Sharp Hessian integrability for fully nonlinear elliptic supersolutions in low dimensions
Abstract
We settle the planar sharp Hessian-integrability conjecture of Armstrong, Silvestre, and Smart [\emph{Comm. Pure Appl. Math.} \textbf{65} (2012), 1169--1184] for viscosity supersolutions of fully nonlinear uniformly elliptic equations. If κ=Λ/λ, then the optimal exponent in the W2,ε regularity theory in the plane is exactly ε2(κ)=2/(κ+1). The same mechanism reaches the known upper obstruction in dimension three throughout the full range 1≤ κ≤ 4 and therefore gives ε3(κ)=3/(2κ+1) (1≤κ≤4). Beyond this threshold, it yields a closed algebraic lower bound and identifies the precise spectral configuration responsible for the remaining gap. The proof introduces a spectrally resolved continuum-in-opening mechanism for contact geometry. Throughout the contact evolution, the vertex Jacobian retains the negative index of the Hessian and its mean negative curvature---data erased by the classical pointwise reduction---and these variables are optimized only after integration over all openings. This reduces the analysis to a finite family of explicit one-dimensional kernels whose critical exponents recover the sharp planar threshold, identify the exact three-dimensional regime above, and expose a broader principle for extracting second-order regularity from the spectral geometry of contact sets.
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BibTeX
@misc{nascimento2026sharp,
title = {{Sharp Hessian integrability for fully nonlinear elliptic supersolutions in low dimensions}},
author = {Thialita M. Nascimento and Eduardo V. Teixeira},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06817},
eprint = {2609.06817},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06817},
}
RIS
TY - UNPB
TI - Sharp Hessian integrability for fully nonlinear elliptic supersolutions in low dimensions
AU - Thialita M. Nascimento
AU - Eduardo V. Teixeira
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06817
UR - https://arxiv.org/abs/2609.06817
AB - We settle the planar sharp Hessian-integrability conjecture of Armstrong, Silvestre, and Smart [\emph{Comm. Pure Appl. Math.} \textbf{65} (2012), 1169--1184] for viscosity supersolutions of fully nonlinear uniformly elliptic equations. If κ=Λ/λ, then the optimal exponent in the W^2,ε regularity theory in the plane is exactly ε_2(κ)=2/(κ+1). The same mechanism reaches the known upper obstruction in dimension three throughout the full range 1≤ κ≤ 4 and therefore gives ε_3(κ)=3/(2κ+1) (1≤κ≤4). Beyond this threshold, it yields a closed algebraic lower bound and identifies the precise spectral configuration responsible for the remaining gap. The proof introduces a spectrally resolved continuum-in-opening mechanism for contact geometry. Throughout the contact evolution, the vertex Jacobian retains the negative index of the Hessian and its mean negative curvature---data erased by the classical pointwise reduction---and these variables are optimized only after integration over all openings. This reduces the analysis to a finite family of explicit one-dimensional kernels whose critical exponents recover the sharp planar threshold, identify the exact three-dimensional regime above, and expose a broader principle for extracting second-order regularity from the spectral geometry of contact sets.
ER -