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Elliptic Theory for Sets with Higher Co-dimensional Boundaries

About this Title

G. David, J. Feneuil and S. Mayboroda

Publication: Memoirs of the American Mathematical Society
Publication Year: 2021; Volume 274, Number 1346
ISBNs: 978-1-4704-5043-4 (print); 978-1-4704-6915-3 (online)
DOI: https://doi.org/10.1090/memo/1346
Published electronically: November 16, 2021
Keywords: Harmonic measure, boundary of co-dimension higher than 1, trace theorem, extension theorem, degenerate elliptic operators, maximum principle, Hölder continuity of solutions, de Giorgi-Nash-Moser estimates, Green functions, comparison principle, homogeneous weighted Sobolev spaces

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Table of Contents

Chapters

  • 1. Introduction
  • 2. The Harnack Chain Condition and the Doubling Property
  • 3. Traces
  • 4. Poincaré Inequalities
  • 5. Completeness and Density of Smooth Functions
  • 6. The Chain Rule and Applications
  • 7. The Extension Operator
  • 8. Definition of Solutions
  • 9. Harmonic Measure
  • 10. Green Functions
  • 11. The Comparison Principle

Abstract

Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1.

To this end, we turn to degenerate elliptic equations. Let $\Gamma \subset \mathbb {R}^n$ be an Ahlfors regular set of dimension $d<n-1$ (not necessarily integer) and $\Omega = \mathbb {R}^n \setminus \Gamma$. Let $L = - \operatorname {div} A\nabla$ be a degenerate elliptic operator with measurable coefficients such that the ellipticity constants of the matrix $A$ are bounded from above and below by a multiple of $\operatorname {dist}(\cdot , \Gamma )^{d+1-n}$. We define weak solutions; prove trace and extension theorems in suitable weighted Sobolev spaces; establish the maximum principle, De Giorgi-Nash-Moser estimates, the Harnack inequality, the Hölder continuity of solutions (inside and at the boundary). We define the Green function and provide the basic set of pointwise and/or $L^p$ estimates for the Green function and for its gradient. With this at hand, we define harmonic measure associated to $L$, establish its doubling property, non-degeneracy, change-of-the-pole formulas, and, finally, the comparison principle for local solutions.

In another article to appear, we will prove that when $\Gamma$ is the graph of a Lipschitz function with small Lipschitz constant, we can find an elliptic operator $L$ for which the harmonic measure given here is absolutely continuous with respect to the $d$-Hausdorff measure on $\Gamma$ and vice versa. It thus extends Dahlberg’s theorem to some sets of codimension higher than 1.

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