Multivalued nonmonotone dynamic boundary condition

In this paper, we introduce a new class of hemivariational inequalities, called dynamic boundary hemivariational inequalities reflecting the fact that the governing operator is also active on the boundary. In our context, it concerns the Laplace operator with Wentzell (dynamic) boundary conditions perturbed by a multivalued nonmonotone operator expressed in terms of Clarke subdifferentials. We will show that one can reformulate the problem so that standard techniques can be applied. We will use the well-established theory of boundary hemivariational inequalities to prove that under growth and general sign conditions, the dynamic boundary hemivariational inequality admits a weak solution. Moreover, in the situation where the functionals are expressed in terms of locally bounded integrands, a"filling in the gaps"procedure at the discontinuity points is used to characterize the subdifferential on the product space. Finally, we prove that, under a growth condition and eventually smallness conditions, Faedo-Galerkin approximation sequence converges to a desired solution.


Introduction
Let Ω ⊂ R d with enough smooth boundary Γ , and A = −∇.(a∇) be a uniformly elliptic differential operator. Then, the operator A with Wentzell boundary conditions is given by the system where b and c are nonnegative bounded measurable functions on Γ and ∂ a ν u = (a∇u).ν is the co-normal derivative of u with respect to a. The fact that the boundary condition in (1.1) could involve the operator A ( eventually with jumps) goes back to the pioneering paper of Wentzell [44]. What make Wentzell boundary condition relevant in the applications is the time-derivative introduced in the boundary conditions. More explicitly, the heat equation with the Wentzell boundary condition becomes u(x, 0) = u 0 (x). (1. 2) The heat equation (1.2) corresponds to the situation where there is a heat source acting on the boundary [16,17]. Moreover, in the model of vibrating membrane, the Wentzell boundary condition arises if we assume the boundary Γ can be affected by vibrations in Ω and thus contributes to the kinetic energy of the system [12,25]. In [26] dynamic boundary conditions are derived for a solid in contact with thin layer of stirred liquid with a heat exchange coefficient represented here by c. A detailed derivation of dynamic boundary conditions can also be found in [21] and references therein. In many physical situations the exchange rate of heat diffusion can be nonlinear or nonmonotone. In this situation, the Wentzell boundary problem becomes u(x, 0) = u 0 (x). (1. 3) The nonlinear, eventually nonmonotone, dynamic boundary conditions have been extensively studied in recent years. In the case where β 1 = ∂j 1 and β 2 = ∂j 2 are the subdifferentials, in the sense of convex analysis, of proper, convex and lower semicontinuous functionals j 1 and j 2 respectively (hence with maximal monotone graphs), the problem (1.3) generates, by a result from Minty [29], a unique solution described by a strongly continuous nonlinear semigroup [16,17,42,41,43]. In the case of nonmonotone but single valued β 1 and β 2 such problems were considered in [18,19,20], and references therein. The functions β 1 and β 2 was supposed to be of classe C 1 and satisfy a sign-growth conditions in the spirit of critical point theory.
The main tool in studying the above problems is to work on a product space instead of the state space itself. This trick, now a standard procedure, provide a good insight into the structure of the problem. In fact, consider functions of the form U = (u, u |Γ ) defined on a suitable product space and define the operator A by AU = (Au, b ∂ a ν u + cu), then the problem (1.2) can be formulated as follows where f = (f 1 , f 1 ). The idea to incorporate boundary conditions into a product space goes back to Greiner [22] and has been used by Amann and Escher [1], Arendt et al. [2,Chapter 6] and in [40] in the context of Dirichlet forms. In the context of heat equations, Wentzell boundary conditions were introduced by A. Favini et al. [16], see also [11,30].
In the situation of nonlinear multivalued dynamic boundary conditions (1.3), the product space procedure leads to (1.5) In the framework of convex functionals, the regularity, in the sense of [12], of j 1 or j 2 at all points allows us to write the inclusion where the functional j is defined by j(U ) = j 1 (u) + j 2 (u |Γ ). Moreover, by Proposition 2 the equality holds in (1.6). This is due to the structure of j as a variable separated functional. The problem (1.5) is then equivalent to the following problem and can be solved by the nonlinear semigroups theory or the variational inequality theory. In the nonconvex functionals framework, the inclusion (1.6) still holds but not the equality except for the regular case. Hence, generally, one can only say that the solvability of (1.7) implies the solvability of (1.5), which, in addition, can not be expressed as a variational inequality due to a lake of monotonicity. It is the aim of this paper to prove that the problem (1.7) have a weak solution by using the theory of hemivariational inequality. This theory was initiated by Panagiotopoulus as a generalization of the variational inequality theory [33,34,35], see also [7,23,8]. Hemivariational inequalities are suitable to model physical and engineering problems where multivalued and nonmonotone constitutive laws are involved. The main tool in this formulations is the generalized gradient of Clarke and Rockafellar [13,12,39]. As a subclass of this theory, we can mention boundary hemivariational inequalities, which are boundary value problems where the boundary condition is multivalued, nonmonotone and of subdifferential form (cf. [31,32,37] and references therein).
