On the lower and upper solutions method for mean curvature problems with general boundary conditions

Abstract

We discuss existence, localisation, regularity, and stability issues of the bounded variation solutions of the prescribed mean curvature equation \begin{equation} -{\rm div } \Big( {\nabla u}/{ \sqrt{1+|\nabla u|^2}}\Big) = g(x,u) \quad \hbox{\, in $\Omega$}, \end{equation} in the presence of a couple of bounded variation lower and upper solutions \a and \b satisfying the ordering condition \a(x)\le \b(x) almost everywhere in $\Omega$. The equation is supplemented with general non-homogeneous boundary conditions, which incorporate, possibly mixed, Dirichlet, Neumann, and, seemingly for the first time in this context, Robin-type ones. Our findings are new and extend to a more general setting results previously established in the literature