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Quantitative unique continuation for elliptic equations with Hölder continuous potentials

Long Teng, Zhiwei Wang, Jiuyi Zhu

Abstract

We study quantitative unique continuation for second order elliptic equations with lower-order terms of Hölder regularity via a weighted frequency function method. We establish quantitative three-ball inequalities and corresponding vanishing-order bounds for Schrödinger equations with Hölder potentials and Hölder gradient terms, and corresponding results for elliptic equations with variable leading coefficients. Our results are quantitative with explicit dependence of Hölder norms in the three-ball inequalities. These fill in the gap for quantitative unique continuation between bounded potentials and $C^1$ potentials.

Quantitative unique continuation for elliptic equations with Hölder continuous potentials

Abstract

We study quantitative unique continuation for second order elliptic equations with lower-order terms of Hölder regularity via a weighted frequency function method. We establish quantitative three-ball inequalities and corresponding vanishing-order bounds for Schrödinger equations with Hölder potentials and Hölder gradient terms, and corresponding results for elliptic equations with variable leading coefficients. Our results are quantitative with explicit dependence of Hölder norms in the three-ball inequalities. These fill in the gap for quantitative unique continuation between bounded potentials and potentials.
Paper Structure (5 sections, 15 theorems, 219 equations)

This paper contains 5 sections, 15 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Three-ball inequality for classical Schrödinger equations
  3. Three-ball inequality for elliptic equations with drift term
  4. Three-ball inequality for elliptic equation with variable coefficients
  5. Appendix

Key Result

Theorem 1

Let $u$ be the solution of (shrod-1) with $R\geq 1$. Assume $\|V\|_{C^{0,\beta}(B_R)}\leq M$ for $0<\beta<1$ and some large constant $M$. Let $0<r_1<r_2<2r_2<r_3< \frac{R}{2}$. Then there exists $C=C(n,\beta)>0$ such that for $0<\theta=\frac{\log \frac{r_3}{2r_2}}{\log \frac{r_3}{r_1} }<1$.

Theorems & Definitions (28)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 18 more