تقرير
Payne nodal set conjecture for the fractional $p$-Laplacian in Steiner symmetric domains
| العنوان: | Payne nodal set conjecture for the fractional $p$-Laplacian in Steiner symmetric domains |
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| المؤلفون: | Bobkov, Vladimir, Kolonitskii, Sergey |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs, Mathematics - Spectral Theory, 35J92, 35R11, 35B06, 49K30 |
| الوصف: | Let $u$ be either a second eigenfunction of the fractional $p$-Laplacian or a least energy nodal solution of the equation $(-\Delta)^s_p \, u = f(u)$ with superhomogeneous and subcritical nonlinearity $f$, in a bounded open set $\Omega$ and under the nonlocal zero Dirichlet conditions. Assuming only that $\Omega$ is Steiner symmetric, we show that the supports of positive and negative parts of $u$ touch $\partial\Omega$. As a consequence, the nodal set of $u$ has the same property whenever $\Omega$ is connected. The proof is based on the analysis of equality cases in certain polarization inequalities involving positive and negative parts of $u$, and on alternative characterizations of second eigenfunctions and least energy nodal solutions. Comment: 22 pages, 3 figures |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2405.06936 |
| رقم الأكسشن: | edsarx.2405.06936 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |