On a multivalued prescribed mean curvature problem and inclusions defined on dual spaces

Abstract

This article addresses two main objectives. First, it establishes a functional analytic framework and presents existence results for a quasilinear inclusion describing a prescribed mean curvature problem with homogeneous Dirichlet boundary conditions, involving a multivalued lower order term. The formulation of the problem is done in the space of functions with bounded variation. The second objective is to introduce a general existence theory for inclusions defined on nonreflexive Banach spaces, which is specifically applicable to the aforementioned prescribed mean curvature problem. This problem can be formulated as a multivalued variational inequality in the space of functions with bounded variation, which, under suitable conditions, is equivalent to an inclusion involving a maximal monotone mapping of type (D) and a generalized pseudomonotone mapping. We prove an abstract existence theorem for inclusions of this form, under some coercivity conditions involving both the maximal monotone and the generalized pseudomonotone mappings.