The atomic decomposition of harmonic functions satisfying certain conditions of integrability.
The boundary-value problems for Laplace equation and domains with nonsmooth boundary
Dirichlet, Neumann and Robin problem for the Laplace equation is investigated on the open set with holes and nonsmooth boundary. The solutions are looked for in the form of a double layer potential and a single layer potential. The measure, the potential of which is a solution of the boundary-value problem, is constructed.
The converse of the Fatou theorem for smooth measures.
The integral equation methods for the perturbed Helmholtz eigenvalue problems.
The multiple layer potential for the biharmonic equation in variables
The definition of multiple layer potential for the biharmonic equation in is given. In order to represent the solution of Dirichlet problem by means of such a potential, a singular integral system, whose symbol determinant identically vanishes, is considered. The concept of bilateral reduction is introduced and employed for investigating such a system.
The Neumann problem for the Laplace equation on general domains
The solution of the weak Neumann problem for the Laplace equation with a distribution as a boundary condition is studied on a general open set in the Euclidean space. It is shown that the solution of the problem is the sum of a constant and the Newtonian potential corresponding to a distribution with finite energy supported on . If we look for a solution of the problem in this form we get a bounded linear operator. Under mild assumptions on a necessary and sufficient condition for the solvability...
The problem and singular integrals on Orlicz spaces
The transmission problem with boundary conditions given by real measures
The unique solvability of the problem Δu = 0 in G⁺ ∪ G¯, u₊ - au_ = f on ∂G⁺, n⁺·∇u₊ - bn⁺·∇u_ = g on ∂G⁺ is proved. Here a, b are positive constants and g is a real measure. The solution is constructed using the boundary integral equation method.
Thinness and non-tangential limit associated to coupled PDE
In this paper, we study the reduit, the thinness and the non-tangential limit associated to a harmonic structure given by coupled partial differential equations. In particular, we obtain such results for biharmonic equation (i.e. ) and equations of type.
Three-dimensional mathematical problems of thermoelasticity of anisotropic bodies. I.