On solvability of boundary integral equations of potential theory for a multidimensional cusp domain

The Dirichlet and the Neumann problems for the Laplace equation on a multidimensional cusp domain are considered. The unique solvability of the boundary integral equation for the internal Dirichlet problem for harmonic double layer potential is established. We also prove the unique solvability of the boundary integral equation for the external Neumann problem for harmonic single layer potential. Bibliography: 13 titles.     

Integral equations for the Dirichlet and Neumann boundary value problems in a plane domain with a cusp on the boundary

The boundary equations of the logarithmic potential theory corresponding to the interior Dirichlet problem and the exterior Neumann problem for a plane domain with a cusp on the boundary are studied. Solvability theorems are proved for these integral equations in the spacesL ^p. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 881–892, June, 1996.     

The dirichlet problem in a domain whose boundary is partly degenerated

Summary In this paper we bring a formulation of the Dirichlet problem for strongly elliptic equations in domains vhose boundaries may include manifolds of different dimensions. It is shown that, under certain regularity conditions, this problem is equivalent to the generalized Dirichlèt problem, with respect to existence and uniquennes of solutions. This paper represents part of a thesis submitted to the Senate of the Technion, Israel Institute of Technology, in partial fulfillment of the requirements for the degree of Doctor of Science. The author wishes to tank ProfessorS. Agmon for his gudance and help in the preparation of this work.     

Representation of a solution of the dirichlet problem in a planar cusp domain as a logarithmic single-layer potential

The problem of representing the solution of the Dirichlet problem for the Laplace equation as a single-layer potential V _ϱ with unknown density ϱ is known to lead to the equation V _ϱ = f for density ϱ, where f is the Dirichlet boundary data. Let Γ be the boundary of a bounded planar domain with an outward or inward peak and T(Γ) be the space of the traces on Γ of functions with finite Dirichlet integral over R ². It is shown that the operator $ L_2 \left( \Gamma \right) \ominus 1 \mathrel\backepsilon \varrho \to V\left. \varrho \right|\Gamma \in T\left( \Gamma \right) $ L_2 \left( \Gamma \right) \ominus 1 \mathrel\backepsilon \varrho \to V\left. \varrho \right|\Gamma \in T\left( \Gamma \right) is continuous, and the operator $ \varrho \to V\varrho - \overline {V\varrho } $ \varrho \to V\varrho - \overline {V\varrho } (where $ \bar u $ \bar u denotes u averaged over Γ) can be uniquely extended to the isomorphism      

Asymptotics of the solution of a dirichlet problem in an angular domain with a periodically changing boundary

Translated from Matematicheskie Zametki, Vol. 49, No. 5, pp. 86–96, May, 1991.     

Functions integral at points of a geometric progression

Translated from Matematicheskie Zametki, Vol. 46, No. 3, pp. 68–73, September, 1989.     

On the number of integral points on a norm-form variety in a cube-like domain

Specialising the results of [1] we obtain an asymptotic formula for the number of integral points in a cube-like domain on the algebraic set defined by a system of norm-form equations with integral rational coefficients.     

An integral representation and boundary behavior of functions defined in a domain with a peak

We establish an invertible characteristic of boundary behavior of functions in Sobolev spaces defined in a space domain with a vertex of a peak on the boundary.     

Size of the set of singular points on the boundary of a non-Smirnov domain

One improves the necessary conditions of G. Ts. Tumarkin and H. Shapiro for a Jordan domain to belong to the Smirnov class.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 176–183, 1989.     

The essential spectrum of boundary value problems for systems of differential equations in a bounded domain with a cusp

Simple algebraic conditions are found for the existence of essential spectrum of the Neumann problem operator for a formally self-adjoint elliptic system of differential equations in a domain with a cuspidal singular point. The spectrum is discrete in the scalar case.     

Estimate of the modulus of continuity of a cauchy-type integral in a domain and on its boundary

We estimate the modulus of continuity of a Cauchy-type integral in a closed domain and its limit values on the boundary in the case where the boundary of the domain is an arbitrary closed rectifiable Jordan curve.     

On singular solutions of the initial boundary value problem for the Stokes system in a power cusp domain

The initial boundary value problem for the non-steady Stokes system is considered in bounded domains with the boundary having a peak-type singularity (power cusp singularity). The case of the boundary value with a nonzero time-dependent flow rate is studied. The formal asymptotic expansion of the solution near the singular point is constructed. This expansion contains both the outer asymptotic expansion and the boundary-layer-in-time corrector with the ‘fast time’ variable depending on the distance to the cusp point. The solution of the problem is constructed as the sum of the asymptotic expansion and the term with finite energy.