Coarse-grained ellipticity and De Giorgi-Nash-Moser theory

Abstract

We prove local boundedness and a Harnack inequality for nonnegative weak solutions of the equation $-\nabla \cdot (\mathbf{a}(x)\nabla u)=0$ under a coarse-grained ellipticity assumption on the symmetric coefficient field $\mathbf{a}$. Coarse-grained ellipticity is a scale-dependent condition, defined for fields with only $\mathbf{a},{\mathbf{a}}^{-1}\in L^1$, in terms of families of effective diffusion matrices on triadic cubes of all sizes, and our estimates depend quantitatively on a corresponding coarse-grained ellipticity ratio. We show that coarse-grained ellipticity can be enforced by purely negative Sobolev regularity hypotheses: if $\mathbf{a}\in L^1\cap W^{-s,p}(U)$ and ${\mathbf{a}}^{-1}\in L^1\cap W^{-t,q}(U)$ for exponents $p,q\in [1,\infty ]$ and $s,t\in [0,1)$ satisfying $s<1-\frac{1}{p}$, $t<1-\frac{1}{q}$ and \[ \frac{s+t}{2} + \frac{d}{2}\Bigl(\frac{1}{p}+\frac{1}{q}\Bigr) < 1, \] then $\mathbf{a}$ is coarse-grained elliptic in $U$ and every nonnegative solution satisfies a quantitative unit-scale Harnack inequality. In particular, when $s=t=0$ we recover Trudinger's classical result under the integrability condition $\mathbf{a}\in L^p$, ${\mathbf{a}}^{-1}\in L^q$ with $\frac{1}{p}+\frac{1}{q}<\frac{2}{d}$, and we obtain the sharp scaling of the Harnack constant in terms of $\Vert \mathbf{a}{\Vert }_{L^p}$ and $\Vert {\mathbf{a}}^{-1}{\Vert }_{L^q}$. More importantly, our criteria apply to new classes of degenerate and singular coefficient fields for which $\mathbf{a},{\mathbf{a}}^{-1}\notin L^{1+\delta }$ for all $\delta >0$, including examples generated by singular fractal measures and Gaussian multiplicative chaos, beyond the reach of previous approaches based solely on integrability assumptions.

39 pages