Almost Lipschitz regularity for solutions of elliptic equations with discontinuous coefficients
Abstract
We are interested in the local higher integrability of solutions to elliptic equations with linear growth of the form − divA(x,Du)=f(x). Under a Besov regularity on the partial map x ↦ A(x,ξ) and a summability condition on the datum f in the scale of Lebesgue spaces, we prove that the solutions are almost Lipschitz continuous, i.e. their gradients belong locally to Lq, for any finite exponent q. In turn, solutions are locally γ-Hölder continuous, for every γ∈ (0,1). The difficulty arising from the lack of an explicit second variation for the problem is overcome by testing the equation with a function proportional to a power of the finite difference quotient of the solution. To the best of our knowledge, this technique is used in this context for the first time. We also provide an example showing the sharpness of our result in the scale of Lebesgue spaces.
Related
- Existence, Uniqueness, and Pathwise Regularity for Multidimensional Semilinear SPDEs with Locally Lipschitz Coefficients and Rough Initial Data
- Stochastic heat equations with non-Lipschitz coefficients driven by symmetric α-stable Lévy white noise
- Regularity theorems for Lipschitz viscosity solutions to the minimal surface system
- Wave Equations with Lipschitz Nonlinearity of Combined Power and Logarithmic Type: Existence of Global Solutions
- Wasserstein regularity of vorticity solutions to the 2D Navier-Stokes equations
BibTeX
@misc{giova2026almost,
title = {{Almost Lipschitz regularity for solutions of elliptic equations with discontinuous coefficients}},
author = {Raffaella Giova and Antonio Giuseppe Grimaldi and Stefania Russo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.26642},
eprint = {2606.26642},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.26642},
}
RIS
TY - UNPB
TI - Almost Lipschitz regularity for solutions of elliptic equations with discontinuous coefficients
AU - Raffaella Giova
AU - Antonio Giuseppe Grimaldi
AU - Stefania Russo
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2606.26642
UR - https://arxiv.org/abs/2606.26642
AB - We are interested in the local higher integrability of solutions to elliptic equations with linear growth of the form − divA(x,Du)=f(x). Under a Besov regularity on the partial map x ↦ A(x,ξ) and a summability condition on the datum f in the scale of Lebesgue spaces, we prove that the solutions are almost Lipschitz continuous, i.e. their gradients belong locally to L^q, for any finite exponent q. In turn, solutions are locally γ-Hölder continuous, for every γ∈ (0,1). The difficulty arising from the lack of an explicit second variation for the problem is overcome by testing the equation with a function proportional to a power of the finite difference quotient of the solution. To the best of our knowledge, this technique is used in this context for the first time. We also provide an example showing the sharpness of our result in the scale of Lebesgue spaces.
ER -