p-Harmonic functions with boundary data having jump discontinuities and Baernstein's problem |
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Authors: | Anders Björn |
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Institution: | Department of Mathematics, Linköpings universitet, SE-581 83 Linköping, Sweden |
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Abstract: | For p-harmonic functions on unweighted R2, with 1<p<∞, we show that if the boundary values f has a jump at an (asymptotic) corner point z0, then the Perron solution Pf is asymptotically a+barg(z−z0)+o(|z−z0|). 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 . 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 Rn and metric spaces. |
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Keywords: | primary 31C45 secondary 31E05 35J65 |
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