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)

Comments

See [a2] for an additional classical reference, and [a1] for irregular points in axiomatic potential theory.

References

[a1]	L.L. Helms, "Introduction to potential theory" , Wiley (1969) (Translated from German)
[a2]	C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972)