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International Journal of Differential Equations
Volume 2014, Article ID 245350, 8 pages
http://dx.doi.org/10.1155/2014/245350
Research Article

Mixed Boundary Value Problem on Hypersurfaces

A. Razmadze Mathematical Institute, Tbilisi State University, Tamarashvili Street 6, 0177 Tbilisi, Georgia

Received 22 February 2014; Revised 31 May 2014; Accepted 2 June 2014; Published 17 August 2014

Academic Editor: Ioannis G. Stratis

Copyright © 2014 R. DuDuchava et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of the present paper is to investigate the mixed Dirichlet-Neumann boundary value problems for the anisotropic Laplace-Beltrami equation on a smooth hypersurface with the boundary in . is an bounded measurable positive definite matrix function. The boundary is decomposed into two nonintersecting connected parts and on the Dirichlet boundary conditions are prescribed, while on the Neumann conditions. The unique solvability of the mixed BVP is proved, based upon the Green formulae and Lax-Milgram Lemma. Further, the existence of the fundamental solution to is proved, which is interpreted as the invertibility of this operator in the setting , where is a subspace of the Bessel potential space and consists of functions with mean value zero.