Sharp maximum estimates for linear uniformly elliptic equations with measurable coefficients

Shibing Chen; Neil S. Trudinger; Xu-Jia Wang

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arXiv · 2610.23596

Sharp maximum estimates for linear uniformly elliptic equations with measurable coefficients

Shibing Chen·Neil S. Trudinger·Xu-Jia Wang

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Abstract

In this paper we obtain sharp exponents in the Aleksandrov-Bakelman-Pucci maximum estimate for linear uniformly elliptic equations with measurable coefficients, originally conjectured by Pucci in 1966 \cite{Pucci}, together with corresponding sharp results under the more natural quantification of uniform ellipticity in more than two dimensions. The proofs combine a Hessian integral inequality obtained from boundary Morse inequalities with an obstacle argument. The estimates also hold on arbitrary open sets, of finite volume, without regularity assumptions on their boundaries.

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Shibing Chen, Neil S. Trudinger, Xu-Jia Wang. 2026-10-16. Sharp maximum estimates for linear uniformly elliptic equations with measurable coefficients. https://arxiv.org/abs/2610.23596

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