Regular boundary point
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.
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
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 .
|||M.V. Keldysh, "On the solvability and stability of the Dirichlet problem" Uspekhi Mat. Nauk , 8 (1941) pp. 171–232 (In Russian)|
|||N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)|
|||W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976)|
The polarity of the set of irregular boundary points is contained in the Kellogg–Evans theorem. See, e.g., [a1] for irregular boundary points in abstract potential theory.
|[a1]||J. Bliedtner, W. Hansen, "Potential theory. An analytic and probabilistic approach to balayage" , Springer (1986)|
|[a2]||H. Lebesgue, "Sur des cas d'impossibilité du problème de Dirichlet ordinaire" C.R. Séances Soc. Math. France , 41 (1913) pp. 17|
|[a3]||H. Lebesgue, "Conditions de régularité, conditions d'irrégularité, conditions d'impossibilité dans le problème de Dirichlet" C.R. Acad. Sci. Paris , 178 (1924) pp. 349–354|
|[a4]||N. Wiener, "The Dirichlet problem" J. Math. Phys. , 3 (1924) pp. 127–146|
|[a5]||M. Tsuji, "Potential theory in modern function theory" , Chelsea, reprint (1975)|
|[a6]||J. Wermer, "Potential theory" , Lect. notes in math. , 408 , Springer (1981)|
Regular boundary point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_boundary_point&oldid=15319