In this paper we introduce a new class of problems within this theory. It concerns dynamic boundary conditions where the boundary condition is multivalued, nonmonotone and of the subdifferential form. The dynamic boundary hemivariational inequalities can model problems where the boundary contains a thermostat regulating the temperature within certain specified bounds. It can also be incorporated into Navier-Stokes equations to obtain a variant of Boussinesq model describing the behaviour of a heat conducting liquid with boundaries participating in the total energy. The exchange of the heat with the boundary can then be expressed, in a general way, with a multivalued, nonmonotone and of the subdifferential form functional.
The structure of this paper is as follows. In Section 2, we present the preliminary material needed later. In section 3, we state our problem in a suitable functional spaces and we prove the existence of weak solutions in section 4. In section 5 we present a seemingly new result related to partial generalized gradient for nonregular locally Lipschitz functions and we prove a Chang's type lemma related to locally bounded functions. Finally, we devote section 6 to the convergence of the Faedo-Galerkin approximation to desired solutions.

Preliminaries
Let E be a reflexive Banach space with its dual E * and A : D(A) ⊂ E → 2 E * be a multivalued function, where D(A) = {u ∈ E : Au = ∅} stands for the domain of A. We say that it is monotone and it has a maximal graph in the sense of inclusion among all monotone operators, namely, the inequality (a) for each u ∈ E, the set Au is nonempty, closed and convex in E * . (b) A is upper semicontinuous from each finite dimensional subspace of E into E * endowed with its weak topology; (c) if un → u weakly weakly in E, u * n ∈ Aun and lim sup For a single-valued operator A : E → E * , we say that: (v) A is demicontinuous if it is continuous from E to E * endowed with weak topology. (vi) A is pseudomonotone if for each sequence {un} ⊂ E such that it converges weakly to u ∈ E and lim sup Now let ϕ : E → R := R ∪ {+∞} be a proper, convex and lower semicontinuous functional. The mapping ∂cϕ : E → 2 E * defined by is called the subdifferential of ϕ. Any element u * ∈ ∂cϕ(u) is called a subgradient of ϕ at u. It is a well know fact that ∂ϕc is a maximal monotone operator.
Let J : E → R be a locally Lipschitz continuous functional and u, v ∈ E. We denote by J • (u; v) the generalized Clarke directional derivative of J at the point u in the direction v defined by We collect the following properties (a) the function v → J • (u; v) is positively homogeneous, subadditive and satisfies is a nonempty, convex and weakly * compact subset of E * with ξ E * ≤ Lu for all ξ ∈ ∂J(u).
(e) Let F be another Banach spaces and A ∈ L(F, E). Then for u ∈ E and where A * ∈ L(E * , F * ) denotes the ajoint operator to A.
The following surjectivity result for operators which are L−pseudomonotone will be used in our existence theorem in section 4 (cf. [36, Theorem 2.1]). Theorem 1 If E is a reflexive strictly convex Banach space, L : D(L) ⊂ E → E * is a linear maximal monotone operator, and A : E → 2 E * is a multivalued operator , which is bounded, coercive and L−pseudomonotone.
It is worth to mention that one can drop the strict convexity of the reflexive Banach space E. It suffices to invoke the Troyanski renorming theorem to get an equivalent norm so that the space itself and its dual are strictly convex(cf. [45, Proposition 32.23, p.862]).

