PreprintCC BY 4.0arXiv
Sharpness and Planar Stability of the Alexandrov-Bakelman-Pucci Estimate: a Convex Geometric Approach
Abstract
Based on a 1981 work by G. Talenti, which established a comparison principle for smooth solutions of the Monge-Ampère equation alongside related two-dimensional a priori estimates, this paper explores the connection to the Blaschke-Santaló inequality and its reverse form by K. Mahler (1939). More specifically, in this work, a quantitative version of one of Talenti's a priori estimates is proved. We exploit the interplay between Talenti's analytical framework and Mahler/Blaschke-Santaló inequalities via convex geometry, demonstrating how this link can be utilized to gain sharp insights into the Alexandrov-Bakelman-Pucci (ABP) maximum principle for elliptic PDEs.
Versions
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BibTeX
@misc{porricelli2026sharpness,
title = {{Sharpness and Planar Stability of the Alexandrov-Bakelman-Pucci Estimate: a Convex Geometric Approach}},
author = {Antonio Porricelli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.24388},
eprint = {2607.24388},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.24388},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharpness and Planar Stability of the Alexandrov-Bakelman-Pucci Estimate: a Convex Geometric Approach
AU - Antonio Porricelli
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2607.24388
UR - https://arxiv.org/abs/2607.24388
AB - Based on a 1981 work by G. Talenti, which established a comparison principle for smooth solutions of the Monge-Ampère equation alongside related two-dimensional a priori estimates, this paper explores the connection to the Blaschke-Santaló inequality and its reverse form by K. Mahler (1939). More specifically, in this work, a quantitative version of one of Talenti's a priori estimates is proved. We exploit the interplay between Talenti's analytical framework and Mahler/Blaschke-Santaló inequalities via convex geometry, demonstrating how this link can be utilized to gain sharp insights into the Alexandrov-Bakelman-Pucci (ABP) maximum principle for elliptic PDEs.
ER -