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

We study the Ekeland–Nirenberg variational problem in the two-dimensional diagonal family $$J_{a,c,d}(u)={\int }_{{\text{Rp}}^2}(u_{xy}^2+a{u}_x^2+c{u}_y^2+d{u}^2)\text{dd}x\text{dd}y,a,c,d>0,$$ 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 $$K_{a,c,d}>0\text{ on }{\text{Rp}}^2⟺u_{a,c,d}>0\text{ on }{\text{Rp}}^2⟺d\le ac\mathrm{.}$$ 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.