Some results on the boundary regularity of harmonic functions

Abstract

Let θ be an open set in the plane which contains the interval I: {(x, y)∣y=0, -1?x?1}. We consider functions u, which are harmonic in θ-I and continuous in θ. Then, without additional smoothness conditions on f(x)=u(x,0),-1?x$le;1, the one sided normal derivatives(?u/?y)⁺lim_h?0+u(x,h)-f(x)/h and (?u/?y)^-=lim_h?0-u(x,h)-f(x)/h, may not exist at any point of I. Here we assume only that f(x) is continuous. We show that in this case the normal derivatives will exist in a "Sobolev-like" Space of distributions.

Notes

This research was supported in part by the National Science Foundations under Grant GP 12838 with the university of Maryland.