Continuity for Sobolev mappings with null Lagrangian bounds

Abstract

We prove the continuity of Sobolev functions $\ddot{I}†\in W_{\mathrm{loc}}^{1,n}(\hat{I}\text{c◯})$, $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$, that satisfy \[ \lvert\nabla φ(x)\rvert^n \le K(x)\bigl(\langle \nabla φ(x), ξ(x)\rangle + A(x)\bigr), \] where $\hat{I}¾\in L_{\mathrm{loc}}^{n/(n-1)}(\hat{I}\text{c◯},{\mathbb{R}}^n)$ is weakly divergence-free, and $K\in L_{\mathrm{loc}}^p(\hat{I}\text{c◯})$, $A\in L_{\mathrm{loc}}^q(\hat{I}\text{c◯})$ 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 $f\in W_{\mathrm{loc}}^{1,n}(\hat{I}\text{c◯},{\mathbb{R}}^n)$ satisfying the distortion inequality with defect $|Df(x){|}^n\le K(x)\det Df(x)+\hat{I}\pounds (x)$; 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.

25 pages, 1 figure. Replaces the previous version which was entitled "Values of finite distortion: continuity"; the updated title reflects the fact that the results are stated in higher generality