Regularizing sets of irregular points. (English) Zbl 0701.31007

Hansen, W.

J. Reine Angew. Math. 409, 205-218 (1990).

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