Lopatinskii–Shapiro condition

Summary

The Lopatinskii–Shapiro condition, also called the complementing condition, is an algebraic condition on boundary operators. It is what a boundary value problem for an elliptic operator needs to be well behaved. It is checked at each boundary point by freezing coefficients and looking at an ordinary differential equation in the normal direction. When it holds, the problem has a priori estimates in Sobolev spaces and is a Fredholm problem. It plays for boundary conditions the role that ellipticity plays for the operator itself.