The Shapiro–Lopatinskij condition for elliptic boundary-value problems

Abstract: Elliptic boundary value problems are well posed in suitable Sobolev spaces, if the boundary conditions satisfy the Shapiro–Lopatinskij condition. The authors of this paper propose a criterion (which also covers overdetermined elliptic systems) for checking this condition. They present a constructive method for computing the compatibility operator for the given boundary value problem operator, which is also necessary when checking the criterion. In the case of two independent variables they give a formulation of the criterion for the Shapiro–Lopatinskij condition which can be checked in a finite number of steps. Their approach is based on formal theory of PDEs, and they use constructive module theory and polynomial factorisation in their test. Actual computations were carried out with the computer algebra systems SINGULAR and MUPAD.