The mixed harmonic problem in a bounded cracked domain with Dirichlet condition on cracks

Крутицкий, П. А.

doi:10.1016/j.jde.2003.09.007

Cited by 6 publications

(4 citation statements)

References 12 publications

(34 reference statements)

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“…Note that can be taken small enough. Consider the representation for u(x) in D ∩ S(d, /3) given by formula (6). Consider such a sector with the center in x(d) and of radius /3 and of angle β = (β 2 − β 1 ) > 0,…”

Section: Nonexistence Of a Weak Solutionmentioning

confidence: 99%

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On existence of a classical solution and nonexistence of a weak solution to the Dirichlet problem for the Laplacian with discontinuous boundary data

Крутицкий

2009

Quart. Appl. Math.

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Abstract. The Dirichlet problem for the Laplacian in a planar multiply connected interior domain bounded by smooth closed curves is considered in the case when the boundary data is piecewise continuous; i.e., it may have jumps in certain points of the boundary. It is assumed that the solution to the problem may not be continuous at the same points. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integral representation for a classical solution is obtained. The problem is reduced to a uniquely solvable Fredholm integral equation of the second kind and of index zero. It is shown that a weak solution to the problem does not exist typically, though the classical solution exists. Introduction.It is known that if the Dirichlet problem for the Laplacian is considered in a planar domain bounded by sufficiently smooth closed curves, and if the function specified in the boundary condition is smooth enough, then the existence of a classical solution follows from the existence of a weak solution. In the present paper we consider the Dirichlet problem for the Laplacian in a planar multiply connected interior domain bounded by closed curves on the assumption that the boundary data is piecewise continuous; i.e., it may have jumps in certain points of the boundary. We look for a solution to the problem which may be discontinuous at these points. We give the well-posed formulation of the problem. We prove that there exists a unique classical solution to this problem and obtain an integral representation for the classical solution in the form of a double layer potential with some additions. Moreover, we reduce the problem to a uniquely solvable Fredholm integral equation of the second kind and of index zero for the density of a potential.In addition, we prove that a weak solution to this problem may not exist. This result follows from the fact that the square of the gradient of the classical solution, basically, is

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Section: Nonexistence Of a Weak Solutionmentioning

confidence: 99%

“…It is shown in Appendix 3 (see Theorem A3) that the integral term in (12 ) belongs to C 0,1/3 (Γ) in s; therefore µ 0 (s) ∈ C 0,1/3 (Γ). Now we consider the function u[µ 0 ](x) introduced in (6). This function satisfies the following homogeneous Dirichlet problem for the Laplace equation:…”

Section: Nonexistence Of a Weak Solutionmentioning

confidence: 99%

On existence of a classical solution and nonexistence of a weak solution to the Dirichlet problem for the Laplacian with discontinuous boundary data

Крутицкий

2009

Quart. Appl. Math.

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“…The gradient of the solution has to satisfy estimate (2.7). Both the existence and the uniqueness of this problem have been investigated in [17]. However, in proving the existence of a solution other types of integral equations were used in [17] that are not suitable for our numerical implementation.…”

Section: Numerical Solution Of the Mixed Problems By The Layer Potential Approachmentioning

confidence: 99%

“…Both the existence and the uniqueness of this problem have been investigated in [17]. However, in proving the existence of a solution other types of integral equations were used in [17] that are not suitable for our numerical implementation. We shall therefore give an alternative proof of the existence based on the integral equations we use for the numerical investigations.…”

Section: Numerical Solution Of the Mixed Problems By The Layer Potential Approachmentioning

confidence: 99%

An Alternating Potential-Based Approach To The Cauchy Problem For The Laplace Equation In A Planar Domain With A Cut

Chapko

Johansson

2008

Comput. Methods Appl. Math.

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We consider a Cauchy problem for the Laplace equation in a bounded region containing a cut, where the region is formed by removing a sufficiently smooth arc (the cut) from a bounded simply connected domain D. The aim is to reconstruct the solution on the cut from the values of the solution and its normal derivative on the boundary of the domain D. We propose an alternating iterative method which involves solving direct mixed problems for the Laplace operator in the same region. These mixed problems have either a Dirichlet or a Neumann boundary condition imposed on the cut and are solved by a potential approach. Each of these mixed problems is reduced to a system of integral equations of the first kind with logarithmic and hypersingular kernels and at most a square root singularity in the densities at the endpoints of the cut. The full discretization of the direct problems is realized by a trigonometric quadrature method which has super-algebraic convergence. The numerical examples presented illustrate the feasibility of the proposed method.

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Riemann-Hilbert Problems for Multiply Connected Domains and Circular Slit Maps

Mityushev

2012

Comput. Methods Funct. Theory

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The mixed harmonic problem in a bounded cracked domain with Dirichlet condition on cracks

Cited by 6 publications

References 12 publications

On existence of a classical solution and nonexistence of a weak solution to the Dirichlet problem for the Laplacian with discontinuous boundary data

On existence of a classical solution and nonexistence of a weak solution to the Dirichlet problem for the Laplacian with discontinuous boundary data

An Alternating Potential-Based Approach To The Cauchy Problem For The Laplace Equation In A Planar Domain With A Cut

Riemann-Hilbert Problems for Multiply Connected Domains and Circular Slit Maps

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