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

Dagmar Medková

Czechoslovak Mathematical Journal (1997)

  • Volume: 47, Issue: 4, page 651-679
  • ISSN: 0011-4642

Abstract

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The paper investigates the third boundary value problem ∂ u ∂ n + λ u = μ 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 ν is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure T ν . Denote by T ν → T ν the corresponding operator on the space of signed measures on the boundary of the investigated domain G . If there is α ≠ 0 such that the essential spectral radius of ( α I - T ) is smaller than | α | (for example, if G ⊂ R 3 is a domain “with a piecewise smooth boundary” and the restriction of the Newtonian potential 𝒰 λ on ∂ G is a finite continuous functions) then the third problem is uniquely solvable in the form of a single layer potential (1) with the only exception which occurs if we study the Neumann problem for a bounded domain. In this case the problem is solvable for the boundary condition μ ∈ for which μ ( ∂ G ) = 0 .

How to cite

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Medková, Dagmar. "The third boundary value problem in potential theory for domains with a piecewise smooth boundary." Czechoslovak Mathematical Journal 47.4 (1997): 651-679. <http://eudml.org/doc/30390>.

@article{Medková1997,
abstract = {The paper investigates the third boundary value problem $\frac\{\partial u\}\{\partial n\}+\lambda u=\mu $ 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 $\nu $ is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure $\{T\}\nu $. Denote by $\{T\}\:\nu \rightarrow \{T\}\nu $ the corresponding operator on the space of signed measures on the boundary of the investigated domain $G$. If there is $\alpha \ne 0$ such that the essential spectral radius of $(\alpha I-\{T\})$ is smaller than $|\alpha |$ (for example, if $G\subset R^3$ is a domain “with a piecewise smooth boundary” and the restriction of the Newtonian potential $\{\mathcal \{U\}\}\lambda $ on $\partial G$ is a finite continuous functions) then the third problem is uniquely solvable in the form of a single layer potential (1) with the only exception which occurs if we study the Neumann problem for a bounded domain. In this case the problem is solvable for the boundary condition $\mu \in $ for which $\mu (\partial G)=0$.},
author = {Medková, Dagmar},
journal = {Czechoslovak Mathematical Journal},
keywords = {Laplace equation; single layer potential; nonsmooth domains; third boundary value problem},
language = {eng},
number = {4},
pages = {651-679},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The third boundary value problem in potential theory for domains with a piecewise smooth boundary},
url = {http://eudml.org/doc/30390},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Medková, Dagmar
TI - The third boundary value problem in potential theory for domains with a piecewise smooth boundary
JO - Czechoslovak Mathematical Journal
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 47
IS - 4
SP - 651
EP - 679
AB - The paper investigates the third boundary value problem $\frac{\partial u}{\partial n}+\lambda u=\mu $ 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 $\nu $ is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure ${T}\nu $. Denote by ${T}\:\nu \rightarrow {T}\nu $ the corresponding operator on the space of signed measures on the boundary of the investigated domain $G$. If there is $\alpha \ne 0$ such that the essential spectral radius of $(\alpha I-{T})$ is smaller than $|\alpha |$ (for example, if $G\subset R^3$ is a domain “with a piecewise smooth boundary” and the restriction of the Newtonian potential ${\mathcal {U}}\lambda $ on $\partial G$ is a finite continuous functions) then the third problem is uniquely solvable in the form of a single layer potential (1) with the only exception which occurs if we study the Neumann problem for a bounded domain. In this case the problem is solvable for the boundary condition $\mu \in $ for which $\mu (\partial G)=0$.
LA - eng
KW - Laplace equation; single layer potential; nonsmooth domains; third boundary value problem
UR - http://eudml.org/doc/30390
ER -

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Citations in EuDML Documents

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  1. Dagmar Medková, Solution of the Neumann problem for the Laplace equation
  2. Dagmar Medková, Solution of the Robin problem for the Laplace equation
  3. Dagmar Medková, The boundary-value problems for Laplace equation and domains with nonsmooth boundary
  4. Dagmar Medková, Continuous extendibility of solutions of the Neumann problem for the Laplace equation
  5. Dagmar Medková, Continuous extendibility of solutions of the third problem for the Laplace equation
  6. Dagmar Medková, Boundedness of the solution of the third problem for the Laplace equation
  7. Dagmar Medková, Solution of the Dirichlet problem for the Laplace equation

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