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arXiv · 1412.2915

Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities and spectral estimates

Jean Dolbeault·Michal Kowalczyk
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Abstract

This paper is devoted to the Lin-Ni conjecture for a semi-linear elliptic equation with a super-linear, sub-critical nonlinearity and homogeneous Neumann boundary conditions. We establish a new rigidity result, that is, we prove that the unique positive solution is a constant if the parameter of the problem is below an explicit bound that we relate with an optimal constant for a Gagliardo-Nirenberg-Sobolev interpolation inequality and also with an optimal Keller-Lieb-Thirring inequality. Our results are valid in a sub-linear regime as well. The rigidity bound is obtained by nonlinear flow methods inspired by recent results on compact manifolds, which unify nonlinear elliptic techniques and the carr{é} du champ method in semi-group theory. Our method requires the convexity of the domain. It relies on integral quantities, takes into account spectral estimates and provides improved functional inequalities.

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Jean Dolbeault, Michal Kowalczyk. 2016-07-01. Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities and spectral estimates. https://arxiv.org/abs/1412.2915

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