Higher integrability of solutions to elliptic equations under additional sign constraints

Abstract

Solutions to elliptic equations often exhibit higher regularity properties such as \emph{higher integrability}. That is, for instance, a solution $u$ to a system that a priori only satisfies $ u \in W^{1,r}$ is more regular and even in the Sobolev space $W^{1,s}$ for some $s>r$. Under additional constraints of the sign of specific terms such as $(\partial_i u)$ this improvement of regularity can be sharpened further. In this work, we consider two examples of such higher integrability results: First, we show a version of Müller's result on the higher integrability of the determinant for maps $u \in W^{1,n} $ such that $\mathrm{det}(\nabla u) \geq 0$ (or $ \mathrm{det}_-(\nabla u) \in L \log L$). Second, we consider (very weak) solutions to the $p$-Laplace equation that satisfy sign constraints for their partial derivatives, i.e. that $(\partial_i u)_- $ is of higher integrability than $(\partial_i u)_+$. To prove our results, we use the method of Lipschitz truncation; for the second example we further develop a variation of this technique, the \emph{asymmetric} Lipschitz truncation.

Figures (1)

• Figure 1: Rough outline of the start of the staircase construction: The elliptic equation is seen as a differential inclusion, i.e as a pair $(\nabla u,\sigma)$ where $\mathop{\mathrm{div}}\nolimits \sigma=0$ and $\sigma= \vert \nabla u \vert^{p-2} \nabla u$. At the end of the construction (up to some small error), we obtain a pair $(\nabla u,\sigma)$ such that $(\nabla u(x),\sigma(x))$ is contained in the red points almost everywhere. For this staircase construction (which is based on rank-one connections) to work it is crucial that $\partial_1 u$ is allowed to have (unbounded) positive and negative values.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 2.1: Lipschitz truncation
• Lemma 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Lemma 2.6