The purpose of the present paper is to investigate the mixed Dirichlet-Neumann boundary value problems for the anisotropic Laplace-Beltrami equationdivC(A∇Cφ)=fon a smooth hypersurfaceCwith the boundaryΓ=∂CinRn.A(x)is ann×nbounded measurable positive definite matrix function. The boundary is decomposed into two nonintersecting connected partsΓ=ΓD∪ΓNand onΓDthe Dirichlet boundary conditions are prescribed, while onΓNthe 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 todivS(A∇S)is proved, which is interpreted as the invertibility of this operator in the settingHp,#s(S)→Hp,#s-2(S), whereHp,#s(S)is a subspace of the Bessel potential space and consists of functions with mean value zero.
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