Regularity Theory

Abstract

The weak solutions of the p-harmonic equation are, by definition, members of the Sobolev space $W^{1,p}_{\text {loc}}(\Omega )$. In fact, they are also of class $C_{\text {loc}}^\alpha (\Omega )$. More precisely, a weak solution can be redefined in a set of Lebesgue measure zero, so that the new function is locally Hölder continuous with exponent $\alpha =\alpha (n, p)$.