An L∞-variational problem involving the fractional Laplacian

Simone Carano; Roger Moser

doi:10.48550/arXiv.2510.14476

An L^∞-variational problem involving the fractional Laplacian

Simone Carano
, Roger Moser

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

For s ∈ (0, 1) and an open bounded set Ω⊂Rⁿ, we prove existence and uniqueness of absolute minimisers of the supremal functional E_∞(u)=∥(−Δ)su∥_{L_∞_{₍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 _∞ satisfies the (fractional) PDE (−Δ)^su_∞=E_∞(u_∞)sgnf_∞inΩ, for some analytic function f _∞ ∈ L ¹(Ω) obtained as the restriction of an s -harmonic measure μ in Ω.

Original language	English
Article number	114194
Number of pages	16
Journal	Nonlinear Analysis
Volume	272
Early online date	5 Jun 2026
DOIs	https://doi.org/10.48550/arXiv.2510.14476 https://doi.org/10.1016/j.na.2026.114194
Publication status	E-pub ahead of print - 5 Jun 2026

Data Availability Statement

Our manuscript has no associated data. No data was used for the research described in the article

Acknowledgements

S.C. is grateful to Lorenzo Brasco, Alessandro Carbotti, Serena Dipierro and Enrico Valdinoci for useful discussion and suggestions. S.C. is a member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) of Italy.

Funding

S.C and R.M. acknowledge partial financial support through the EPSRC grant EP/X017206/1. S.C acknowledges partial financial support through the EPSRC grant EP/X017109/1.

Keywords

Calculus of variations in 𝐿∞
Euler-Lagrange equations
Fractional Laplacian
Gamma convergence
�-harmonic measures

Access to Document

10.48550/arXiv.2510.14476
10.1016/j.na.2026.114194Licence: CC BY

Cite this

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title = "An L∞-variational problem involving the fractional Laplacian",

abstract = "For s ∈ (0, 1) and an open bounded set Ω⊂Rn, we prove existence and uniqueness of absolute minimisers of the supremal functional E∞(u)=∥(−Δ)su∥L∞(Rn), 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 ∞ satisfies the (fractional) PDE (−Δ)su∞=E∞(u∞)sgnf∞inΩ, for some analytic function f ∞ ∈ L 1(Ω) obtained as the restriction of an s -harmonic measure μ in Ω.",

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N2 - For s ∈ (0, 1) and an open bounded set Ω⊂Rn, we prove existence and uniqueness of absolute minimisers of the supremal functional E∞(u)=∥(−Δ)su∥L∞(Rn), 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 ∞ satisfies the (fractional) PDE (−Δ)su∞=E∞(u∞)sgnf∞inΩ, for some analytic function f ∞ ∈ L 1(Ω) obtained as the restriction of an s -harmonic measure μ in Ω.

AB - For s ∈ (0, 1) and an open bounded set Ω⊂Rn, we prove existence and uniqueness of absolute minimisers of the supremal functional E∞(u)=∥(−Δ)su∥L∞(Rn), 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 ∞ satisfies the (fractional) PDE (−Δ)su∞=E∞(u∞)sgnf∞inΩ, for some analytic function f ∞ ∈ L 1(Ω) obtained as the restriction of an s -harmonic measure μ in Ω.

KW - Calculus of variations in 𝐿∞

KW - Euler-Lagrange equations

KW - Fractional Laplacian

KW - Gamma convergence

KW - �-harmonic measures

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