p-harmonic functions with boundary data having jump discontinuities and Baernstein´s problem
2009 (English)Report (Other academic)
For p-harmonic functions on unweighted R-2, with 1 andlt; p andlt; 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 Baernsteins 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 andgt; 2 these results are new also for continuous functions. Finally we look at generalizations to R-n and metric spaces.
Place, publisher, year, edition, pages
Linköping: Linköpings universitet , 2009. , 32 p.
LiTH-MAT-R, ISSN 0348-2960 ; 2009:4
IdentifiersURN: urn:nbn:se:liu:diva-51092OAI: oai:DiVA.org:liu-51092DiVA: diva2:272749