the literature. We also give applications of the technique to the nodal set of solutions, and to give estimates on the corresponding eigenvalue problem.'
author:
- 'Aaron Naber and Daniele Valtorta [^1]'
bibliography:
- 'nava\_bib.bib'
title: Volume estimates on the critical sets of solutions to elliptic PDEs
---

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Introduction:
=============

In this paper, we study solutions $u$ to second order linear homogeneous elliptic equations with Lipschitz leading coefficients. That is, we will study on ${\mathds{R}}^n$ solutions $u$ to the equation $$\begin{gathered}
\label{eq_Lu}
{\mathcal{L}}(u)=\partial_i(a^{ij}(x)\partial_j u) + b^i(x) \partial _i u + c(x) u=0\, ,\end{gathered}$$ where the coefficients $a$ are Lipschitz and the coefficients $b,c$ are bounded. More effectively, we assume

$$\begin{aligned}
\label{e:coefficient_estimates}
&(1+\lambda)^{-1}\delta^{ij}\leq a^{ij}\leq (1+\lambda)\delta^{ij}, \,\notag\\
&\text{Lip}(a^{ij}), \,\, {\left|b^i\right|},\,\,{\left|c\right|}\leq \lambda\, .\end{aligned}$$

We will call the equation [*critical*]{} if $c\equiv 0$. Given the local nature of the estimates and the techniques involved in their proof, it is not restrictive to assume for simplicity that $u$ is defined on the ball $B_1(0)\subset {\mathds{R}}^n$. With simple modifications the results are easily extensible also to more general domains in Riemannian manifolds. Using a new covering argument we prove new $n-2$-[*Minkowski*]{} 