Hypoellipticity of certain degenerate elliptic boundary value problems

Abstract

The concept of hypoellipticity for degenerate elliptic boundary value problems is defined, and its relation with the hypoellipticity of certain pseudo-differential operators on the boundary is discussed (for second order equations). A theorem covering smoothness of solutions of boundary value problems such as a ( x ) ∂ u / ∂ n + b ( x ) u = f ( x ) a(x)\partial u/\partial n + b(x)u = f(x) for the Laplace equation is proved. An almost complete characterization of hypoelliptic boundary value problems for elliptic second order equations in two dimensions is given via analysis of hypoelliptic pseudo-differential operators in one variable.