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$L^q$-Carleman estimates with boundary observations and applications to inverse problems

Elena-Alexandra Melnig

Abstract

We consider coupled linear parabolic systems and we establish estimates in $L^q$-norm for the sources in terms of observations on the corresponding solutions on a part of the boundary. The main tool is a family of Carleman estimates in $L^q$-norm with boundary observations.

$L^q$-Carleman estimates with boundary observations and applications to inverse problems

Abstract

We consider coupled linear parabolic systems and we establish estimates in -norm for the sources in terms of observations on the corresponding solutions on a part of the boundary. The main tool is a family of Carleman estimates in -norm with boundary observations.
Paper Structure (3 sections, 6 theorems, 58 equations)

This paper contains 3 sections, 6 theorems, 58 equations.

Table of Contents

  1. Preliminaries and main results
  2. $L^q-L^2$-Carleman estimates with boundary observations
  3. Source stability for linear systems. Proof of Theorem \ref{['ThLq']}

Key Result

Theorem 1

Consider the system sysinitial with hypotheses $(H1)-(H5)$. Then for $2\le q<\infty$, $g\in \mathcal{G}_{q,\tilde{\delta},\tilde{G}}$ and the corresponding solution $y\in W^{2,1}_q(Q)$, there exists $C=C(q,\tilde{\delta},\tilde{G} )>0$ such that Concerning the $L^\infty$ estimates, for sources $g\in \mathcal{G}_{q,\tilde{\delta},\tilde{G}}$ and corresponding solutions $y\in W^{2,1}_q(Q)$ for all

Theorems & Definitions (9)

  • Theorem 1
  • Remark 1
  • Proposition 1
  • Proposition 2
  • Lemma 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Proposition 3