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

TL;DR

The paper develops a unit-scale De Giorgi–Nash–Moser theory under a coarse-grained ellipticity framework, where scale-dependent effective diffusion matrices encode the large-scale regularity of solutions to $-\nabla\cdot(\mathbf{a}\nabla u)=0$. By introducing Besov-type and negative Sobolev regularity hypotheses on the coefficients, the authors derive coarse-grained Sobolev–Poincaré, Caccioppoli, reverse Hölder, and logarithmic estimates, and then implement a classical De Giorgi–Moser iteration and Bombieri–Giusti crossover to obtain local boundedness and a Harnack inequality with explicit dependence on the coarse-grained ellipticity ratio $\Theta_{s,t}$. The framework accommodates highly singular or degenerate coefficients, including fractal- and Gaussian multiplicative chaos–generated fields, where traditional $L^{1+\delta}$-type integrability fails, thus broadening applicability to new classes of random and irregular media while preserving quantitative control at unit scale. The results recover Trudinger’s classical results in the appropriate limit and deliver sharp scaling of the Harnack constant, while providing explicit mechanisms to verify coarse-grained ellipticity via negative Sobolev norms. Overall, the work offers a robust, scale-aware pathway to De Giorgi–Nash–Moser theory beyond uniformly elliptic regimes, with concrete examples illustrating the reach to fractal- and multifractal-driven coefficients.

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 $\|\mathbf{a}\|_{L^p}$ and $\|\mathbf{a}^{-1}\|_{L^q}$. More importantly, our criteria apply to new classes of degenerate and singular coefficient fields for which $\mathbf{a},\mathbf{a}^{-1}\notin L^{1+δ}$ for all $δ>0$, including examples generated by singular fractal measures and Gaussian multiplicative chaos, beyond the reach of previous approaches based solely on integrability assumptions.

Theorems & Definitions (48)

• Theorem 1.1: Local boundedness principle
• Theorem 1.2: Harnack inequality
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof