Work

Coarse ellipticity and De Giorgi-Nash-Moser theory in the optimal range

Year: 2026

Type: preprint

Abstract: We extend the theory of De Giorgi-Nash-Moser to elliptic equations with symmetric coefficients $\mathbf{a}(x)$ which are possibly degenerate and unbounded but satisfy a coarse ellipticity condition. We prove local upper bounds for weak subsolutions, a weak Harnack inequality for nonnegative supersolutions, and a Harnack inequality for nonnegative solutions. The coarse ellipticity hypothesis requires spatial moments of coarse-grained matrices, suitably discounted and summed across scales, to be finite. In particular, it holds if, for some $α,β\geq0$ and $10$. For $α=β=0$, it corresponds to the results of Bella and Schäffner [8].

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.11879

Authors: Armstrong, Scott, Avelin, Benny, Kuusi, Tuomo, Turpeinen, Aatu

Institutions: Centre National de la Recherche Scientifique, Courant Institute of Mathematical Sciences, Sorbonne Université, Laboratoire Jacques-Louis Lions, New York University, Uppsala University, University of Helsinki

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: European Commission, HORIZON EUROPE Framework Programme

OpenAlex ID: W7221150062

Cites: 5

Cited by: 1

Related to: 10