Partial Differential Inclusions

Abstract

The present chapter is devoted to partial differential inclusions described by the semielliptic set-valued partial differential operators \(\mathbb{L}_{FG}\) generated by given set-valued mappings F and G. Such inclusions will be investigated by stochastic methods. As in the theory of ordinary differential inclusions, the existence of solutions of such inclusions follows from continuous selections theorems and existence theorems for partial differential equations. Therefore, Sects. 2 and 3 are devoted to existence and representation theorems for elliptic and parabolic partial differential equations. Some selection theorems and existence and representation theorems for such partial differential inclusions follow. It will be proved that solutions of initial and boundary value problems for partial differential inclusions can be described by weak solutions of stochastic functional inclusions SFI(F, G), as considered in Chap. 4.