The Ekeland--Nirenberg Variational Problem:A Sharp Positivity Threshold and Extensions

TL;DR

This paper characterizes the positivity threshold for solutions to a specific variational problem, establishing a sharp criterion based on parameters and analyzing stability and extensions in multiple dimensions.

Contribution

It provides a complete classification of when the minimizer's kernel remains positive, introduces stability results under perturbations, and extends the analysis to higher dimensions.

Findings

Positivity of the kernel is equivalent to the parameter condition d ≤ ac.

Supercritical parameters d > ac lead to sign changes in the minimizer.

Local stability of sign change is established under small perturbations.

Abstract

We study the Ekeland--Nirenberg variational problem in the two-dimensional diagonal family \[ J_{a,c,d}(u)=\int_{\Rp^2}\bigl(u_{xy}^2+a u_x^2+c u_y^2+d u^2\bigr)\dd x\dd y, \qquad a,c,d>0, \] under the constraint $u(0,0)=1$. If $u_{a,c,d}$ is the unique minimizer and $K_{a,c,d}$ is its cosine kernel, we prove the sharp classification \[ K_{a,c,d}>0 \hbox{ on } \Rp^2\quad\Longleftrightarrow\quad u_{a,c,d}>0 \hbox{ on } \Rp^2\quad\Longleftrightarrow\quad d\le ac . \] Thus every supercritical triple $d>ac$ produces sign change. We also prove local sign-change stability under small two-dimensional non-diagonal perturbations and a sharp product-type $n$-dimensional diagonal threshold. The domain and evolution results are stated in precise auxiliary settings: a free-boundary capacity formulation for domains and a selected decaying branch of the second-order evolution equation.