The Liouville theorem for $p$-harmonic functions and quasiminimizers with finite energy

TL;DR

This paper establishes Liouville-type theorems for $p$-harmonic functions and quasiminimizers with finite energy in metric measure spaces, showing nonexistence under certain geometric conditions and characterizing finite-energy quasiminimizers on the weighted real line.

Contribution

It provides new conditions under which nonconstant quasiminimizers with finite energy cannot exist and characterizes finite-energy quasiminimizers on weighted real lines.

Findings

No nonconstant quasiminimizers with finite energy under strong volume growth or annular quasiconvexity.

On weighted real lines, finite-energy quasiminimizers are exactly the bounded ones.

$p$-harmonic functions are included in these Liouville-type results.

Abstract

We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite $p$th power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global $p$-Poincar\'e inequality. The geometric conditions are that either (a) the measure has a sufficiently strong volume growth at infinity, or (b) the metric space is annularly quasiconvex (or its discrete version, annularly chainable) around some point in the space. Moreover, on the weighted real line $\mathbf{R}$, we characterize all locally doubling measures, supporting a local $p$-Poincar\'e inequality, for which there exist nonconstant quasiminimizers of finite $p$-energy, and show that a quasiminimizer is of finite $p$-energy if and only if it is bounded. As $p$-harmonic functions are quasiminimizers they are covered by these results.