Divergence-form equations admitting nowhere $C^1$ Lipschitz weak solutions

TL;DR

This paper demonstrates that certain divergence-form PDEs can have highly irregular Lipschitz weak solutions that are nowhere differentiable, using convex integration techniques adapted from irregular diffusion equations.

Contribution

It extends convex integration methods to divergence-form equations, showing the existence of nowhere $C^1$ Lipschitz solutions under structural conditions.

Findings

Existence of nowhere $C^1$ Lipschitz solutions for specific divergence-form equations.

Application of convex integration to divergence equations with irregular solutions.

Construction of solutions satisfying zero boundary conditions in bounded domains.

Abstract

We study a class of partial differential equations in divergence form that admit highly irregular Lipschitz weak solutions. By reformulating these divergence-form equations as a first-order partial differential relation and adapting the convex integration scheme recently developed in \cite{GKY26} for irregular diffusion equations, we show that the same structural Condition~$O_N$ introduced there also ensures the existence of Lipschitz weak solutions that are nowhere $C^1$ for the corresponding time-independent equations in bounded domains, under suitable boundary data. In particular, for the smooth strongly polyconvex functions on $\mathbb{R}^{2\times n}$ constructed in that paper for all $n \ge 2$, the associated Euler--Lagrange equations admit Lipschitz weak solutions that are nowhere $C^1$ and satisfy zero boundary conditions in any bounded domain of $\mathbb{R}^n$. Our approach relies on new building blocks constructed from the same wave cone and $\mathcal{T}_N$-configurations employed in the analysis of diffusion equations.