Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Regularity for elliptic systems of differential forms and applications

Swarnendu Sil

TL;DR

This work develops up-to-boundary Hölder and $L^{p}$ regularity for weak solutions of the elliptic system $\delta(A d\omega)+B^{T}d\delta(B\omega)=\lambda B\omega+f$ under tangential or normal boundary data. It adapts the Campanato method to differential-form systems, using boundary flattening and coefficient freezing to avoid potential theory and LS/ADN condition verifications. The results unify and extend regularity theory to a broad class of Hodge-type, Maxwell, Stokes, and div-curl problems, including spectral analyses and solvability up to harmonic fields. The approach provides a robust, intrinsic framework for up-to-boundary regularity that applies to both second-order and first-order elliptic systems with mixed boundary conditions, without relying on componentwise scalar reductions. The techniques yield sharp Gaffney-type inequalities and spectral information, with explicit regularity up to $W^{r+2,p}$ or $C^{r+2,\\gamma}$ depending on the data and domain geometry.

Abstract

We prove existence and up to the boundary regularity estimates in $L^{p}$ and Hölder spaces for weak solutions of the linear system $$ δ\left( A dω\right) + B^{T}dδ\left( Bω\right) = λBω+ f \text{ in } Ω, $$ with either $ ν\wedge ω$ and $ν\wedge δ\left( Bω\right)$ or $ν\lrcorner Bω$ and $ν\lrcorner \left( A dω\right)$ prescribed on $\partialΩ.$ The proofs are in the spirit of `Campanato method' and thus avoid potential theory and do not require a verification of Agmon-Douglis-Nirenberg or Lopatinskiĭ-Shapiro type conditions. Applications to a number of related problems, such as general versions of the time-harmonic Maxwell system, stationary Stokes problem and the `div-curl' systems, are included.

Regularity for elliptic systems of differential forms and applications

TL;DR

This work develops up-to-boundary Hölder and regularity for weak solutions of the elliptic system under tangential or normal boundary data. It adapts the Campanato method to differential-form systems, using boundary flattening and coefficient freezing to avoid potential theory and LS/ADN condition verifications. The results unify and extend regularity theory to a broad class of Hodge-type, Maxwell, Stokes, and div-curl problems, including spectral analyses and solvability up to harmonic fields. The approach provides a robust, intrinsic framework for up-to-boundary regularity that applies to both second-order and first-order elliptic systems with mixed boundary conditions, without relying on componentwise scalar reductions. The techniques yield sharp Gaffney-type inequalities and spectral information, with explicit regularity up to or depending on the data and domain geometry.

Abstract

We prove existence and up to the boundary regularity estimates in and Hölder spaces for weak solutions of the linear system with either and or and prescribed on The proofs are in the spirit of `Campanato method' and thus avoid potential theory and do not require a verification of Agmon-Douglis-Nirenberg or Lopatinskiĭ-Shapiro type conditions. Applications to a number of related problems, such as general versions of the time-harmonic Maxwell system, stationary Stokes problem and the `div-curl' systems, are included.

Paper Structure

This paper contains 24 sections, 22 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Boundary estimates
  4. Preliminary lemmas
  5. Estimates for constant coefficient operator
  6. Boundary Caccioppoli inequality
  7. $L^{2}$ estimates
  8. Decay estimates for the constant coefficient operator
  9. $\mathcal{L}^{2, \mu}$ estimates
  10. $L^{p}$ estimates
  11. Schauder and $L^{p}$ estimates
  12. Applications
  13. Second order Hodge type systems: A special case
  14. Maxwell operator: A simple case
  15. Stokes type operator
  16. ...and 9 more sections

Key Result

Lemma 3.1

Let $B:\Lambda^{k}\rightarrow \Lambda^{k}$ satisfy for some constant $\gamma_{B} > 0.$ Then for every $\gamma_{A} >0,$ there exist constants $c_{1},c_{2} > 0,$ depending only on $\gamma_{A}$ and $B,$ such that

Theorems & Definitions (46)

  • Definition 2.1
  • Lemma 3.1: ellipticity lemma
  • Lemma 3.2: Gårding inequality
  • Remark 3.3
  • Lemma 3.4
  • Theorem 3.5: Caccioppoli inequality
  • Remark 3.6
  • Theorem 3.7: $L^{2}$ estimates for constant coefficients
  • Remark 3.8
  • Theorem 3.9
  • ...and 36 more