Problem statement
Let Ω ⊂ R N be a bounded domain with Lipschitz boundary Γ := ∂Ω. Let λ N denote the N −dimensional Lebesgue measure and σ the surface measure on Γ . For simplicity, we take b = 1 and a the N −dimensional matrix identity. Define the following product space and the induced natural norm |.| := ., .
and endow it with the norm It is easy to see that we can identify V with W 1,2 (Ω) ⊕ L 2 (Γ ) under this norm. Moreover, we emphasize that V is not a product space and since W 1,2 (Ω) ֒→ L 2 (Γ ) by trace theory V is topologically isomorphic to W 1,2 (Ω) in the obvious way. It is also immediate that V is compactly embedded into H. We have then the Gelfand triple with continuous and compact embeddings. The embedding Λ : V → H is defined in a natural way by Λ(U ) = (i(u), γ(u)), where i : W 1,2 (Ω) → L 2 (Ω) is the natural embedding and γ is the trace operator. It is obvious that Λ is continuous and compact from V into H. Consider the Laplacian operator with multivalued nonmonotone dynamic boundary conditions described as follows where a ∈ L ∞ (∂Ω) with a ≥ a 0 > 0 for some constant a 0 and j 1 , j 2 are locally Lipschitz functions on R. Let ., . denotes the duality between V and V * . The system (3.1) can be written as follows Lemma 2.111], it is clear that the operator A is pseudomonotone. The continuity and coercivity of A can be proved in the same way as for Robin boundary conditions. Now, by using the definition of Clarke subdifferential, the system (3.2) leads to Then the problem (3.1) has a solution if the following problem has one It is clear that the equivalence holds, if j 1 is regular at u or j 2 at u |Γ . An equivalent formulation to (3.4) reads: for every V ∈ V, In what follows we need the spaces for f ∈ V * and V ∈ V . It is know [45] that the embedding W ⊂ C(0, T ; H) is continuous. The problem under consideration is as follows: find U ∈ W such that for all V ∈ V and a.e. t ∈ (0, T ) We will prove the existence of solutions to the heat problem with multivalued nonmonotone dynamic boundary condition by considering functionals defined in L 2 (Ω). Define the functional J : Let us consider the following hypotheses One can see that if j 1 and j 2 satisfy assumptions similar to H(j), then H(j) holds. The following lemma will be proved in the same way as the similar one for functionals on Γ (cf. [31,37]).
Then the functional J give by (3.7) is well defined and locally Lipschitz in the second variable(in fact, Lipschitz in the second variable on bounded subsets of H, its generalized gradient satisfies the linear growth condition with c ′ > 0 and for its generalized directional derivative we have Proof It is clear that (Ω, F , µ) is a positive complete measure space. Then, it follows from [13, Theorem 2.7.5] that the functional J is well posed and uniformly Lipschitz on bounded subsets of L 2 (Ω). By definition of the Clarke directional derivative and Fatou's lemma, we have Now let ξ = (ξ 1 , ξ 1 ) ∈ ∂J(t, V ) and by the definition of the generalized Clarke gradient of J, we have On the other hand, we have Finally by [13, Theorem 2.7.5 ], we have ∂(J • Λ)(U ) = Λ * ∂J(ΛU ).

Existence result
From (1.6), it is clear that in order to obtain the solvability of the problem (3.6), it is enough to show that the problem admits a solution. The proofs are inspired from [31,37].
Proposition 1 Suppose that hypotheses H(j) and H 0 hold and U is a solution of (4.1), then there exists a constant C > 0 such that Proof Let U be a solution of (4.1). Multiplying by U and integrating over (0, T ), one obtain with η(t) ∈ Λ * ∂J(t, ΛU ) for a.e. t ∈ (0, T ). Hence there exists w(t) ∈ ∂J(t, ΛU ) such that η(t) = Λ * w(t). It follows w(t), ξ ≤ J • (t, ΛU (t); ξ(t)) and hence Using the coercivity of A and Young inequality one obtains It follows Using the relation (4.3) and integrating over (0, T ) we obtain the result.
By the density of V in H, we may assume that U 0 ∈ V. Define the Nemytskii operators corresponding to A : V → V * and N : V → 2 V * as follow We note that Z ∈ W is a solution to problem (4.4) if and only if Z + U 0 ∈ W is a solution to problem (4.1). Let L : D(L) ⊂ V → V * be the operator defined by LV = V ′ with D(L) = {w ∈ W : w(0) = 0}. It is well known (cf. [45]) that L is a linear densely defined and maximal monotone operator. As a consequence, from(4.4) we obtain the problem where T : V → 2 V * is the operator given by T = A + N . It is clear that problems (4.5) and (4.1) are equivalent. To prove the existence of solutions to problem (4.5), we will use theorem 1 and standard techniques from [31,37].
Theorem 2 If hypotheses H(j) and H 0 hold, then problem (4.1) has a solution.
