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Regularizing sets of irregular points. (English) Zbl 0701.31007
Let U be a relatively compact open subset of $${\mathbb{R}}^ n$$ and, for every $$f\in {\mathcal C}(U^*)$$, let $$H_ Uf: x\mapsto \epsilon_ x^{\complement U}(f)$$ denote the corresponding generalized solution of the classical Dirichlet problem. A subset A of the set $$U^*_{irr}$$ of irregular boundary points is called regularizing if, given any $$f\in {\mathcal C}(U^*)$$, the relation $$\lim_{x\to z}H_ Uf(x)=f(z)$$ holds for every $$z\in U^*_{irr}$$ provided it holds for every $$z\in A$$. An example shows that a regularizing set A may satisfy $$\lim_{x\to z,x\in A}\epsilon_ x^{\complement U}=\epsilon_ z$$ for some $$z\in U^*_{irr}$$, i.e., there are regularizing sets which are no “piquetage faible” (negative answer to a question raised by G. Choquet in 1968). A subset B of $$U^*_{irr}$$ is called $$\lambda$$-closed if $$\lim_{x\to z,x\in B}\epsilon_ x^{\complement U}=\epsilon_ z$$ for every $$z\in \bar B\setminus B$$. It is shown that a set A is regularizing if and only if A is $$\lambda$$-dense in $$U^*_{irr}$$. In fact this holds for any U in any harmonic space satisfying $$\epsilon_ x^{\complement U}(U^*_{irr})=0$$ for every $$x\in U$$. For the heat equation in $${\mathbb{R}}^ 2$$ a counterexample is given.
Reviewer: W.Hansen

##### MSC:
 31B20 Boundary value and inverse problems for harmonic functions in higher dimensions 35K05 Heat equation 31B25 Boundary behavior of harmonic functions in higher dimensions
##### Keywords:
Dirichlet problem; irregular boundary points; regularizing
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