Regular Inversion of the Divergence Operator with Dirichlet Boundary Conditions on a Polygon

摘要

Consider the existence of regular solutions to the boundary value problem div U = f on a plane polygonal domain Omega with the Dirichlet boundary condition U = g on del Omega. We formulate simultaneously necessary and sufficient conditions on f and g in order that a solution U exist in the Sobolev space W (over s+1 to p) (Omega). In addition to the obvious regularity and integral conditions these consist of at most one compatibility condition at each vertex of the polygon. In the special case of homogeneous boundary data, it is necessary and sufficient that f belong to W (over s to p) (Omega), have mean value zero, and vanish at each vertex. (The latter condition only applies if s is large enough that the point values make sense.) We construct a solution operator which is independent of s and p. Various new trace theorems for Sobolev spaces on polygons are obtained.