Duality for the gradient of a $p$-harmonic function and the existence of gradient curves

TL;DR

This paper establishes a duality framework for the gradient of $p$-harmonic functions in metric measure spaces, introducing a generalized modulus problem and proving the existence of gradient curves and a dual metric current.

Contribution

It formulates a dual problem for the $p$-Dirichlet problem in metric spaces, demonstrating exact duality, existence of minimizers, and the construction of gradient curves and a dual metric current.

Findings

Dual problem for $p$-harmonic functions is formulated.

Existence of gradient curves supported on minimizers.

Counterexample to local $p$-harmonicity in metric spaces.

Abstract

Every convex optimization problem has a dual problem. The $p$-Dirichlet problem in metric measure spaces is an optimization problem whose solutions are $p$-harmonic functions. What is its dual problem? In this paper, we give an answer to this problem in the following form. We give a generalized modulus problem whose solution is the gradient of the $p$-harmonic function for metric measure spaces. Its dual problem is an optimization problem for measures on curves and we show exact duality and the existence of minimizers for this dual problem under appropriate assumptions. When applied to $p$-harmonic functions the minimizers of this dual problem are supported on gradient curves, yielding a natural concept associated to such functions that has yet to be studied. This process defines a natural dual metric current and proves the existence of gradient curves. These insights are then used to construct a counter example answering the old ``sheaf problem'' on metric spaces: in contrast to Euclidean spaces, in general metric spaces being $p$-harmonic is not strictly speaking a local property.