Holder continuity of green's functions [electronic resource] /
ABSTRACT: We investigate local properties of the Green function of the complement of a compact set$E$. First we consider the case $E\subset [0,1]$ in the extended complex plane. We extend a result of V. Andrievskii which claims that if the Green function satisfies the H\"older-$1/2$ condition locally at the origin, then the density of $E$ at $0$, in terms of logarithmic capacity, is the same as that of the whole interval $[0,1]$. We give an integral estimate on the density in terms of the Green function, which also provides a necessary condition for the optimal smoothness. Then we extend the results to the case $Esubset [-1,1]. In this case the maximal smoothness of the Green function is "older-1 and a similar integral estimate and necessary condition hold as well.In the second part of the paper we consider the case when $E$ is acompact set in R, >2. We give a Wiener type characterization for the "older continuity of the Green function, thus extending a result of L.
School:University of South Florida
School Location:USA - Florida
Source Type:Master's Thesis
Keywords:logarithmic capacity newtonian potential equilibrium measure boundary behavior wiener s criterion
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