Elliptic boundary value problem

Summary

An elliptic boundary value problem asks for a solution of an elliptic equation in a domain, subject to conditions on the boundary of the domain. The model case is the Dirichlet problem for the Laplacian: $\Delta u = f$ inside and $u = g$ on the boundary. In general, the operator must be elliptic and the boundary conditions must satisfy the Lopatinskii–Shapiro condition. Such problems have a priori estimates, regularity up to the boundary and a Fredholm solution theory.