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Non-uniqueness and failure of Calderón-Zygmund estimates below the critical exponent for non-monotone PDE with linear growth

Akshara Vincent

TL;DR

The authors construct a smooth, uniformly elliptic operator A with linear growth that is non-monotone and not asymptotically Uhlenbeck, and show that both uniqueness and Calderón–Zygmund estimates can fail in the near-linear regime. Using convex integration with laminates and piecewise affine maps, they produce nontrivial solutions to div(A(∇u))=0 that defy uniqueness below a threshold exponent bar q(θ), and they also produce counterexamples to a priori CZ estimates for r<bar q. Moreover, they extend the argument to produce CZ failure above 2, revealing a nuanced relationship between uniqueness and CZ theory in nonlinear elliptic equations. The results highlight sharp limits of CZ-type regularity and demonstrate the potential for non-uniqueness and CZ breakdown even when A is smooth, close to linear, and elliptic.

Abstract

We provide counterexamples to uniqueness of solutions as well as a priori Calderón-Zygmund estimates for solutions below $L^2$ using convex integration argument for equations of the type $$ \text{div} (A (\nabla u)) = 0 \quad \text{in } \mathbb{B}^2, $$ where $A: \mathbb{R}^{2} \to \mathbb{R}^2$ is smooth, uniformly elliptic and has essentially linear growth, but fails to be monotone and asymptotically Uhlenbeck.

Non-uniqueness and failure of Calderón-Zygmund estimates below the critical exponent for non-monotone PDE with linear growth

TL;DR

The authors construct a smooth, uniformly elliptic operator A with linear growth that is non-monotone and not asymptotically Uhlenbeck, and show that both uniqueness and Calderón–Zygmund estimates can fail in the near-linear regime. Using convex integration with laminates and piecewise affine maps, they produce nontrivial solutions to div(A(∇u))=0 that defy uniqueness below a threshold exponent bar q(θ), and they also produce counterexamples to a priori CZ estimates for r<bar q. Moreover, they extend the argument to produce CZ failure above 2, revealing a nuanced relationship between uniqueness and CZ theory in nonlinear elliptic equations. The results highlight sharp limits of CZ-type regularity and demonstrate the potential for non-uniqueness and CZ breakdown even when A is smooth, close to linear, and elliptic.

Abstract

We provide counterexamples to uniqueness of solutions as well as a priori Calderón-Zygmund estimates for solutions below using convex integration argument for equations of the type where is smooth, uniformly elliptic and has essentially linear growth, but fails to be monotone and asymptotically Uhlenbeck.
Paper Structure (7 sections, 22 theorems, 213 equations)

This paper contains 7 sections, 22 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Piecewise affine maps
  4. Laminates.
  5. CZ estimate
  6. non uniqueness
  7. CZ estimate

Key Result

Theorem 1.1

Let $\bar{q}$ as in eq for q bar. For any $r \in (1,\bar{q})$ there exists a non-zero function $u\in W^{1,r}_0(\mathbb{B}^2)$ such that

Theorems & Definitions (42)

  • Theorem 1.1: Non-uniqueness
  • Theorem 1.2: Failure of Calderón-Zygmund a priori estimate
  • Remark 1.3
  • Theorem 1.4: Failure of Calderón-Zygmund a priori estimate above 2
  • Remark 1.5
  • Definition 2.1: Piecewise affine maps
  • Definition 2.2
  • Lemma 2.3: Gluing argument
  • Definition 2.4: Elementary splitting and laminates of finite order
  • Proposition 2.5
  • ...and 32 more