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.

The purpose of the present research is to investigate the mixed Dirichlet-Neumann boundary value problems for the Helmholtz equation in a 2D domain R with finite number of non-cuspidal angular points on the boundary. Using localization, the problem is reduced in [Proc. A. Razmadze Math. Inst. 162 (2013), 37-44] to model problems in plane angles of magnitude˛j 2 OE0; 2 , j D 1; : : : ; m.In the present paper, we apply the potential method and reduce the model mixed BVP (with Dirichlet-Neumann conditions on the boundary) to an equivalent boundary integral equation (BIE) of Mellin convolution type. Applying the recent results on Mellin convolution equations with meromorphic kernels in Bessel potential and Sobolev-Slobodeckij (Besov) spaces obtained by V. Didenko and R. Duduchava and by R. Duduchava, the unique solvability criteria (Fredholm criteria) of the above mentioned mixed BVP are obtained in classical finite energy space H 1 .@ ˛/ D W 1 .@ ˛/ and also in non-classical Bessel potential spaces H 1 p .@ ˛/ when 1 < p < 1. The problem has been tackled before only for angular domains of magnitude˛for rational angles˛D m=n.

We investigate the mixed Dirichlet-Neumann boundary value problems for the Laplace-Beltrami equation on a smooth bounded surface C with a smooth boundary in non-classical setting in the Bessel potential spaceTo the initial BVP we apply a quasi-localization and obtain a model BVP for the Laplacian. The model mixed BVP on the half plane is reduced to an equivalent system of Mellin convolution equation (MCE) in Sobolev-Slobodečkii space (potential method). MCE is ivestigated in both Bessel potential and Sobolev-Slobodečkii spaces. The symbol of the obtained system is written explicitly and is responsible for the Fredholm properties and the index of the system. An explicit criterion for the unique solvability of the initial BVP in the non-classical setting is derived as well.

AbstractThe purpose of the present research is to investigate model mixed boundary value problems (BVPs) for the Helmholtz equation in a planar angular domain {\Omega_{\alpha}\subset\mathbb{R}^{2}} of magnitude α. These problems are considered in a non-classical setting when a solution is sought in the Bessel potential spaces {\mathbb{H}^{s}_{p}(\Omega_{\alpha})}, {s>\frac{1}{p}}, {1<p<\infty}. The investigation is carried out using the potential method by reducing the problems to an equivalent boundary integral equation (BIE) in the Sobolev–Slobodečkii space on a semi-infinite axis {\mathbb{W}^{{s-1/p}}_{p}(\mathbb{R}^{+})}, which is of Mellin convolution type. Applying the recent results on Mellin convolution equations in the Bessel potential spaces obtained by V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731], explicit conditions of the unique solvability of this BIE in the Sobolev–Slobodečkii {\mathbb{W}^{r}_{p}(\mathbb{R}^{+})} and Bessel potential {\mathbb{H}^{r}_{p}(\mathbb{R}^{+})} spaces for arbitrary r are found and used to write explicit conditions for the Fredholm property and unique solvability of the initial model BVPs for the Helmholtz equation in the non-classical setting. The same problem was investigated in a previous paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in arbitrary 2D-sectors, Georgian Math. J. 20 2013, 3, 439–467], but there were made fatal errors. In the present paper, we correct these results.

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