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

Sylvester Eriksson-Bique, Saara Sarsa

TL;DR

The work establishes a duality framework for the $p$-Dirichlet problem in metric measure spaces by connecting $p$-harmonic gradients to a generalized modulus problem. The primal problem minimizes an $L^p$-density subject to a curve-integral bound, while the dual problem maximizes a boundary-term over measures on curves, with the transpose operator linking the two. Under suitable compactness and positivity assumptions, the authors prove exact duality and construct a measure $\eta^*$ supported on gradient curves, thereby obtaining the existence of gradient curves for $p$-harmonic functions in this non-smooth setting. This duality also enables a counterexample to the sheaf property in metric spaces, illustrating that $p$-harmonicity is not purely local in general. Altogether, the paper provides a robust geometric-analytic framework tying gradient-curves, currents, and modulus into a coherent theory in metric measure 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.

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

TL;DR

The work establishes a duality framework for the -Dirichlet problem in metric measure spaces by connecting -harmonic gradients to a generalized modulus problem. The primal problem minimizes an -density subject to a curve-integral bound, while the dual problem maximizes a boundary-term over measures on curves, with the transpose operator linking the two. Under suitable compactness and positivity assumptions, the authors prove exact duality and construct a measure supported on gradient curves, thereby obtaining the existence of gradient curves for -harmonic functions in this non-smooth setting. This duality also enables a counterexample to the sheaf property in metric spaces, illustrating that -harmonicity is not purely local in general. Altogether, the paper provides a robust geometric-analytic framework tying gradient-curves, currents, and modulus into a coherent theory in metric measure spaces.

Abstract

Every convex optimization problem has a dual problem. The -Dirichlet problem in metric measure spaces is an optimization problem whose solutions are -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 -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 -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 -harmonic is not strictly speaking a local property.

Paper Structure

This paper contains 28 sections, 46 theorems, 233 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Gradient curves
  4. Dual problem
  5. Sheaf property
  6. Structure of paper
  7. Preliminaries
  8. Notation for basic function spaces
  9. Rectifiable curves
  10. Newtonian space, gradient curves and p-harmonic functions
  11. Space of rectifiable curves as a metric space
  12. Properties of CurvX
  13. Borel measures and the space of Radon measures
  14. Path integration operator A and its formal transpose AT
  15. Lower semi-continuity results for A and AT
  16. ...and 13 more sections

Key Result

Theorem 1.2

Let $X=(X,d,\mu)$ be a complete, doubling $p$-Poincaré space, $1<p<\infty$. Suppose that $\Omega'\subset X$ is a domain and $u\in N^{1,p}_{\textnormal{loc}}(\Omega')$ a $p$-harmonic function. Let us fix a bounded subdomain $\Omega\subset\subset\Omega'$ such that $\mu(\partial\Omega)=0$ and $u$ is no such that the following hold:

Figures (2)

  • Figure 1: The domain $\Omega$ that is used in Example \ref{['ex:non-Radonness']}.
  • Figure 2: Figure of the definition of $\phi$.

Theorems & Definitions (95)

  • Remark 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Example 1.5
  • Theorem 1.6
  • Lemma 2.1: Fuglede's lemma
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4: Bjorn2011*Definition 7.7
  • ...and 85 more