[Paper Review] Volume growth, capacity estimates, $p$-parabolicity and sharp integrability properties of $p$-harmonic Green functions

Anders Björn, Jana Björn+1 more|arXiv (Cornell University)|Jan 27, 2021

Nonlinear Partial Differential Equations41 references4 citations

TL;DR

This paper establishes sharp integrability and growth estimates for $p$-harmonic Green functions and their gradients in metric measure spaces with doubling measures and $p$-Poincaré inequalities. It introduces a novel capacity estimate for annuli that characterizes $p$-parabolicity and determines integrability based on measure growth near singularities, generalizing results to weighted $\mathbf{R}^n$ and manifolds with precise thresholds for $L^\tau$ and $L^t$ integrability.

ABSTRACT

In a complete metric space equipped with a doubling measure supporting a $p$-Poincaré inequality, we prove sharp growth and integrability results for $p$-harmonic Green functions and their minimal $p$-weak upper gradients. We show that these properties are determined by the growth of the underlying measure near the singularity. Corresponding results are obtained also for more general $p$-harmonic functions with poles, as well as for singular solutions of elliptic differential equations in divergence form on weighted $\mathbf{R}^n$ and on manifolds. The proofs are based on a new general capacity estimate for annuli, which implies precise pointwise estimates for $p$-harmonic Green functions. The capacity estimate is valid under considerably milder assumptions than above. We also use it, under these milder assumptions, to characterize singletons of zero capacity and the $p$-parabolicity of the space. This generalizes and improves earlier results that have been important especially in the context of Riemannian manifolds.

Motivation & Objective

• To characterize the $L^\tau$-integrability and volume growth of $p$-harmonic Green functions in metric measure spaces.
• To establish sharp thresholds for integrability of $p$-harmonic functions with poles and their minimal $p$-weak upper gradients.
• To generalize results on $p$-parabolicity and capacity to metric spaces under minimal assumptions.
• To unify and improve existing results on singular solutions in weighted $\mathbf{R}^n$ and Riemannian manifolds.
• To provide a general capacity estimate for annuli that captures the interplay between measure growth and $p$-parabolicity.

Proposed method

• Develops a new general capacity estimate for annuli in metric measure spaces under mild assumptions, valid even without global Poincaré inequality.
• Uses the capacity estimate to derive precise pointwise bounds for $p$-harmonic Green functions and their gradients.
• Applies the capacity estimate to characterize singletons of zero capacity and $p$-parabolicity of the space.
• Relies on the theory of $p$-weak upper gradients and $p$-energy minimization in the sense of upper gradients.
• Adapts and extends results from weighted $\mathbf{R}^n$ and manifolds by linking integrability to the measure's growth rate near the singularity.
• Employs Caccioppoli-type inequalities and truncation techniques to analyze local integrability of gradients.

Experimental results

Research questions

• RQ1 How does the growth of the measure near a singularity determine the $L^\tau$-integrability of $p$-harmonic Green functions?
• RQ2 What is the precise threshold $\tau_p$ for $L^\tau$-integrability of $p$-harmonic Green functions in terms of the measure's homogeneity dimension $\overline{s}_0$?
• RQ3 How do capacity estimates for annuli relate to $p$-parabolicity and the existence of zero-capacity points?
• RQ4 To what extent do the results on $p$-harmonic functions with poles extend to solutions of general elliptic equations in divergence form?
• RQ5 Can sharp integrability results for gradients be obtained under minimal geometric assumptions, such as doubling and Poincaré inequality?

Key findings

• The $L^\tau$-integrability of $p$-harmonic Green functions is sharp and determined by $\tau_p = \overline{s}_0(p-1)/\overline{s}_0$, where $\overline{s}_0$ is the critical growth exponent of the measure near the singularity.
• The minimal $p$-weak upper gradient of the Green function belongs to $L^t_{\rm loc}$ for all $t < t_p$, with $t_p = \overline{s}_0(p-1)/\overline{s}_0$, and fails to be in $L^{t_p}$, establishing sharpness.
• A new capacity estimate for annuli holds under only doubling and Poincaré inequality assumptions, enabling characterization of $p$-parabolicity and zero-capacity singletons.
• The results generalize to weighted $\mathbf{R}^n$ and Riemannian manifolds, with the same integrability thresholds determined by the measure's local growth.
• The capacity estimate implies that $p$-parabolicity is equivalent to the critical measure growth $\mu(B(x_0,r)) \lesssim r^{\overline{s}_0}$ with $\overline{s}_0 \leq p-1$.
• For solutions of elliptic equations in divergence form with Dirac-type sources, the same integrability and growth properties hold as for $p$-harmonic Green functions.