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Doubling Inequality and Strong Unique Continuation for an Elliptic Transmission Problem

Tianrui Dai, Elisa Francini, Sergio Vessella

TL;DR

The work addresses strong unique continuation for elliptic transmission problems with jump discontinuities across a regular interface. It combines a flattening transformation with a Carleman estimate using a singular weight to obtain a doubling inequality, which in turn implies SUCP at the interface. This provides a quantitative propagation-of-smallness result and lays groundwork for inverse boundary problems aimed at estimating the size of an unknown inclusion from boundary measurements. The approach reinforces the connection between Carleman-based inequalities and energy-based size estimates in composite media, and it extends SUCP results to settings with measureable inclusions and piecewise Lipschitz coefficients.

Abstract

We investigate the Strong Unique Continuation Property (SUCP) for elliptic equations with piecewise Lipschitz coefficients exhibiting jump discontinuities across a regular interface. We prove SUCP at the interface using a doubling inequality derived from a Carleman estimate with a singular weight. This result is intended as a first step toward solving the inverse problem of estimating the size of an unknown, merely measurable, inclusion inside a conductor from boundary measurements.

Doubling Inequality and Strong Unique Continuation for an Elliptic Transmission Problem

TL;DR

The work addresses strong unique continuation for elliptic transmission problems with jump discontinuities across a regular interface. It combines a flattening transformation with a Carleman estimate using a singular weight to obtain a doubling inequality, which in turn implies SUCP at the interface. This provides a quantitative propagation-of-smallness result and lays groundwork for inverse boundary problems aimed at estimating the size of an unknown inclusion from boundary measurements. The approach reinforces the connection between Carleman-based inequalities and energy-based size estimates in composite media, and it extends SUCP results to settings with measureable inclusions and piecewise Lipschitz coefficients.

Abstract

We investigate the Strong Unique Continuation Property (SUCP) for elliptic equations with piecewise Lipschitz coefficients exhibiting jump discontinuities across a regular interface. We prove SUCP at the interface using a doubling inequality derived from a Carleman estimate with a singular weight. This result is intended as a first step toward solving the inverse problem of estimating the size of an unknown, merely measurable, inclusion inside a conductor from boundary measurements.

Paper Structure

This paper contains 7 sections, 10 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Notation and main result for flat interface
  3. Proof of Theorem \ref{['teopiatta']}
  4. Pointwise estimates for $f$
  5. Integral estimates for $f$
  6. Proof of Theorem \ref{['teopiatta']}
  7. Doubling inequality

Key Result

Theorem 2.1

Let $g$ satisfy assumptions ellipticity, lipg, ginzero, and gatzero, let $\gamma^+\in C^{0,1}(B_1^+)$ and $\gamma^-\in C^{0,1}(B_1^-)$ such that $\gamma^\pm\geq \gamma_0>0$ in $B_1^\pm$ and let $w$ as in w. Then, there exist $\overline{r}\in(0,1)$ and $\tau_0>0$, depending only on $\lambda$, $\Lambd where $\gamma=\gamma^+\chi_{B_1^+}+\gamma^-\chi_{B_1^-}$ and $[\cdot]$ is defined as in jump. Notic

Theorems & Definitions (20)

  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 10 more