A comparison principle for the Lane-Emden equation and applications to geometric estimates
Analysis of PDEs
2022-02-24 v2 Optimization and Control
Abstract
We prove a comparison principle for positive supersolutions and subsolutions to the Lane-Emden equation for the Laplacian, with subhomogeneous power in the right-hand side. The proof uses variational tools and the result applies with no regularity assumptions, both on the set and the functions. We then show that such a comparison principle can be applied to prove: uniqueness of solutions; sharp pointwise estimates for positive solutions in convex sets; localization estimates for maximum points and sharp geometric estimates for generalized principal frequencies in convex sets.
Keywords
Cite
@article{arxiv.2111.09603,
title = {A comparison principle for the Lane-Emden equation and applications to geometric estimates},
author = {Lorenzo Brasco and Francesca Prinari and Anna Chiara Zagati},
journal= {arXiv preprint arXiv:2111.09603},
year = {2022}
}
Comments
43 pages. In Theorem 4.1, the assumption "non-negative" has been replaced by "positive". Some bibliographical items have been added