Parabolic variational inequalities with generalized reflecting directions

Abstract We study, in a Hilbert framework, some abstract parabolic variational inequalities, governed by reflecting subgradients with multiplicative perturbation, of the following type: y´(t)+ Ay(t)+0.t Θ(t,y(t)) ∂φ(y(t))∋f(t,y(t)),y(0) = y0,t ∈[0,T] where A is a linear self-adjoint operator, ∂φ is the subdifferential operator of a proper lower semicontinuous convex function φ defined on a suitable Hilbert space, and Θ is the perturbing term which acts on the set of reflecting directions, destroying the maximal monotony of the multivalued term. We provide the existence of a solution for the above Cauchy problem. Our evolution equation is accompanied by examples which aim to (systems of) PDEs with perturbed reflection.


Introduction
Multivalued deterministic and stochastic evolution equations featuring a transformed subdifferential operator are introduced and studied by Gassous, Rȃşcanu, Rotenstein [1] (for the convex framework) and Rȃşcanu, Rotenstein [2] (for the non-convex setup). The novelty of these type of equations is given by the presence of a time-state operator beside the subdifferential one, a couple which destroys the maximal monotony of the new term obtained. As a consequence, a new approach for the qualitative analysis of the equation is mandatory. In the finitedimensional setting used in the mentioned papers, the study of the smooth problem is continued with the analysis of some generalized Skorohod problems, which lead to stochastic variational inequalities. The convergence of the approximating equations uses some arguments unavailable in the present infinite-dimensional setup. The aim is to obtain the existence and uniqueness of a solution when we translate the problem to infinite-dimensional spaces. Consistent results concerning parabolic variational inequalities, with or without a singular input, are produced by Lions, Sznitman [3], Barbu, Rȃşcanu [4], Rȃşcanu [5], Pardoux, Rȃşcanu [6] and the references within. Their papers cover also a wide horizon of applications and examples in the PDEs' field. A new approach for dealing with multivalued differential equations is introduced in Rȃşcanu, Rotenstein [7]. The authors establish a one-to-one correspondence between the solutions of different type of variational inclusions and the solutions of some convex optimization problems. The Fitzpatrick function is the main ingredient which offers interesting perspectives for numerical approximations.
While our generalization is focused on transforming the reflection direction at the frontier of the domain into a new one, which is no longer normal at the frontier, different research directions aim at perturbing the Laplacian operator (see Eidus [8], Barbu,Favini [9] or Altamore, Milella, Musceo [10]). Multiplicative perturbations of the Laplacian play an important role in the theory of wave propagation in nonhomogeneous media whose density is related to the perturbation coefficient.
The article is organized as follows. Section 2 presents the main notations and some basic assumptions on the spaces and the problem approached in the current study. Additional hypothesis on the terms of our evolution equation will be introduced in the following section, when we present also some particular problems. Section 3 proves the existence of a solution for a smooth multivalued problem with oblique reflection at the frontier of the domain and some applications to systems of PDEs are also provided.

