Graduation Year

2005

Document Type

Dissertation

Degree

Ph.D.

Degree Granting Department

Mathematics and Statistics

Major Professor

Vilmos Totik, Ph.D.

Committee Member

Mourad E. H. Ismail, Ph.D.

Committee Member

Evguenii A. Rakhmanov, Ph.D.

Committee Member

Boris Shekhtman, Ph.D.

Keywords

logarithmic capacity, Newtonian potential, equilibrium measure, boundary behavior, Wiener’s criterion

Abstract

We investigate local properties of the Green function of the complement of a compact set E.

First we consider the case E ⊂ [0, 1] in the extended complex plane. We extend a result of V. Andrievskii which claims that if the Green function satisfies the Hölder1/2 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 E ⊂ [−1, 1]. In this case the maximal smoothness of the Green function is H¨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 a compact set in Rd , d > 2. We give a Wiener type characterization for the Hölder continuity of the Green function, thus extending a result of L. Carleson and V. Totik. The obtained density condition is necessary, and it is sufficient as well, provided E satisfies the cone condition. It is also shown that the Hölder condition for the Green function at a boundary point can be equivalently stated in terms of the equilibrium measure and the solution to the corresponding Dirichlet problem. The results solve a long standing open problem - raised by Maz’ja in the 1960’s - under the simple cone condition.

Download

Share

COinS