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

Menglan Liao, Baisheng Yan

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.

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

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~ introduced there also ensures the existence of Lipschitz weak solutions that are nowhere for the corresponding time-independent equations in bounded domains, under suitable boundary data. In particular, for the smooth strongly polyconvex functions on constructed in that paper for all , the associated Euler--Lagrange equations admit Lipschitz weak solutions that are nowhere and satisfy zero boundary conditions in any bounded domain of . Our approach relies on new building blocks constructed from the same wave cone and -configurations employed in the analysis of diffusion equations.
Paper Structure (9 sections, 8 theorems, 208 equations, 1 figure)

This paper contains 9 sections, 8 theorems, 208 equations, 1 figure.

Table of Contents

  1. Introduction and Main Results
  2. Convex Integration, $\mathcal{T}_N$-configurations and Condition $O_N$
  3. Notation
  4. Convex Integration Lemma
  5. $\mathcal{T}_N$-Configurations and Condition $O_N$
  6. Proof of Theorem \ref{['mainthm']}
  7. Main stage theorem
  8. Constructions of $\mathcal{U}_\nu$ and $(u_\nu,V_\nu)$
  9. Proof of Theorem \ref{['mainthm']}

Key Result

Theorem 1.1

Let $m,n \ge 1$, and let $\sigma\colon \mathbb R^{m\times n}\to \mathbb R^{m\times n}$ be locally Lipschitz and satisfy Condition $O_N$ for some $N\ge 2$. Let $\Sigma(1)$ denote the open set defined in Definition O-N. Assume that $\bar{u} \in C^1(\bar{\Omega};\mathbb R^m)$ and $\bar{V} \in C(\bar{\O Here and throughout, the divergence-free condition is understood in the sense of distributions. The

Figures (1)

  • Figure 1: Illustration of Theorem \ref{['lem1']} for $N=5$. Here $i=3$, and $Y$ lies on the (ultra-thick) line segment $[\zeta_3(\lambda',\rho), \pi_3(\rho')]$. The sets $\mathcal{T}(X_1,\dots,X_N)$ and $\mathcal{T}(X'_1,\dots,X'_N)$ are represented by the corresponding line segments. The points $X_j, X'_j$ lie in $S^r_j(\mu)$, and all segments shown lie in $\Sigma^r(\mu)$. Only these points and segments are used in the proof.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • proof : Proof of Theorem \ref{['mainthm2']}
  • Lemma 2.1
  • proof
  • Definition 2.1
  • Remark 2.1
  • Theorem 2.2
  • proof
  • ...and 9 more