Regularity of the stress field for degenerate and/or singular elliptic problems (Q6652171)

This article investigates locally Lipschitz solutions of the equation\N\[\N\mathrm{div}\, G(\nabla u)=f,\N\]\Nwith \(G\colon \mathbb R^N \to\mathbb R^N\) a monotone field, and gives various sufficient conditions on \(N\), \(G\) and \(f\) which ensure that the stress-field \(G(\nabla u)\) is continuous. These equations are of elliptic type, but for merely monotone fields \(G\), ellipticity might be very degenerate.\N\NMost results presented here assume that \(N=2\) and \(f\) is constant, and establish continuity of the stress field for a broad class of monotone fields \(G\), for instance gradients of convex functions with some regularity or symmetry requirements. For general dimension \(N\geq 2\) and smooth enough right-hand side \(f\), it is assumed that the complement of the set where \(G\) is locally elliptic (in the sense that the symmetric part of \(\nabla G\) is locally bounded and positive definite) is contained in a plane and satisfies some topological constraint.