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Analyticity up to the boundary for the divergence equation

Igor Kukavica, Qi Xu

Abstract

We address analytic regularity for the divergence equation $\text{div}\, u = f$ in $Ω$, with $u=0$ on $\partialΩ$, where $Ω$ is an arbitrary bounded analytic domain and $\int_Ω f\,dx=0$. If $f$ is analytic on the closure of $Ω$, then we prove that there exists a solution that is analytic on the closure of $Ω$.

Analyticity up to the boundary for the divergence equation

Abstract

We address analytic regularity for the divergence equation in , with on , where is an arbitrary bounded analytic domain and . If is analytic on the closure of , then we prove that there exists a solution that is analytic on the closure of .

Paper Structure

This paper contains 14 sections, 6 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Analytic vector fields in a bounded domain
  3. Analytic vector fields
  4. Equivalent norm of analytic functions
  5. Reduction of the Divergence Equation to the Stokes System
  6. Derivative reductions for the Stokes system
  7. Normal derivative reductions
  8. Tangential derivative reduction
  9. Leibniz-type formulae
  10. Analyticity for the stationary Stokes system
  11. The sum $S_1$
  12. The $S_2$ term
  13. The $S_3$ term
  14. Conclusion of the proof

Key Result

Theorem 1.1

Assume that $f$ is a real-analytic function on $\overline\Omega$ satisfying the compatibility condition $\int_{\Omega}f\,d x=0$. Then there exists a solution to EQ01 that is real analytic on $\overline{\Omega}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 2.1
  • Proposition 2.2: K2
  • Remark 2.3
  • Theorem 3.1: Stokes equations
  • Lemma 4.1
  • proof : Proof of Lemma \ref{['L01']}
  • Lemma 4.2
  • proof : Proof of Lemma \ref{['L02']}
  • Lemma 4.3
  • ...and 1 more