A Survey on the Div-Curl Lemma and Some Extensions to Fractional Sobolev Spaces

2025/09/15 by Maicol Caponi, Caponi, Maicol
Mathematics · #Differential Equations and Boundary Problems #Nonlinear Partial Differential Equations #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.2509.11734

Abstract

This survey is a chapter of a forthcoming book. This chapter recalls the classical formulation of the Div-Curl lemma along with its proof, and presents some possible generalizations in the fractional setting, within the framework of the Riesz fractional gradient and divergence introduced by Shieh and Spector (2015) and further developed by Comi and Stefani (2019).

Citations

• Quasiconvexity in the fractional calculus of variations: Characterization of lower semicontinuity and relaxation
• A short proof of local regularity of distributional solutions of Poisson's equation
• Fractional Piola identity and polyconvexity in fractional spaces
• On a new class of fractional partial differential equations II
• H-convergence result for nonlocal elliptic-type problems via Tartar's method
• On a new class of fractional partial differential equations

Related

• The Div-Curl Lemma Revisited
• A Functional Analytic Perspective to the div-curl Lemma
• Endpoint for the div-curl lemma in Hardy spaces
• On New Families of Fractional Sobolev Spaces
• A gentle invitation to the fractional world