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

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 in-equality 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.

Automated summary

This article mainly introduces the existence results for a quasilinear inclusion describing a prescribed mean curvature problem and establishes a functional analytic framework. It also introduces a general existence theory for inclusions defined on nonreflexive Banach spaces.