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Unique continuation for a gradient inequality with $L^n$ potential

Adam Coffman, Yifei Pan, Yuan Zhang

TL;DR

This work establishes sharp unique continuation properties for solutions of $|\nabla u| \le V|u|$ with $V$ in the critical space $L^n_{loc}$, clarifying the boundary between propagation and vanishing for real-analytic and Lipschitz functions. The authors combine Hardy-type inequalities, Sobolev embeddings, and a $\bar{\partial}$-based reduction in two dimensions to prove strong unique continuation for $H^1_{loc}$ solutions, and extend the analysis to locally Lipschitz functions where the zero set propagates under the same inequality. They also derive finite-order vanishing results for exponentials of $W^{1,n}$ functions via a Moser-Trudinger framework and develop several applications, including logarithmic obstructions and extension properties. A dedicated discussion in the last section connects these real-variable results to the $\bar{\partial}$-problem, showing that replacing gradient by $\bar{\partial}$ can destroy UCP and characterizing vanishing orders for holomorphic extensions across boundaries. Together, the results illuminate the critical role of the $L^n$-threshold for potentials and reveal intrinsic obstructions to smooth analytic extension in complex and several complex variable settings.

Abstract

We establish a unique continuation property for solutions of the differential inequality $|\nabla u|\leq V|u|$, where $V$ is locally $L^n$ integrable on a domain in $\mathbb R^n$. A stronger uniqueness result is obtained if in addition the solutions are locally Lipschitz. One application is a finite order vanishing property in the $L^2$ sense for the exponential of $W^{1,n}$ functions. We further discuss related results for the Cauchy-Riemann operator $\bar\partial$ and characterize the vanishing order for smooth extension of holomorphic functions across the boundary.

Unique continuation for a gradient inequality with $L^n$ potential

TL;DR

This work establishes sharp unique continuation properties for solutions of with in the critical space , clarifying the boundary between propagation and vanishing for real-analytic and Lipschitz functions. The authors combine Hardy-type inequalities, Sobolev embeddings, and a -based reduction in two dimensions to prove strong unique continuation for solutions, and extend the analysis to locally Lipschitz functions where the zero set propagates under the same inequality. They also derive finite-order vanishing results for exponentials of functions via a Moser-Trudinger framework and develop several applications, including logarithmic obstructions and extension properties. A dedicated discussion in the last section connects these real-variable results to the -problem, showing that replacing gradient by can destroy UCP and characterizing vanishing orders for holomorphic extensions across boundaries. Together, the results illuminate the critical role of the -threshold for potentials and reveal intrinsic obstructions to smooth analytic extension in complex and several complex variable settings.

Abstract

We establish a unique continuation property for solutions of the differential inequality , where is locally integrable on a domain in . A stronger uniqueness result is obtained if in addition the solutions are locally Lipschitz. One application is a finite order vanishing property in the sense for the exponential of functions. We further discuss related results for the Cauchy-Riemann operator and characterize the vanishing order for smooth extension of holomorphic functions across the boundary.
Paper Structure (5 sections, 26 theorems, 91 equations)

This paper contains 5 sections, 26 theorems, 91 equations.

Table of Contents

  1. Introduction and Results
  2. Unique continuation for $H^1$ solutions
  3. Uniqueness for Lipschitz functions
  4. Further applications
  5. In relation to $\bar{\partial}$

Key Result

Theorem 1.1

Let $\Omega$ be a domain in $\mathbb R^n, n\ge 2$ and $V\in L_{loc}^n(\Omega)$. Suppose $u=(u_1, \ldots, u_M): \Omega\rightarrow \mathbb R^M$ with $u\in H^1_{loc}(\Omega)$ and satisfies $|\nabla u|\le V|u|$ a.e. on $\Omega$. If $u$ vanishes to infinite order at some $x_0\in \Omega$, then $u\equiv 0$

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof : Proof of Theorem \ref{['main']}:
  • Example 2.4
  • ...and 45 more