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 ([lowercase nu ? 1)-dimensional bounded surface of class C[superscript 2] in [double-struck R][superscript n] and let [uppercase delta]S be the Laplace-Beltrami operator on S. In this thesis, under suitable geometric assumptions, we prove a priori estimates in the W[superscript 2,2] (S) Sobolev space for solutions u to the Poisson problem [uppercase delta][subscript S]u = f in S and u satisfying homogeneous Dirichlet, Neumann, or mixed type boundary conditions, in terms of the L[superscript 2] 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.