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The boundary behaviour of solutions of the Dirichlet problem. (English) Zbl 0783.31004
Potential theory, Proc. Int. Conf., Nagoya/Jap. 1990, 75-92 (1992).
[For the entire collection see Zbl 0745.00008.]
This survey discusses various aspects of the boundary behaviour of solutions of the Dirichlet problem in harmonic spaces, e.g. regular points and sets, semiregular points and sets, weakly and strongly irregular points (related to the dichotomy that as \(x\in U\) tends to a boundary point \(z\) of \(U\) either the accumulation points of the harmonic measures \(\varepsilon_ x^{\complement U}\) are contained in \(\{\varepsilon_ z,\varepsilon_ z^{\complement U}\}\) or form the interval \(\{\alpha\varepsilon_ z+(1-z)\varepsilon_ z^{\complement U}:0\leq\alpha\leq 1\})\), and maximal sequences.
A long section summarizes recent results on regularizing sets of irregular points. A subset \(A\) of the set \(U^*_{\text{irr}}\) of irregular points of an open relatively compact set \(U\) is called regularizing provided every continuous function \(f\) on the boundary \(U^*\) satisfying \(\lim_{x\to z}H_ Uf(x)=f(z)\) for every \(z\in A\) admits a solution of the classical Dirichlet problem (i.e., satisfies \(\lim_{x\to z}H_ Uf(x)=f(z)\) for every \(z\in U^*)\). The introduction of a suitable topology on \(U^*_{\text{irr}}\) by a closure operation \(\lambda\) leads to the following characterization: \(A\) is regularizing if and only if \(A\) is \(\lambda\)-dense in \(U^*_{\text{irr}}\).
31B25 Boundary behavior of harmonic functions in higher dimensions
31D05 Axiomatic potential theory
35J25 Boundary value problems for second-order elliptic equations
35K20 Initial-boundary value problems for second-order parabolic equations
Zbl 0745.00008