n→∞ S(a n , h n ). Applying again Step 2, we conclude that u * ∈ S(a, h). Therefore, from the lower semicontinuity of the function L 1 (Ω) a → TV(a) ∈ R and the weak lower semicontinuity of W u → ∇u − z L p (Ω;R N ) ∈ R and L p (Γ 2 ) h → h p ,Γ2 ∈ R, it follows that lim inf n→∞ C(a n , h n ) = lim inf n→∞ ∇u n − z L p (Ω;R N ) + κ TV(a n ) + τ h n p ,Γ2 ≥ lim inf n→∞ ∇u n − z L p (Ω;R N ) + lim inf n→∞ κ TV(a n ) + lim inf Hence (4.12) follows.
Step 4: The solution set of Problem 4.1 is nonempty and weakly compact. By the definition of C, we see that C is bounded from below. Let {(a n , h n )} n∈N ⊂ Λ × H be a minimizing sequence of 4.1, namely, inf a∈Λ and h∈H C(a, h) = lim n→∞ C(a n , h n ). (4.14)