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Nonlinear Stein theorem for differential forms

Swarnendu Sil

TL;DR

This work extends Stein’s borderline regularity results to nonlinear systems acting on differential forms by studying the inhomogeneous quasilinear system $\delta\left( a(x)|du|^{p-2}du\right)=f$ with uniformly elliptic, Dini-continuous coefficients. The authors develop a nonlinear Stein-type framework for differential forms, incorporating gauge fixing (Coulomb gauge) and carefully chosen boundary-value problems to obtain precise comparison estimates and excess-decay arguments, culminating in continuity of $du$ and, under $\delta u=0$, local $VMO$ regularity of $\nabla u$. They further establish Campanato-type gradient estimates, yielding Hölder and BMO/VMO regularity for the gradient when $p\ge 2$, and they recover the scalar and vectorial $p$-Laplacian benchmarks while providing corollaries relevant to nonlinear Maxwell-type models. The results advance the regularity theory for nonlinear $d-\delta$-systems on differential forms and offer tools potentially applicable to Maxwell and Stokes-type quasilinear PDEs in areas such as geometry and mathematical physics.

Abstract

We prove that if $u$ is an $\mathbb{R}^{N}$-valued $W^{1,p}_{loc}$ differential $k$-form with $δ\left( a(x) \lvert du \rvert^{p-2} du \right) \in L^{(n,1)}_{loc}$ in a domain of $\mathbb{R}^{n}$ for $N \geq 1,$ $n \geq 2,$ $0 \leq k \leq n-1, $ $1 < p < \infty, $ with uniformly positive, bounded, Dini continuous scalar function $a$, then $du$ is continuous. This generalizes the classical result by Stein in the scalar case and the work of Kuusi-Mingione for the $p$-Laplacian type systems. We also discuss Hölder, BMO and VMO regularity estimates for such systems when $p \geq 2.$

Nonlinear Stein theorem for differential forms

TL;DR

This work extends Stein’s borderline regularity results to nonlinear systems acting on differential forms by studying the inhomogeneous quasilinear system with uniformly elliptic, Dini-continuous coefficients. The authors develop a nonlinear Stein-type framework for differential forms, incorporating gauge fixing (Coulomb gauge) and carefully chosen boundary-value problems to obtain precise comparison estimates and excess-decay arguments, culminating in continuity of and, under , local regularity of . They further establish Campanato-type gradient estimates, yielding Hölder and BMO/VMO regularity for the gradient when , and they recover the scalar and vectorial -Laplacian benchmarks while providing corollaries relevant to nonlinear Maxwell-type models. The results advance the regularity theory for nonlinear -systems on differential forms and offer tools potentially applicable to Maxwell and Stokes-type quasilinear PDEs in areas such as geometry and mathematical physics.

Abstract

We prove that if is an -valued differential -form with in a domain of for with uniformly positive, bounded, Dini continuous scalar function , then is continuous. This generalizes the classical result by Stein in the scalar case and the work of Kuusi-Mingione for the -Laplacian type systems. We also discuss Hölder, BMO and VMO regularity estimates for such systems when

Paper Structure

This paper contains 29 sections, 27 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Preliminary material and notations
  3. Notations
  4. Morrey, Campanato and Lorentz spaces
  5. The auxiliary mapping $V$
  6. Local and boundary regularity for the linear $d-\delta$ system
  7. On existence and weak formulations
  8. Enlarging the space of test functions
  9. Regularity for the homogeneous constant coefficient system
  10. Nonlinear Stein theorem for forms
  11. Homogeneous system with Dini coefficients
  12. General setting for the proofs
  13. Pointwise bound
  14. Proof of continuity
  15. Gauge fixing
  16. ...and 14 more sections

Key Result

Theorem 1

Let $n \geq 2 ,$$N \geq 1$ and $0 \leq k \leq n-1$ be integers and let $\Omega \subset \mathbb{R}^{n}$ be open. Suppose that Let $1 < p < \infty$ and $u \in W_{loc}^{1,p} \left(\Omega; \Lambda^{k}\mathbb{R}^{n}\otimes \mathbb{R}^{N} \right)$ be a local weak solution to the system Then $du$ is continuous in $\Omega.$ Moreover, if in addition $\delta u = 0$ in $\Omega,$ then $\nabla u$ is locally

Theorems & Definitions (32)

  • Theorem 1: Nonlinear Stein theorem
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Lemma 6
  • Theorem 7: local estimates
  • Theorem 8: boundary estimates
  • Remark 9
  • Proposition 10: Poincaré-Sobolev inequality
  • Proposition 11
  • ...and 22 more