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On the Obstacle Problem in Fractional Generalised Orlicz Spaces

Catharine W. K. Lo, José Francisco Rodrigues

TL;DR

The paper addresses obstacle problems for the nonlocal nonlinear anisotropic g-Laplacian $\mathcal{L}_g^s$ in fractional generalized Orlicz spaces. It develops a comprehensive framework: proves strict $T$-monotonicity of $\mathcal{L}_g^s$, establishes Lewy-Stampacchia inequalities for one- and two-obstacle problems, and shows existence, uniqueness, and global $L^\infty$ bounds for the Dirichlet problem; it further provides a semilinear approximation scheme with rigorous convergence and transfers regularity to obstacle problems, including fractional $p(x,y)$-Laplacian cases. The work also introduces a fractional generalised Orlicz capacity, defines the $(s,G_{:})$-capacity and its capacitary potential, and compares it to the $s$-capacity in the Hilbertian setting, connecting obstacle problems with nonlinear potential theory. Overall, the results extend obstacle problem theory to nonlocal, anisotropic, and Musielak-Orlicz spaces, unifying Dirichlet, obstacle, and capacity analyses and paving the way for further potential-theoretic developments in this broad function space setting.

Abstract

We consider the one and the two obstacles problems for the nonlocal nonlinear anisotropic $g$-Laplacian $\mathcal{L}_g^s$, with $0<s<1$. We prove the strict T-monotonicity of $\mathcal{L}_g^s$ and we obtain the Lewy-Stampacchia inequalities. We consider the approximation of the solutions through semilinear problems, for which we prove a global $L^\infty$-estimate, and we extend the local Hölder regularity to the solutions of the obstacle problems in the case of the fractional $p(x,y)$-Laplacian operator. We make further remarks on a few elementary properties of related capacities in the fractional generalised Orlicz framework, with a special reference to the Hilbertian nonlinear case in fractional Sobolev spaces.

On the Obstacle Problem in Fractional Generalised Orlicz Spaces

TL;DR

The paper addresses obstacle problems for the nonlocal nonlinear anisotropic g-Laplacian in fractional generalized Orlicz spaces. It develops a comprehensive framework: proves strict -monotonicity of , establishes Lewy-Stampacchia inequalities for one- and two-obstacle problems, and shows existence, uniqueness, and global bounds for the Dirichlet problem; it further provides a semilinear approximation scheme with rigorous convergence and transfers regularity to obstacle problems, including fractional -Laplacian cases. The work also introduces a fractional generalised Orlicz capacity, defines the -capacity and its capacitary potential, and compares it to the -capacity in the Hilbertian setting, connecting obstacle problems with nonlinear potential theory. Overall, the results extend obstacle problem theory to nonlocal, anisotropic, and Musielak-Orlicz spaces, unifying Dirichlet, obstacle, and capacity analyses and paving the way for further potential-theoretic developments in this broad function space setting.

Abstract

We consider the one and the two obstacles problems for the nonlocal nonlinear anisotropic -Laplacian , with . We prove the strict T-monotonicity of and we obtain the Lewy-Stampacchia inequalities. We consider the approximation of the solutions through semilinear problems, for which we prove a global -estimate, and we extend the local Hölder regularity to the solutions of the obstacle problems in the case of the fractional -Laplacian operator. We make further remarks on a few elementary properties of related capacities in the fractional generalised Orlicz framework, with a special reference to the Hilbertian nonlinear case in fractional Sobolev spaces.
Paper Structure (13 sections, 25 theorems, 133 equations)

This paper contains 13 sections, 25 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Fractional Generalised Orlicz Functional Framework
  4. The Quasilinear Fractional Dirichlet Problem
  5. Quasilinear Fractional Obstacle Problems
  6. T-monotonicity and Comparison Properties
  7. Lewy-Stampacchia Inequalities for Obstacle Problems
  8. Approximation by Semilinear Problems and Regularity
  9. Approximation via Bounded Penalisation
  10. Regularity in Obstacle Problems
  11. Capacities
  12. The Fractional Generalised Orlicz Capacity
  13. The $s$-Capacity in the $H^s_0(\Omega)$ Hilbertian Nonlinear Framework

Key Result

Lemma 2.1

Let $s\in]0,1[$ and $\Omega$ be a bounded open subset of $\mathbb{R}^d$ with a Lipschitz bounded boundary. Then there exists a constant $C=C(s,d,\Omega)>0$ such that for all $u\in W^{s,G_{:}}_0(\Omega)$. Therefore, the embedding is continuous. Furthermore, $[u]_{s,G}$ is an equivalent norm to $\norm{u}_{s,G}$ for the fractional generalised Orlicz space $W^{s,G_{:}}_0(\Omega)$.

Theorems & Definitions (56)

  • Lemma 2.1: Corollary of Theorem 2.3 of AzroulBenkiraneShimiSrati2021-EmbeddingFractionalMusielakSobolev
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Proposition 2.5: Lemma 2.3 of AzroulBenkiraneShimiSrati2021-EmbeddingFractionalMusielakSobolev
  • Corollary 2.6
  • Remark 2.7
  • Proposition 2.8
  • Theorem 2.9
  • Lemma 2.10
  • ...and 46 more