Preliminaries, notations and basic assumptions
If OEa; b is a real, closed interval and Y is a Banach space, then we denote by L p .a; bI Y/ ; C .OEa; b I Y/, BV .OEa; b I Y/ and AC .OEa; b I Y/ the usual spaces of p-integrable, continuous, with bounded variation, and, respectively, absolutely continuous Y-valued function on OEa; b. By W 1;p .OEa; bI Y/ (and, in the same manner, we can define W 2;p .OEa; bI Y /) we shall denote the space of y 2 L p .a; bI Y/ such that y 0 2 L p .a; bI Y/; where y 0 is the derivative in the sense of distributions. Equivalently, according to Barbu [11], we know that Throughout this article we shall situate our work in the Gelfand triple framework. More precisely, we consider two real separable Hilbert spaces V and H such that V H Š H V , with continuous and dense embeddings, where V denotes the dual of V. Moreover, assume that the inclusion V H is a compact one. The norm from V is given by jj jj, the one from H is j j and V is endowed with the norm jj jj . The scalar product of H is . ; / and the duality pairing between V and V is given by h ; i. Let 1 ; 2 > 0 be some positive constants of boundedness, corresponding to the above inclusions jjyjj Ä 1 jyj Ä 2 jjyjj, 8y 2 V. Remark 2.1. For the situation when H is a finite dimensional space, the problem was studied by Gassous, Rȃşcanu, Rotenstein [1], but with the essential assumption int.Dom.'// ¤ ;. They considered even a more general differential inclusion, featuring a singular input which drives the equation. To analyze such a Skorohod problem in the infinite dimensional setting we should consider also a real separable Banach space .X; jj jj X /, with its separable dual .X ; jj jj X / such that X H X and V \ X is dense in V and X (for more details see Rȃşcanu [5]). In the infinite dimensional case, if int.Dom.'// ¤ ; then H D X, but this assumption is too strong because, for example, even in the case of PDEs with barriers, this assumption is not satisfied. If int.Dom.'// D ; the necessity of using the Banach space X is mandatory. One can see this if we consider, for example, for a non-empty domain D R k , It is obvious that int.Dom.'// is empty and, therefore, one can take X DC .DI R/, which satisfies X H X .
We study the following type of multivalued evolution equation, driven by oblique reflected subgradients: where: is a proper, convex, lower semicontinuous function.

Existence and uniqueness of a solution
Concerning the Cauchy problem (1), we first prove the existence of at least one solution. For its uniqueness we will renounce at the dependence on the state for the perturbing term ‚ and we consider some particular systems of PDEs. Assume .A ' /, A f and .A ‚ / still hold and we enhance them by adding the additional hypothesis: Without losing the generality, for convenience only, we can suppose that '.y/ '.0/ D 0, which easily implies that 0 D ' " .0/ Ä ' " .y/, for every " > 0 and y 2 H. This is not an effective restriction since, using the pair  .
Using (6), there exist a subsequence of " n (we denote it also with " n ) and h 2 L 2 .0; T I H/ such that, as n ! 1, y " n * y, weakly in L 2 .0; T I V/, Ay " n * Ay, weakly* in L 2 .0; T I V/ and r' " n .y " n / * h, weakly in L 2 .0; T I H/.

Parabolic variational inequalities. Uniqueness of the solution
In order to prove the uniqueness of a solution we restrict the study of the general problem (1) and analyze a scenario given by the consideration of a particular system of multivalued PDEs. For doing this, let D R d be a domain with a smooth frontier (for example, of class  The assumptions on ‚ assure the existence of some positive constants c 1 and c 2 such that the inequality (9) leads to v .t/ Ä v .s/ C C R t s v.r/dr, for all 0 Ä s Ä t. Apply now the Gronwall's inequality and the uniqueness of the solution for (8) follows.
Within the framework constructed in this section we present some particular cases of semilinear systems of multivalued parabolic PDEs.
Examples. Let K be a non-empty closed convex set from H and L DfL.t /g t2R , L.t / W H ! H a strongly continuous group of linear operators, with B W D.B/ H ! H being its infinitesimal generator. Suppose that L satisfies the hypothesis imposed on ‚ along the previous section and consider the convex, l.s.c. function ' which appears in (8) to be the convexity indicator of the time-dependent convex set L.t /K. The system (8) now becomes: u.x; t / D 0, on @D .0; T /: .a/ Let ‚ Á I k k 2 R k k and, for every t , L.t / D l.t / I H , with l W R ! R C a continuous invertible function. By a suitable change of variable, we can reduce the problem to a reflected PVI with a time-independent convex domain. For example, for the situation of a contracting, time-dependent set characterized by the decreasing function l.t/ D e t , > 0, we have Denote v.x; t/ WD e t u.x; t / and we obtain that u is a solution for the constructed problem iff v is a solution for the following system of PDEs v.x; 0/ D u 0 .x/; on D; v.x; t / D 0; on @D .0; T /: According to Barbu,Rȃşcanu [4], the transformed problem admits a unique solution.
.b/ If ‚ is no longer the identity matrix or/and L is the group of linear operators defined at the beginning of the Example, the situation from point .a/ changes because we have