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Continuous extendibility of solutions of the third problem for the Laplace equation. (English) Zbl 1080.35009
Summary: A necessary and sufficient condition for the continuous extendibility of a very weak solution of the third problem for the Laplace equation is given. Here the boundary condition is a real measure on the boundary of the domain. It is shown that the solution is continuously extendible to the closure of the domain if and only if the single layer potential corresponding to the boundary condition is continuously extendible to the closure of the domain. For such boundary condition a general form of a solution is given.

##### MSC:
 35B65 Smoothness and regularity of solutions to PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35J25 Boundary value problems for second-order elliptic equations 31B10 Integral representations, integral operators, integral equations methods in higher dimensions
##### Keywords:
third problem; Laplace equation; continuous extendibility
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##### References:
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