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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.08700 |
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| _version_ | 1866917476528291840 |
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| author | Guo, Qi Huang, Xueping Huang, Yi |
| author_facet | Guo, Qi Huang, Xueping Huang, Yi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_08700 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Ekeland--Nirenberg Variational Problem:A Sharp Positivity Threshold and Extensions Guo, Qi Huang, Xueping Huang, Yi Analysis of PDEs 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. |
| title | The Ekeland--Nirenberg Variational Problem:A Sharp Positivity Threshold and Extensions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.08700 |