Annales Academiĉ Scientiarum Fennicĉ
Mathematica
Volumen 37, 2012, 339-355

MODULUS OF CONTINUITY OF p-DIRICHLET SOLUTIONS IN A METRIC MEASURE SPACE

Tsubasa Itoh

Hokkaido University, Department of Mathematics
Sapporo 060-0810, Japan; tsubasa 'at' math.sci.hokudai.ac.jp

Abstract. Let 1 < p < \infty and let X be a metric measure space with a doubling measure and a (1,p)-Poincaré inequality. Let Omega be a bounded domain in X. For a function f on \partial Omega we denote by P_Omegaf the p-Dirichlet solution of f over Omega. It is well known that if Omega is p-regular and f \in C(\partial Omega), then P_Omegaf is p-harmonic in Omega and continuous in \overline Omega. We characterize the family of domains Omega such that improved continuity of boundary functions f ensures improved continuity of P_Omegaf. We specify such improved continuity if X is Ahlfors regular and X \ Omega is uniformly p-fat.

2010 Mathematics Subject Classification: Primary 31E05, 31C45, 35J60.

Key words: Modulus of continuity, p-harmonic, p-Dirichlet solution, metric measure space, p-capacity.

Reference to this article: T. Itoh: Modulus of continuity of p-Dirichlet solutions in a metric measure space. Ann. Acad. Sci. Fenn. Math. 37 (2012), 339-355.

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doi:10.5186/aasfm.2012.3741

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