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Hölder Continuity of the Dirichlet Solution for a General Domain
Author(s) -
Aikawa Hiroaki
Publication year - 2002
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609302001522
Subject(s) - mathematics , bounded function , hölder condition , domain (mathematical analysis) , exponent , boundary (topology) , function (biology) , mathematical analysis , dirichlet distribution , dirichlet boundary condition , dirichlet problem , combinatorics , pure mathematics , boundary value problem , philosophy , linguistics , evolutionary biology , biology
Let D be a bounded domain in R n . For a function f on the boundary ∂ D , the Dirichlet solution of f over D is denoted by H D f , provided that such a solution exists. Conditions on D for H D to transform a Hölder continuous function on ∂ D to a Hölder continuous function on D with the same Hölder exponent are studied. In particular, it is demonstrated here that there is no bounded domain that preserves the Hölder continuity with exponent 1. It is also also proved that a bounded regular domain D preserves the Hölder continuity with some exponent α, 0<α<1, if and only if ∂ D satisfies the capacity density condition, which is equivalent to the uniform perfectness of ∂ D if n = 2. 2000 Mathematics Subject Classification 31A05, 31A20, 31B05, 31B25.