View source for Regular boundary point

A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806801.png" /> on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806802.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806803.png" /> in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806805.png" />, at which, for any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806806.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806807.png" />, the generalized solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806808.png" /> of the [[Dirichlet problem|Dirichlet problem]] in the sense of Wiener–Perron (see [[Perron method|Perron method]]) takes the boundary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r0806809.png" />, that is,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068010.png" /></td> </tr></table>

The regular boundary points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068011.png" /> form a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068012.png" />, at the points of which the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068013.png" /> is not a [[Thin set|thin set]]; the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068014.png" /> of irregular boundary points (cf. [[Irregular boundary point|Irregular boundary point]]) is a [[Polar set|polar set]] of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068015.png" />. If all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068016.png" /> are regular boundary points, then the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068017.png" /> is called regular with respect to the Dirichlet problem.

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068018.png" /> to be a regular boundary point it is necessary and sufficient that in the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068020.png" /> with any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068022.png" /> there is a superharmonic [[Barrier|barrier]] (a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068025.png" />, Lebesgue's criterion for a barrier). It was first shown by H. Lebesgue in 1912 that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068026.png" /> the vertex of a sufficiently acute angle lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068027.png" /> need not be a regular boundary point.

Let

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and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068029.png" /> be the [[Capacity|capacity]] of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068030.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068031.png" /> to be a regular boundary point it is necessary and sufficient that the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068032.png" /></td> </tr></table>

diverges, or for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068033.png" />, that the series

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diverges, where

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(Wiener's criterion).

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068036.png" />, a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068037.png" /> is a regular boundary point if there is a continuous path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068039.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068040.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068041.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068042.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068043.png" />, a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068044.png" /> is a regular boundary point if it can be reached by the vertex of a right circular cone belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068045.png" /> in a sufficiently small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068046.png" />. In the case of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068047.png" /> in the compactified space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068049.png" />, the point at infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068050.png" /> is always a regular boundary point; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068051.png" />, the point at infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068052.png" /> is a regular boundary point if there is a continuous path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068054.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068055.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068056.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080680/r08068057.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.V. Keldysh,   "On the solvability and stability of the Dirichlet problem"  ''Uspekhi Mat. Nauk'' , '''8'''  (1941)  pp. 171–232  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.S. Landkof,   "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.K. Hayman,   P.B. Kennedy,   "Subharmonic functions" , '''1''' , Acad. Press  (1976)</TD></TR></table>



====Comments====
The polarity of the set of irregular boundary points is contained in the [[Kellogg–Evans theorem|Kellogg–Evans theorem]]. See, e.g., [[#References|[a1]]] for irregular boundary points in abstract potential theory.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Bliedtner,   W. Hansen,   "Potential theory. An analytic and probabilistic approach to balayage" , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Lebesgue,   "Sur des cas d'impossibilité du problème de Dirichlet ordinaire"  ''C.R. Séances Soc. Math. France'' , '''41'''  (1913)  pp. 17</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Wiener,   "The Dirichlet problem"  ''J. Math. Phys.'' , '''3'''  (1924)  pp. 127–146</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Tsuji,   "Potential theory in modern function theory" , Chelsea, reprint  (1975)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Wermer,   "Potential theory" , ''Lect. notes in math.'' , '''408''' , Springer  (1981)</TD></TR></table>