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An $L^\infty$-variational problem involving the Fractional Laplacian

Simone Carano, Roger Moser

TL;DR

The paper addresses a nonlocal L∞-variational problem for the fractional Laplacian by proving the existence and uniqueness of an absolute minimiser $u_∞$ of $E_∞(u)=\|(-Δ)^s u\|_{L^∞}$ under exterior data. It uses a Gamma-convergence approach from the $p$-norm energies to the supremal energy, and derives a limiting fractional Euler–Lagrange system in which $(-Δ)^s u_∞ = e_∞ \mathrm{sgn} f_∞$ in $Ω$, with the density $f_∞ \in L^1(Ω)$ real-analytic and given as the restriction of an $s$-harmonic measure $μ$. A key feature is the appearance of the $s$-harmonic measure and the analytic density in the optimality condition, together with nonlocal boundary considerations that require whole-space analysis. The framework extends to general supremands $F(x,(-Δ)^s u)$ under structural conditions, preserving existence, uniqueness, and the analytic representation of the density. Overall, the work provides a fractional, nonlocal analogue of Aronsson-type results in $L^∞$-calculus and lays groundwork for further regularity and boundary-behavior studies.

Abstract

For $s\in(0,1)$ and an open bounded set $Ω\subset\mathbb R^n$, we prove existence and uniqueness of absolute minimisers of the supremal functional $$E_\infty(u)=\|(-Δ)^s u\|_{L^\infty(\mathbb R^n)},$$ where $(-Δ)^s$ is the Fractional Laplacian of order $s$ and $u$ has prescribed Dirichlet data in the complement of $Ω$. We further show that the minimiser $u_\infty$ satisfies the (fractional) PDE $$ (-Δ)^s u_\infty=E_\infty(u_\infty)\,\mathrm{sgn}f_\infty \qquad\mbox{in }Ω, $$ for some analytic function $f_\infty\in L^1(Ω)$ obtained as the restriction of an $s$-harmonic measure $μ$ in $Ω$.

An $L^\infty$-variational problem involving the Fractional Laplacian

TL;DR

The paper addresses a nonlocal L∞-variational problem for the fractional Laplacian by proving the existence and uniqueness of an absolute minimiser of under exterior data. It uses a Gamma-convergence approach from the -norm energies to the supremal energy, and derives a limiting fractional Euler–Lagrange system in which in , with the density real-analytic and given as the restriction of an -harmonic measure . A key feature is the appearance of the -harmonic measure and the analytic density in the optimality condition, together with nonlocal boundary considerations that require whole-space analysis. The framework extends to general supremands under structural conditions, preserving existence, uniqueness, and the analytic representation of the density. Overall, the work provides a fractional, nonlocal analogue of Aronsson-type results in -calculus and lays groundwork for further regularity and boundary-behavior studies.

Abstract

For and an open bounded set , we prove existence and uniqueness of absolute minimisers of the supremal functional where is the Fractional Laplacian of order and has prescribed Dirichlet data in the complement of . We further show that the minimiser satisfies the (fractional) PDE for some analytic function obtained as the restriction of an -harmonic measure in .
Paper Structure (7 sections, 7 theorems, 113 equations)

This paper contains 7 sections, 7 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Proof of the main results
  4. Existence
  5. PDE derivation
  6. Uniqueness
  7. More general supremands

Key Result

Theorem 1.1

Fix $s\in (0,1)$ and $n\in\mathbb{N}$, $n>2s$. Let $\Omega\subset\mathbb{R}^n$ be an open bounded set and assume that $u_0\in C^{2s+\gamma}_c(\mathbb{R}^n)$, for some $\gamma>0$, with $u_0\not\equiv0$ in $\mathbb{R}^n\setminus\Omega$. Then the problem admits a unique solution $u_\infty\in\mathcal{W}_{u_0}^{2s,\infty}(\Omega)$. In particular, $(-\Delta)^s u_\infty\in C^\gamma_{\mathrm{loc}}(\mathb

Theorems & Definitions (12)

  • Theorem 1.1
  • Lemma 3.1
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • proof
  • Remark 3.5
  • Lemma 3.6
  • proof
  • ...and 2 more