Dagmar Medková

Solution of the Neumann problem for the Laplace equation

For fairly general open sets it is shown that we can express a solution of the Neumann problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series. If the open set is simply connected and bounded then the solution of the Dirichlet problem is the double layer potential with a density given by a similar series.

Integral representation of a solution of the Neumann problem for the Stokes system

The Neumann problem for the Stokes system is studied on a domain in R3 with Ljapunov bounded boundary. We construct a solution of this problem in the form of appropriate potentials and determine unknown source densities via integral equation systems on the boundary of the domain. The solution is given explicitly in the form of a series.

Boundary value problems for the Stokes equations with jumps in open sets

A boundary value problem for the Stokes system is studied in a cracked domain in ℝ n , n > 2, where the Dirichlet condition is specified on the boundary of the domain. The jump of the velocity and the jump of the stress tensor in the normal direction are prescribed on the crack. We construct a solution of this problem in the form of appropriate potentials and determine unknown source densities via integral equation systems on the boundary of the domain. The solution is given explicitly in the form of a series. As a consequence, a maximum modulus estimate for the Stokes system is proved.

Transmission problem for the Brinkman system

-solutions of the transmission problem, the Robin-transmission problem and the Dirichlet-transmission problem for the Brinkman system are studied by the integral equation method. Necessary and sufficient conditions for the solvability are given. The uniqueness of a solution is also studied.

Bounded solutions of the Dirichlet problem for the Stokes resolvent system

The paper studies the Dirichlet problem for the Stokes resolvent system for bounded boundary data on bounded and unbounded domains with compact Lyapunov boundary. (The boundary might be disconnected.) For a bounded domain, we prove the existence of a unique solution of the problem such that the velocity part is bounded. For an unbounded domain, we prove the existence of such a solution. But this solution is not unique. We characterize all solutions of the problem. Then we study bounded solutions of the nonlinear Dirichlet problem , in , on , where F is bounded. As a consequence, we study bounded solutions of the Dirichlet problem for the Darcy-Forchheimer-Brinkman system , . At last we prove a generalized maximum modulus principle for a solution of the Stokes resolvent system such that the velocity part is bounded.

The harmonic dirichlet problem for a cracked domain with jump conditions on cracks

The harmonic problem in a cracked domain is studied in R m , m > 2. The boundary of the domain is assumed to be nonsmooth, while cracks are smooth. The Dirichlet condition is specified on the boundary of the domain. Jumps of the unknown function and its normal derivative are specified on the cracks. Uniqueness and solvability results are obtained. The problem is reduced to the uniquely solvable integral equation, its solution is given explicitely in the form of a series. The estimates of the solution of the problem depending on the boundary data are obtained.

ON THE NEUMANN-POINCARE OPERATOR

AbstractLet Γ be a rectifiable Jordan curve in the finite complex plane

The third boundary value problem in potential theory for domains with a piecewise smooth boundary

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