Irregular boundary point

A point $y _ {0}$ on the boundary $\Gamma$ of a domain $D$ at which there is a continuous boundary function $f ( y)$ on $\Gamma$ such that the Perron–Wiener–Brélot generalized solution (cf. Perron method) of the Dirichlet problem, $u ( x)$, does not take the boundary value $f ( y _ {0} )$ at $y _ {0}$, i.e. either the limit

$$\lim\limits _ {\begin{array}{c} x \rightarrow y _ {0} \\ x \in L \end{array} } u ( x)$$

does not exist, or it does not coincide with $f ( y _ {0} )$. For domains $D$ in the plane every isolated point of the boundary $\Gamma$ is irregular. In the case of a domain $D$ in a Euclidean space $\mathbf R ^ {n}$, $n \geq 3$, it was H. Lebesgue who first discovered that the vertex of a very acute angle in $D$ is an irregular boundary point. E.g., the coordinate origin $0 \in \mathbf R ^ {3}$ is an irregular boundary point if the boundary of the domain has, in a neighbourhood of $0$, the shape of the entering acute angle obtained by rotating the curve $y = e ^ {-} 1/x$, $x > 0$, around the positive $x$- axis (Lebesgue spine). The generalized solution of the Dirichlet problem does not take the boundary value $f ( y _ {0} )$ at an irregular boundary point if $f ( y _ {0} )$ is the least upper or greatest lower bound of the values of $f ( y)$ on $\Gamma$; the classical solution does not exist in this case. The set of irregular boundary points is thin, in a certain sense: it has type $F _ \delta$, is a polar set and has zero capacity. See also Barrier; Regular boundary point.

References

 [1] N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian) [2] M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)