p-Harmonic functions with boundary data having jump discontinuities and Baernstein's problem

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:249

Article

Bjorn, Anders

关键词: Baernstein's problem;    Boundary regularity;    Dirichlet problem;    Invariance;    Jump discontinuity;    Metric space;    Nonlinear;    Perron solution;    Perturbation;    Quasiminimizer;    p-Harmonic;    Potential theory;    Resolutive;    Semicontinuous;   

DOI  :  10.1016/j.jde.2010.03.002

来源: Elsevier

【 摘 要 】

For p-harmonic functions on unweighted R(2), with 1 < p < infinity, we show that if the boundary values f has a jump at an (asymptotic) corner point zo, then the Perron solution Pf is asymptotically a + b arg(z - z(0)) + o(vertical bar z z(0)vertical bar). We use this to obtain a positive answer to Baernstein's problem on the equality of the p-harmonic measure of a union G of open arcs on the boundary of the unit disc, and the p. harmonic measure of (G) over bar. We also obtain various invariance results for functions with jumps and perturbations on small sets. For p > 2 these results are new also for continuous functions. Finally we look at generalizations to R(n) and metric spaces. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free