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

Abstract

We extend the theory of De Giorgi-Nash-Moser to elliptic equations with symmetric coefficients 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≥0 and 1<p,q<∞, a∈ W^−α,p∩ L¹, a⁻¹∈ W^−β,q∩ L¹ and (α+β)/2+(d−1)/2(1/p+1/q)<1. We show that the coefficient range is sharp, including its boundary, in every dimension d≥3, for α=β=0 and for the corresponding Besov-type quasi-norms of negative order when α,β>0. For α=β=0, it corresponds to the results of Bella and Schäffner [8].

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BibTeX RIS

BibTeX

@misc{armstrong2026coarse,
  title = {{Coarse ellipticity and De Giorgi-Nash-Moser theory in the optimal range}},
  author = {Scott Armstrong and Benny Avelin and Tuomo Kuusi and Aatu Turpeinen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11879},
  eprint = {2610.11879},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11879},
}

RIS

TY  - UNPB
TI  - Coarse ellipticity and De Giorgi-Nash-Moser theory in the optimal range
AU  - Scott Armstrong
AU  - Benny Avelin
AU  - Tuomo Kuusi
AU  - Aatu Turpeinen
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11879
UR  - https://arxiv.org/abs/2610.11879
AB  - We extend the theory of De Giorgi-Nash-Moser to elliptic equations with symmetric coefficients 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≥0 and 1<p,q<∞, a∈ W^−α,p∩ L^1, a^−1∈ W^−β,q∩ L^1 and (α+β)/2+(d−1)/2(1/p+1/q)<1. We show that the coefficient range is sharp, including its boundary, in every dimension d≥3, for α=β=0 and for the corresponding Besov-type quasi-norms of negative order when α,β>0. For α=β=0, it corresponds to the results of Bella and Schäffner [8].
ER  -

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