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

Abstract

We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite pth power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global p-Poincaré 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 R, we characterize all locally doubling measures, supporting a local p-Poincaré 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.

Bounded Geometry and p-Harmonic Functions Under Uniformization and Hyperbolization

“…The characterization is given in terms of the nonexistence of two disjoint compact sets of positive p-capacity in the boundary of the uniformized space, see Theorem 10.5. This characterization complements our results in [12].…”

“…Note that when w ≡ 1, both (10.6) and its analogue for z − fail, showing that the unweighted strip R×[−1, 1] satisfies the finite-energy Liouville theorem. This special case was obtained in Björn-Björn-Shanmugalingam [12] by a more direct method, without the use of uniformization.…”

“…In [12], this question was studied by different methods and under local assumptions on w. It follows from the results in Björn-Björn-Shanmugalingam [11] on local A p weights, that the condition…”

Classification of metric measure spaces and their ends using $p$-harmonic functions

“…Under global assumptions, the Liouville theorem (Theorem 3.2) for positive quasiharmonic functions on metric spaces was obtained in Kinnunen-Shanmugalingam [45]. In [13], we proved the so-called finite-energy Liouville theorem for noncomplete spaces with global assumptions under various additional assumptions. We can now deduce the Liouville theorem for finite-energy quasiharmonic functions without those additional assumptions, as a direct consequence of our identity O p QBD = O p QD (and Theorem 3.2) provided that X is complete, see Corollary 6.2.…”

“…However, those functions are solutions to ∆u = 1, while our quasiharmonic functions are continuous quasiminimizers of the p-energy. Such quasiminimizers were introduced in Giaquinta-Giusti [23], [24] as a unified treatment of variational inequalities, elliptic partial differential equations and quasiregular mappings, see [13] and [16] for further discussion and references.…”