A priori estimates for solutions of elliptic partial differential equations on surfaces

Abstract

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Let S be an (n-1)-dimensional bounded surface of class C² in R[real number] [superscript n] and let [set theory][subscript s] be the Laplace-Beltrami operator on S. In this thesis, under suitable geometric assumptions, we prove a priori estimates in the W²̇²(S) Sobolev space for solutions [u] to the Poisson problem [set theory][subscript s] [superscript u] = [integral] in S and [u] satisfying homogeneous Dirichlet, Neumann, or mixed type boundary conditions, in terms of the L² norm of the datum f. The geometric assumptions S has to satisfy are related to the mean curvatures of the boundary of the surface S.