On essential norm of the Neumann operator

summary:One of the classical methods of solving the Dirichlet problem and the Neumann problem in $\bold R^m$ is the method of integral equations. If we wish to use the Fredholm-Radon theory to solve the problem, it is useful to estimate the essential norm of the Neumann operator with respect to a norm on the space of continuous functions on the boundary of the domain investigated, where this norm is equivalent to the maximum norm. It is shown in the paper that under a deformation of the domain investigated by a diffeomorphism, which is conformal (i.e. preserves angles) on a precisely specified part of boundary, for the given norm there exists a norm on the space of continuous functions on the boundary of the deformated domain such that this norm is equivalent to the maximum norm and the essential norms of the corresponding Neumann operators with respect to these norms are the same

The boundary-value problems for Laplace equation and domains with nonsmooth boundary

summary: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

Continuous extendibility of solutions of the Neumann problem for the Laplace equation

summary:A necessary and sufficient condition for the continuous extendibility of a solution of the Neumann problem for the Laplace equation is given

The Neumann problem for the Laplace equation on general domains

summary: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 $G$ 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 $\partial G$. If we look for a solution of the problem in this form we get a bounded linear operator. Under mild assumptions on $G$ a necessary and sufficient condition for the solvability of the problem is given and the solution is constructed

Conditions ensuring T ⎺¹ (Y ) ⊂ Y

The following theorem is the main result of the paper: Let X be a complex Banach space and ∈ L(X). Suppose that 0 lies at the unbounded component of the set of those λ such that λI − is a Fredholm operator. Let Y be a dense subspace of the dual space X′ and S be a closed operator from Y to X such that ′( Y ) ⊂ Y and S = ST ′ for each ∈ Y . Then for each vector ∈ X′, ′ ∈ Y if and only if ∈ Y .The paper is supported by the grant no. KSK 1019101.peerReviewe

Which solutions of the third problem for the Poisson equation are bounded?

This paper deals with the problem Δu=g on G and ∂u/∂n+uf=L on ∂G. Here, G⊂ℝm, m>2, is a bounded domain with Lyapunov boundary, f is a bounded nonnegative function on the boundary of G, L is a bounded linear functional on W1,2(G) representable by a real measure μ on the boundary of G, and g∈L2(G)∩Lp(G), p>m/2. It is shown that a weak solution of this problem is bounded in G if and only if the Newtonian potential corresponding to the boundary condition μ is bounded in G

Neumann problem for the Stokes system

the Neumann problem for the Stokes systems is studied by the boundary integral equation method. The successive approximation converges for bounded Lipschitz domains

The solution of the third problem for the Laplace equation on planar domains with smooth boundary and inside cracks and modified jump conditions on cracks

This paper studies the third problem for the Laplace equation on a bounded planar domain with inside cracks. The third condition ∂u/∂n+hu=f is given on the boundary of the domain. The skip of the function u+−u−=g and the modified skip of the normal derivatives (∂u/∂n)+−(∂u/∂n)−+hu+=f are given on cracks. The solution is looked for in the form of the sum of a modified single-layer potential and a double-layer potential. The solution of the corresponding integral equation is constructed