Nonlocal gradients: Fundamental theorem of calculus, Poincaré inequalities and embeddings

2024/02/26 by José C. Bellido, Bellido, José Carlos, Carlos Mora‐Corral +3 · 3 citations
Engineering · Mathematics · #26A33 #46E35 #47G20 (Primary) 42B35 #74A70 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems #Thermoelastic and Magnetoelastic Phenomena

paper · pdf · doi:10.48550/arxiv.2402.16487

openalex publication_date 2024/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We address the study of nonlocal gradients defined through general radial kernels ρ. Our investigation focuses on the properties of the associated function spaces, which depend on the characteristics of the kernel function. Specifically, even with minimal assumptions on ρ, we establish Poincaré inequalities and compact embeddings into Lebesgue spaces. Additionally, we present a fundamental theorem of calculus that enables us to recover a function from its nonlocal gradient through a convolution. This is used to demonstrate embeddings into Orlicz spaces and spaces of continuous functions that mirror the well-known Sobolev and Morrey inequalities for classical gradients. Finally, we establish conditions for inclusions and equality of spaces associated to different kernels.

Cited by

• Recovering functions via doubly homogeneous nonlocal gradients
• A Lusin theorem for nonlocal gradients
• Local boundary conditions in nonlocal hyperelasticity via heterogeneous horizons

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