# Regular boundary point

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

A point on the boundary of a domain in a Euclidean space , , at which, for any continuous function on , the generalized solution of the Dirichlet problem in the sense of Wiener–Perron (see Perron method) takes the boundary value , that is, The regular boundary points of form a set , at the points of which the complement is not a thin set; the set of irregular boundary points (cf. Irregular boundary point) is a polar set of type . If all points of are regular boundary points, then the domain is called regular with respect to the Dirichlet problem.

For to be a regular boundary point it is necessary and sufficient that in the intersection of with any neighbourhood of there is a superharmonic barrier (a function in such that , Lebesgue's criterion for a barrier). It was first shown by H. Lebesgue in 1912 that for the vertex of a sufficiently acute angle lying in need not be a regular boundary point.

Let and let be the capacity of the set . For to be a regular boundary point it is necessary and sufficient that the series diverges, or for , that the series diverges, where (Wiener's criterion).

For , a point is a regular boundary point if there is a continuous path , , such that , and for . When , a point is a regular boundary point if it can be reached by the vertex of a right circular cone belonging to in a sufficiently small neighbourhood of . In the case of a domain in the compactified space , , the point at infinity is always a regular boundary point; when , the point at infinity is a regular boundary point if there is a continuous path , , such that for , and .

How to Cite This Entry:
Regular boundary point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_boundary_point&oldid=15319
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article