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Irregular boundary point

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A point on the boundary of a domain at which there is a continuous boundary function on such that the Perron–Wiener–Brélot generalized solution (cf. Perron method) of the Dirichlet problem, , does not take the boundary value at , i.e. either the limit

does not exist, or it does not coincide with . For domains in the plane every isolated point of the boundary is irregular. In the case of a domain in a Euclidean space , , it was H. Lebesgue who first discovered that the vertex of a very acute angle in is an irregular boundary point. E.g., the coordinate origin is an irregular boundary point if the boundary of the domain has, in a neighbourhood of , the shape of the entering acute angle obtained by rotating the curve , , around the positive -axis (Lebesgue spine). The generalized solution of the Dirichlet problem does not take the boundary value at an irregular boundary point if is the least upper or greatest lower bound of the values of on ; the classical solution does not exist in this case. The set of irregular boundary points is thin, in a certain sense: it has type , 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)
How to Cite This Entry:
Irregular boundary point. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Irregular_boundary_point&oldid=12226
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098