Dirihlet-Ventcel bounsdary problem for Laplace equation in an unbounded sector

Abstract

We are concerned with boundary value problems for Laplace equation in an unbounded sector $s_\theta$ with vertex at the origin, the boundary conditions being of mixed type and jumping at corner. The boundary conditions are these: Dirichlet datum on one of the radial lines, while on the other the values of an Ventcel boundary condition is prescribed. We are interested in looking for solutions having a prescribed degree of smoothness up to the origin: more precisely we search for solutions of problem having all the derivatives up to the order that are square integrable with a power weight. This problem has a background in physical modeling of electrostatic or thermal imaging. Determining the geometry and the physical nature of an corrosion within a conducting medium from voltage and current measurements on the accessible boundary of the medium can be modeled as an inverse boundary value problem for the Laplace equation subject to appropriate boundary conditions on the corrosion surface. We are interesting in investigation of a regularity properties of solution to the @direct@ problem. Applying Mellin transform we pass to a finite difference equation.We use the methods of V.A.Solonnikov and E.V.Frolova just as in the case of the analogous finite difference equation obtained under the Dirichlet or the Neumann conditions indstead of the Ventcel condition in our case. We obtain the sulution of homogeneous difference equation in the form of infinite product. Then we find asymptotic formulas for this solution.Returning to nonhomogeneous differerence equation we find its solution in the form of contour integral. we define the solution of the starting problem by the help of the inverse Mellin transform. We estimate this solution in the norm of V.Kondratiev spaces $H^k_\mu(s_\theta$ under some conditions on weight $\mu$, higher order of derivatives $k$ and the opening of the angle $\theta$.