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.
Which solutions of the third problem for the Poisson equation are bounded?
The successive approximation method for the Dirichlet problem in a planar domain
The Dirichlet problem for the Laplace equation for a planar domain with piecewise-smooth boundary is studied using the indirect integral equation method. The domain is bounded or unbounded. It is not supposed that the boundary is connected. The boundary conditions are continuous or p-integrable functions. It is proved that a solution of the corresponding integral equation can be obtained using the successive approximation method.
Conditions ensuring T(Y) ⊂ Y.
The following theorem is the main result of the paper: Let X be a complex Banach space and T belong to L(X). Suppose that 0 lies at the unbounded component of the set of those l such that lI - T 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 T'(Y) is contained in Y and TSy = ST'y for every y belonging to Y. Then for every vector x belonging to X', T'x belongs to Y if and only if x belongs to Y.
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.
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.
The third boundary value problem in potential theory for domains with a piecewise smooth boundary
The paper investigates the third boundary value problem for the Laplace equation by the means of the potential theory. The solution is sought in the form of the Newtonian potential (1), (2), where is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure . Denote by the corresponding operator on the space of signed measures on the boundary of the investigated domain . If there is such that the essential spectral radius of is...
Continuous extendibility of solutions of the Neumann problem for the Laplace equation
A necessary and sufficient condition for the continuous extendibility of a solution of the Neumann problem for the Laplace equation is given.
Continuous extendibility of solutions of the third problem for the Laplace equation
A necessary and sufficient condition for the continuous extendibility of a solution of the third problem for the Laplace equation is given.
Solution of the Dirichlet problem for the Laplace equation
For open sets with a piecewise smooth boundary it is shown that a solution of the Dirichlet problem for the Laplace equation can be expressed in the form of the sum of the single layer potential and the double layer potential with the same density, where this density is given by a concrete series.
Solution of the Robin problem for the Laplace equation
For open sets with a piecewise smooth boundary it is shown that we can express a solution of the Robin problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series.
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.
Remark on Ugaheri's maximum principle
Invariance of the Fredholm radius of the Neumann operator
On a maximum principle in potential theory
Note on contractivity of the Neumann operator
On the convergence of Neumann series for noncompact operators
On the convergence of Neumann series for noncompact operators
On essential norm of the Neumann operator
One of the classical methods of solving the Dirichlet problem and the Neumann problem in 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,...
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...
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