Sharp regularity results for eikonal equations in Banach spaces
Abstract
Viscosity solutions of eikonal equations need not enjoy regularity beyond Lipschitz continuity. In contrast, differentiable solutions exhibit stronger structural properties. In the Euclidean setting, building on observations of Caffarelli and Crandall (Comm. Partial Differential Equations 35 (2010), 391--414), one finds that differentiable solutions of the eikonal equation | Df | ≡ 1 in 𝒰 ⊂ ℝd are of class C1, 1loc(𝒰). Moreover, if 𝒰 = ℝd, then f must be affine. This manuscript revisits these phenomena in the setting of (finite-- and infinite--dimensional) Banach spaces. Our results show how the geometry of the underlying Banach space determines the regularity and affineness of Gateaux--differentiable solutions of eikonal equations. As a geometric application in finite dimensions, we characterize differentiable functions whose gradient ranges are constrained to boundaries of convex bodies.
Related
- Stability and Perron's method for Monge solutions of eikonal equations in metric spaces
- Viscosity Solutions for Hamilton-Jacobi equations on Separable Banach Spaces
- An averaging method for periodic solutions of quasilinear equations in Banach spaces
- A Coupled System of p-Laplacian Langevin Equations with ψ-Hilfer Fractional Derivatives and Lebesgue–Stieltjes Integral Boundary Conditions in Weighted Banach Spaces
- Stochastic maximal Lp-regularity for non-autonomous evolution equations with fractional derivative in UMD spaces
BibTeX
@misc{le2026sharp,
title = {{Sharp regularity results for eikonal equations in Banach spaces}},
author = {Trí Minh Lê and Sebastián Tapia-García},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.20145},
eprint = {2609.20145},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20145},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp regularity results for eikonal equations in Banach spaces
AU - Trí Minh Lê
AU - Sebastián Tapia-García
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2609.20145
UR - https://arxiv.org/abs/2609.20145
AB - Viscosity solutions of eikonal equations need not enjoy regularity beyond Lipschitz continuity. In contrast, differentiable solutions exhibit stronger structural properties. In the Euclidean setting, building on observations of Caffarelli and Crandall (Comm. Partial Differential Equations 35 (2010), 391--414), one finds that differentiable solutions of the eikonal equation | Df | ≡ 1 in 𝒰 ⊂ ℝ^d are of class C^1, 1_loc(𝒰). Moreover, if 𝒰 = ℝ^d, then f must be affine. This manuscript revisits these phenomena in the setting of (finite-- and infinite--dimensional) Banach spaces. Our results show how the geometry of the underlying Banach space determines the regularity and affineness of Gateaux--differentiable solutions of eikonal equations. As a geometric application in finite dimensions, we characterize differentiable functions whose gradient ranges are constrained to boundaries of convex bodies.
ER -