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Propagation of smallness for solutions of elliptic equations in the plane

Yuzhe Zhu

TL;DR

This work develops quantitative propagation of smallness for two-dimensional elliptic equations from sets with positive $\mathcal{H}_\delta$ content, delivering bounds for both solutions and their gradients under rough coefficient regimes. The authors leverage the plane’s quasiregular/holomorphic structure by introducing the $A$-harmonic conjugate and a holomorphic representation $f=F\circ\chi$, applying a holomorphic propagation inequality alongside a Schauder-type perturbation to transfer information from small sets. They establish explicit propagation bounds for $u$ and $\nabla u$ in both divergence-form equations with Hölder continuous coefficients and nondivergence-form equations with merely measurable coefficients, with constants depending only on ellipticity, Hölder exponent, and the $\mathcal{H}_\delta$-content of the relevant set. The results connect to unique continuation, spectral inequalities, and null controllability in the plane, offering sharp, dimension-2 specific quantitative estimates for rough-coefficient elliptic PDEs. These findings have potential implications for observability and control problems where data are available only on highly irregular, lower-dimensional sets.

Abstract

We explore quantitative propagation of smallness for solutions of two-dimensional elliptic equations from sets of positive $δ$-dimensional Hausdorff content for any $δ>0$. In particular, the gradients of solutions to divergence form equations with Hölder continuous coefficients, as well as those of nondivergence form equations with measurable coefficients, can be quantitatively estimated from the small sets.

Propagation of smallness for solutions of elliptic equations in the plane

TL;DR

This work develops quantitative propagation of smallness for two-dimensional elliptic equations from sets with positive content, delivering bounds for both solutions and their gradients under rough coefficient regimes. The authors leverage the plane’s quasiregular/holomorphic structure by introducing the -harmonic conjugate and a holomorphic representation , applying a holomorphic propagation inequality alongside a Schauder-type perturbation to transfer information from small sets. They establish explicit propagation bounds for and in both divergence-form equations with Hölder continuous coefficients and nondivergence-form equations with merely measurable coefficients, with constants depending only on ellipticity, Hölder exponent, and the -content of the relevant set. The results connect to unique continuation, spectral inequalities, and null controllability in the plane, offering sharp, dimension-2 specific quantitative estimates for rough-coefficient elliptic PDEs. These findings have potential implications for observability and control problems where data are available only on highly irregular, lower-dimensional sets.

Abstract

We explore quantitative propagation of smallness for solutions of two-dimensional elliptic equations from sets of positive -dimensional Hausdorff content for any . In particular, the gradients of solutions to divergence form equations with Hölder continuous coefficients, as well as those of nondivergence form equations with measurable coefficients, can be quantitatively estimated from the small sets.
Paper Structure (6 sections, 7 theorems, 53 equations)

This paper contains 6 sections, 7 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Propagation of smallness with rough coefficients
  3. Propagation of smallness for gradients
  4. Elliptic equations in divergence form
  5. Elliptic equations in nondivergence form
  6. Spectral inequality and null controllability

Key Result

Theorem 1.1

Let $\delta>0$ and $\omega\subset B_1\cap l_0$ satisfy $\mathcal{H}_\delta(\omega)>0$ for some line $l_0$ in $\mathbb{R}^2$ with the normal vector ${\boldsymbol{e}}_0$. There exist some constants $C$ and $\alpha>0$ depending only on $\Lambda$, $\delta$ and $\mathcal{H}_\delta(\omega)$ such that for

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • proof : Proof of Theorem \ref{['rough-u']}
  • Proposition A.1
  • ...and 2 more