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Continuity for Sobolev mappings with null Lagrangian bounds

Ilmari Kangasniemi, Jani Onninen

Abstract

We prove the continuity of Sobolev functions $\varphi \in W^{1,n}_{\mathrm{loc}}(Ω)$, $Ω\subset \mathbb{R}^n$, that satisfy \[ \lvert\nabla \varphi(x)\rvert^n \le K(x)\bigl(\langle \nabla \varphi(x), ξ(x)\rangle + A(x)\bigr), \] where $ξ\in L_{\mathrm{loc}}^{n/(n-1)}(Ω, \mathbb{R}^n)$ is weakly divergence-free, and $K \in L^p_{\mathrm{loc}} (Ω)$, $A \in L^q_{\mathrm{loc}} (Ω)$ are non-negative with $p^{-1}+q^{-1}<1$. The result is applicable to a broad class of differential inequalities of null Lagrangian type. As our principal application, we obtain a sharp continuity theorem for $f \in W^{1,n}_{\mathrm{loc}} (Ω, \mathbb{R}^n)$ satisfying the distortion inequality with defect $\lvert Df(x)\rvert^n \le K(x)\det Df(x) + Σ(x)$; this result is new even in the planar case, and closes a significant gap between existing methods and known counterexamples. The proof relies on an overlooked Sobolev-type inequality formulated in terms of measures of superlevel sets.

Continuity for Sobolev mappings with null Lagrangian bounds

Abstract

We prove the continuity of Sobolev functions , , that satisfy where is weakly divergence-free, and , are non-negative with . The result is applicable to a broad class of differential inequalities of null Lagrangian type. As our principal application, we obtain a sharp continuity theorem for satisfying the distortion inequality with defect ; this result is new even in the planar case, and closes a significant gap between existing methods and known counterexamples. The proof relies on an overlooked Sobolev-type inequality formulated in terms of measures of superlevel sets.

Paper Structure

This paper contains 15 sections, 23 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Null Lagrangians and the divergence structure
  3. Distortion inequalities with defects
  4. Quasiregular values and values of finite distortion
  5. The strategy of the proof
  6. The structure of this article
  7. Preliminaries
  8. Piecewise constant approximations
  9. Sobolev functions
  10. Integrals and level set methods
  11. Distribution functions and Sobolev inequalities
  12. Distribution functions
  13. Proof of Proposition \ref{['prop:distr_Sobolev_ineq']}
  14. Proof of Proposition \ref{['prop:weak_weak_mono_cont']}
  15. Proof of Theorem \ref{['thm:continuity_general']}

Key Result

Theorem 1.1

Let $n \ge 2$, let $\Omega \subset \mathbb{R}^n$ be open, let $K, A \colon \Omega \to [0, \infty)$ be measurable, and let $\xi \in L^{n/(n-1)}_\mathrm{loc}(\Omega, \mathbb{R}^n)$ with $\mathop{\mathrm{div}}\nolimits \xi = 0$ weakly. Suppose that $\varphi \in W^{1,n}_\mathrm{loc}(\Omega)$ satisfies e then $\varphi$ admits a continuous representative. Moreover, for each $x_0 \in \Omega$, the local m

Figures (1)

  • Figure 1: A non-decreasing left-continuous function, along with one of its staircase approximations.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.2
  • Definition 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Lemma 2.1: Staircase Lemma
  • proof
  • Lemma 2.2
  • ...and 31 more