Proof We will prove that T is a bounded, coercive and L−pseudomonotone operator. The boundedness follows immediately from the continuity of the operator A and H(j) − (iii). In fact, take D to be a bounded subset of V and Z ∈ D. Hence, every η ∈ T Z is of the form η = AZ + ξ with η ∈ N Z. It follows that we have ξ(t) ∈ Λ * ω(t) and ω(t) ∈ ∂J(t, Λ(Z + U (0))) a.e t ∈ (0, T ). From H(j) − (iii), one can deduce that ξ V * ≤ c(1 + Z V ). Hence by the continuity of A(with constant c A ), we get This proves the boundedness of T . Let Z ∈ V and η ∈ T Z, i.e. η = AZ + ξ where ξ(t) = Λ * ω(t) and ω(t) ∈ ∂J(t, Λ(Z(t) + U 0 )) for a.e. t ∈ (0, T ). It is easy to see that This implies that T is coercive. It is well known that ∂J(t, .) is nonempty, weakly compact and convex subset of L 2 (Ω). Hence, for every Z ∈ V the set N Z is nonempty, convex in V * . Since N Z is closed, convex(hence weakly closed) and bounded, then it is weakly compact in V * . To show that T is upper semicontinuous from V to V * . endowed with the weak topology, we prove that if D is weakly closed in V * then the set T − (D) = {Z ∈ D : T Z ∩ D = ∅} is closed in V . Let {Zn}n ⊂ T − (D) be such that Zn → Z in V . So we can find ηn ∈ T Zn ∩ D for all n ∈ N and by definition ηn = AZn + ξn with ξn ∈ N Zn. Since {Zn}n is bounded and T is a bounded operator, then the sequence {ηn}n is bounded in V * . So we may assume that ηn → η weakly in V * with η ∈ D. For ξn ∈ N Zn, we have ξn(t) = Λ * ωn(t) and ωn(t) ∈ ∂J(t, Λ(Zn(t) + U (0))) a.e. t ∈ (0, T ). From Lemma 1 and the boundedness of Λ, we may suppose that ξn → ξ weakly in H and ξ ∈ H. Next, since Then we can assume that ωn → ω weakly in H . By passing to a subsequence, if necessary, we get ξ = Λ * ω(t). Since Λ * (Zn(t) + U (0)) → Λ * (Z(t) + U (0)) in L 2 (Ω) for a.e. t ∈ (0, T ) and ∂J(., t) is upper semicontinuous with convex values, from the convergence theorem of Aubin and Cellina [3], we have ω(t) ∈ ∂J(t, Λ(Z(t)+U (0))) for a.e. t ∈ (0, T ). Subsequently we obtain that ξ ∈ N Z.
Since A is bounded, coercive and pseudomonotone, then by a result in [6], A is demicontinuous. So by again by [6], we deduce that A is demicontinuous too. Therefore we have AZn → AZ weakly in V * . Hence D ∋ η = AZ + ξ with ξ ∈ N Z which means that η ∈ T Z ∩ D, i.e. Z ∈ T − (D). This proves that T − (D) is closed in V and hence T is upper semicontinuous from V in V * equipped with the weak topology.
To conclude the proof that T is L−pseudomonotone, it is enough to prove condition (c ′ ) in its definition. Let {Zn}n ⊂ D(L), Zn → Z weakly in W , ηn ∈ T Zn, ηn → η weakly in V and assume that lim sup ηn, Zn − Z ≤ 0. Note that ηn = AZn + ξn with ξn ∈ N Zn for n ∈ N. From the fact that N : V → 2 V * is a bounded map, we infer that {ξn}n is bounded in V * , so passing to a subsequence, if necessary, we may assume that ξn → ξ weakly in V * with ξ ∈ V * . Since the embedding V ⊂ H is compact, it follows from a theorem from Denkovski et al. [15], that the embedding W ⊂ H is compact. This entails that We have for a.e. t ∈ (0, T ) that ξn(t) = Λ * ωn(t) with ωn(t) ∈ ∂J(t, Λ(Zn(t)+U (0))). It follows that {ωn} is bounded in H , we may then assume that wn → ω weakly in H . We obtain ξ(t) = Λ * w(t) for a.e t ∈ (0, T ). Moreover, applying a convergence theorem from [3], we get ω(t) ∈ ∂J(t, Λ(Z(t) + U (0))), for a.e. t ∈ (0, T ) and therefore ξ ∈ N Z. Furthermore Since A is pseudomonotone, A is also L−pseudomonotone [6]. Hence AZn → AZ weakly in V * and AZn, Zn → AZ, Z . Because ξ ∈ N Z, we deduce that η ∈ T Z. Furthermore, passing to the limit, we get ηn, Zn → η, Z with η ∈ T Z. This proves that T is L−pseudomonotone.

