Autonomous Evolutionary Inclusions With Applications to Problems With Nonlinear Boundary Conditions

Abstract

We study an abstract class of autonomous differential inclusions in Hilbert spaces and show the well-posedness and causality, by establishing the operators involved as maximal monotone operators in time and space. Then the proof of the well-posedness relies on a well-known perturbation result for maximal monotone operators. Moreover, we show that certain types of nonlinear boundary value problems are covered by this class of inclusions and we derive necessary conditions on the operators on the boundary in order to apply the solution theory. We exemplify our findings by two examples.

Linear wave systems onn-D spatial domains

“…Other work making extensive use of the duality between the divergence and the gradient in the analysis of PDEs is Trostorff (2013Trostorff ( , 2014; this work suggests that there is potential for extending the approach to certain types of non-linearities at the boundary.…”

On integro‐differential inclusions with operator‐valued kernels

“…In , the following type of a differential inclusion was considered: where A ⊆ H ⊕ H is a maximal monotone relation satisfying (0,0) ∈ A , A ν is its extension to given by …”

“…H 0 is a self-adjoint strictly positive definite operator and B, C 2 L 1, .R 0 ; L.H 0 //. It is remarkable that no further assumptions on the kernel C are imposed, although (19) seems to be of the form (18), where C would have to verify the hypotheses (i) and (iii). The reason for that is that (19) can be rewritten as a parabolic system, which is not of the classical form (17).…”

“…According to the solution theory for inclusions of the form (see , Theorem 3.7 or Theorem in this article), it suffices to show the strict positive definiteness of in order to obtain well‐posedness of the problem. Besides existence, uniqueness and continuous dependence, we obtain the causality of the respective solution operators, which enables us to treat initial value problems.…”

“…In the theorem in the preceding text, we have used the solution theory for skew self-adjoint A proved in [12]. However, if one allows A to be maximal monotone, this also covers contact problems of visco-elastic solids with frictional boundary conditions using the theory developed in [19]. Such systems were considered by Migorski et al in [36] (see also [37] for a numerical treatment).…”