Applications of Boundary Harnack Inequalities for p-Harmonic Functions and Related Topics

John Lewis (University of Kentucky, Lexington, USA)

This course will be concerned with applications of recent work - techniques concerning the boundary behavior of positive p-harmonic functions vanishing on a portion of the boundary of Lipschitz, chord arc, and Reifenberg flat domains. An optimistic outline of the course (given time constraints) is as follows:

• (1) Fundamental properties of p-harmonic functions and elliptic measure.
• (2) Two dimensional theorems including the dimension of p-harmonic measure and p-harmonic pseudospheres.
• (3) Boundary Harnack inequalities in Lipschitz - Reifenberg flat domains and the Martin boundary problem for p-harmonic functions.
• (4) Free boundary regularity and inverse type problems for p-harmonic functions.

The lectures concerning (2) will be drawn from the following papers:

• On the Dimension of p Harmonic Measure (with Björn Bennewitz), Ann. Acad. Sci. Fenn. 30 (2005), 459-505.
• Note on p Harmonic Measure, Computational Methods in Function Theory 6 (2006), No.1, 109-144.
• p Harmonic Measure in Simply Connected Domains (with Pietro Poggi Corradini and Kaj Nyström), submitted.
• Non Uniqueness in a Free Boundary problem, PhD Thesis of Björn Bennewitz, to appear in Revista Mat. Iberoamericana.

The lectures involving (3) will be chosen from

• Boundary Behavior for p Harmonic Functions in Lipschitz and Starlike Lipschitz Ring Domains (with Kaj Nyström), Ann. Sc. Ecole Norm. Sup. (4) 40 (2007), no. 4, 765-813.
• Boundary Behaviour and the Martin Boundary Problem for p Harmonic Functions in Lipschitz Domains (with Kaj Nyström), to appear in the Annals of Mathematics.
• Boundary Behavior of p Harmonic Functions in Domains Beyond Lipschitz Domains (with Kaj Nyström), Advances in the Calculus of Variations 1 (2008), 1-38.
• Boundary Harnack Inequalities for Operators of p Laplace Type in Reifenberg Flat Domains (with Kaj Nyström), to appear in Perspectives in PDE, Harmonic Analysis, and Applications, Proceedings of Symposia in Pure Mathematics, Dorina and Marius Mitrea, editors.

Lectures concerning (4) will be based on

• Uniqueness in a free boundary problem (with Andrew Vogel), Comm. in Partial Differental Equations 31 (2006), 1591-1614.
• Symmetry Theorems and Uniform Rectifiability (with Andrew Vogel), Boundary Value Problems 2007 (2007), 1-59.
• Regularity and Free Boundary Regularity for the p Laplacian in Lipschitz and C¹ Domains (with Kaj Nyström), Ann. Acad. Sci. Fenn. 33 (2008), 1-26.
• Regularity of Lipschitz Free Boundaries in Two Phase Problems for the p Laplace Operator (with Kaj Nyström), submitted.
• Regularity of Flat Free Boundaries in Two phase Problems for the p Laplace Operator (with Kaj Nyström), in preparation