Partial generalized gradient
Let X be a Banach space. We say that a function j : X → R is regular at x, if for all v, the usual one-sided directional derivative Banach spaces, and let j : E → R be a locally Lipschitz function. We denote ∂ 1 j(x 1 , x 2 ) the partial generalized gradient of j(., x 2 ) at x 1 , and by ∂ 2 j(x 1 , x 2 ) that of j(x 1 , .) at x 2 . It is a fact that in general neither of the sets ∂j(x 1 , x 2 ) and ∂ 1 j(x 1 , x 2 ) × ∂ 2 j(x 1 , x 2 ) need be contained in the other. To be convinced it suffices to consider the function f : From [13, Example 2.5.2], we have For regular functions, however, a general relationship does hold between these sets. From [13,Proposition 2.3.15] if j is regular at and there is no reason that the equality holds even for regular functions. Next we will give a situation where the inclusion (5.1) holds in a nonregular case, that is for functions with separated variables. Consider two locally Lipschitz functions j 1 : E 1 → R and j 2 : E 2 → R and define the function j : We have the following result Proposition 2 The function j is locally Lipschitz and for every (x 1 , x 2 ) ∈ E, we have Moreover, we have ∂ k j = ∂j k with k = 1, 2. If j 1 is regular at x 1 or j 2 at x 2 , then equality holds.
On the other hand, we get It follows that for every (v 1 , v 2 ) ∈ E, We take v 2 = 0, then for every v 1 ∈ E 1 we have x 2 ). Similarly, if j 2 is regular at x 2 , then equality holds. They are increasing and decreasing functions of µ, respectively. Therefore, the limits for µ → 0 + exist. We denote them by β(t) and β(t), respectively. Then it is proved by Chang that If in addition β(t ± 0) exists for every t ∈ R then, the equality holds, i.e.

Galerkin approximation
Let β 1 , β 2 ∈ L ∞ loc (R) and for k = 1, 2 define j : for all t ∈ R Let p ∈ C ∞ 0 (R) be a positive function with support in [−1, 1] such that R p(ξ) dξ = 1. For ξ ∈ R and ε > 0, define the function pε(ξ) = 1 ε p( ξ ε ) and for k = 1, 2 define We consider a Galerkin basis {Z 1 , Z 2 , . . . } of V and let Vm = span{Z 1 , Z 2 , . . . , Zm} be the resulting m−dimensional subspaces. Let {U m0 }m be an approximation of the initial value U 0 such that U m0 ∈ Vm, U m0 → U 0 in H and {U m0 }m is bounded in V. Let {εm}m be a sequence of real numbers converging to zero as m → ∞. Instead of β kεm we will write β km and we will use the notation We consider the following regularized Galerkin system of finite dimensional differential equations: find Um = (um, u m|Γ ) ∈ L 2 (0, for a.e. t ∈ (0, T ) and for all V = (v, v |Γ ) ∈ V. The problem (6.1) can be written more explicitly as follows 2) for a.e. t ∈ (0, T ) and for all V ∈ V.
For the existence of solutions we will need the following hypothesis H(β) : for k = 1, 2 assume that with c k > 0 and 0 ≤ θ k ≤ 1.
Theorem 4 Let H(β) holds. Moreover, assume that one of the following situations holds where M is the coercivity constant of the operator A. Then problem (4.1) has at least one solution.

Concluding remarks
In this paper, we introduced a new class of hemivariational inequalities, namely dynamic boundary hemivariational inequalities. It concerns dynamic boundary conditions with a Clarke subdifferential perturbation on the boundary. The suitable framework to study such problems is to work on a product space instead of the state space itself. We chose to work with dynamic boundary condition in its simplest form but we nevertheless could work with a general uniformly elliptic operator or even in the L p framework with the p−Laplacian. Moreover, with some changes on the choice of the product spaces, one can incorporate the Laplace-Beltrami operator on the boundary in addition of the usual Laplacian. On the other hand, one can replace the growth condition in section 6 by the Rauch condition expressing the ultimate increase of the graphs of functions β k .
As a continuation of this paper we aim to study the abstract version of the current work and to look at hemivariational inequalities that can be formulated in terms of matrix operators on product spaces. Indeed, this formulation covers a wide range of examples, including second order problems, equations with delay, equations which are memory dependent. As a forthcoming work, we aim to study the abstract version of this problems.