Γ-convergence of energy functionals in fractional Orlicz spaces beyond the Δ₂ condition

2025/02/11 by Ignacio Ceresa Dussel, Dussel, Ignacio Ceresa, Julián Fernández Bonder +3
Mathematics · #35R11 #40E30 #74A70 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Approximation Theory and Sequence Spaces #FOS: Mathematics #Mathematical Inequalities and Applications

paper · pdf · doi:10.48550/arxiv.2502.07568

openalex publication_date 2025/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a Young function A, n≥ 1 and s∈(0,1) we consider the energy functional J_s(u)=(1-s)\iint_{ℝⁿ× ℝⁿ} A((|u(x)-u(y)|)/(|x-y|^s))(dxdy)/(|x-y|ⁿ). Without assuming the Δ₂ condition on A not its conjugated function A, we prove the following liminf inequality: if u∈ E^A(ℝⁿ) and \u_k\_k∈ℕ⊂ E^A(ℝⁿ) is such that u_k→ u in E^A(ℝⁿ), and s_k→ 1, then J(u) ≤ \liminf_k→∞ J_sk(u_k), where J is a limit functional related with the behavior of the fractional Orlicz-Sobolev spaces as s→ 1⁺. As a direct consequence, we obtain the Γ-convergence of the functional J_s. Finally, we extend our result to the study of the so called fractional peridynamic case.

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• Minimizers of abstract generalized Orlicz--bounded variation energy