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Exponential Stability for Maxwell-type Systems Revisited

Marcus Waurick

Abstract

Considering a two-by-two block operator matrix system of Maxwell type, we present an elementary way of deducing exponential stability under minimal smoothness (and boundedness) requirements of the underlying domains when applications are concerned. The approach is based on resolvent estimates using block operator matrices.

Exponential Stability for Maxwell-type Systems Revisited

Abstract

Considering a two-by-two block operator matrix system of Maxwell type, we present an elementary way of deducing exponential stability under minimal smoothness (and boundedness) requirements of the underlying domains when applications are concerned. The approach is based on resolvent estimates using block operator matrices.
Paper Structure (7 sections, 15 theorems, 51 equations)

This paper contains 7 sections, 15 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Reduction to nice coefficients
  3. Well-posedness and the path to exponential stability
  4. A reduction procedure
  5. The special case of one-to-one and onto $C$
  6. Proof of the main theorem
  7. An application to Maxwell's equations with full damping

Key Result

Theorem 1.1

Let $(u_0,v_0)\in (H_0,\beta^{-1}\mathop{\mathrm{ran}}\nolimits(C))$. Then there is $\delta>0$ such that for any mild solution $U \in C[0,\infty;H_0\times H_1)$ of eq:absMax, we have

Theorems & Definitions (32)

  • Theorem 1.1: EKL24 or Tro15DIW24
  • Lemma 2.1
  • remark 2.2
  • remark 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3: Pr84
  • remark 4.3
  • Theorem 4.4
  • proof
  • ...and 22 more