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Regularity of solutions to time-harmonic Maxwell's system with Hölder and various lower than Hölder continuous coefficients

Lei Yu, Basang Tsering-xiao

Abstract

The purpose of this paper is to establish a complete Schauder theory for the second-order linear elliptic equation and the time-harmonic Maxwell's system. We prove global Hölder regularity for the solutions to the time-harmonic anisotropic Maxwell's equations under Hölder continuous coefficients, raising the Hölder index to the interval (0,1)

Regularity of solutions to time-harmonic Maxwell's system with Hölder and various lower than Hölder continuous coefficients

Abstract

The purpose of this paper is to establish a complete Schauder theory for the second-order linear elliptic equation and the time-harmonic Maxwell's system. We prove global Hölder regularity for the solutions to the time-harmonic anisotropic Maxwell's equations under Hölder continuous coefficients, raising the Hölder index to the interval (0,1)

Paper Structure

This paper contains 11 sections, 17 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminary results
  4. Proof of results
  5. Proof of Proposition \ref{['4']}.
  6. Proof of Theorem \ref{['m']}.
  7. Proof of Proposition \ref{['f']}.
  8. Proof of Theorem \ref{['d']}.
  9. Proof of Theorem \ref{['b']}.
  10. Proof of Theorem \ref{['c']}.
  11. Appendix

Key Result

Theorem 1

Assume that 2 holds true and $\varepsilon(x),\mu(x) \in C^{0,\alpha} (\bar{\Omega},\mathbb{R}^3)$, for some $\alpha \in (0,1)$, take $J_{e}, J_{m}\in C^{0,\alpha}(\overline{\Omega},\mathbb{R}^3$) and $G\in C^{1,\alpha}(\operatorname{curl},\Omega)$, Let $(E,H)\in H(curl,\Omega)$ be a weak solution of for some constant C depending only on $\Omega$, $\left\|\varepsilon\right\|_{C^{0,\alpha}\left(\ove

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Remark 3
  • Theorem 4
  • Theorem 5
  • Remark 6
  • Proposition 7
  • Proposition 8
  • Proposition 9
  • Proposition 10
  • ...and 16 more