Payne nodal set conjecture for the fractional $p$-Laplacian in Steiner symmetric domains
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| Format: | Preprint |
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2024
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| _version_ | 1866917336493064192 |
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| author | Bobkov, Vladimir Kolonitskii, Sergey |
| author_facet | Bobkov, Vladimir Kolonitskii, Sergey |
| contents | Let $u$ be either a second eigenfunction of the fractional $p$-Laplacian or a least energy nodal solution of the equation $(-Δ)^s_p \, u = f(u)$ with superhomogeneous and subcritical nonlinearity $f$, in a bounded open set $Ω$ and under the nonlocal zero Dirichlet conditions. Assuming only that $Ω$ is Steiner symmetric, we show that the supports of positive and negative parts of $u$ touch $\partialΩ$. As a consequence, the nodal set of $u$ has the same property whenever $Ω$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_06936 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Payne nodal set conjecture for the fractional $p$-Laplacian in Steiner symmetric domains Bobkov, Vladimir Kolonitskii, Sergey Analysis of PDEs Spectral Theory 35J92, 35R11, 35B06, 49K30 Let $u$ be either a second eigenfunction of the fractional $p$-Laplacian or a least energy nodal solution of the equation $(-Δ)^s_p \, u = f(u)$ with superhomogeneous and subcritical nonlinearity $f$, in a bounded open set $Ω$ and under the nonlocal zero Dirichlet conditions. Assuming only that $Ω$ is Steiner symmetric, we show that the supports of positive and negative parts of $u$ touch $\partialΩ$. As a consequence, the nodal set of $u$ has the same property whenever $Ω$ 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. |
| title | Payne nodal set conjecture for the fractional $p$-Laplacian in Steiner symmetric domains |
| topic | Analysis of PDEs Spectral Theory 35J92, 35R11, 35B06, 49K30 |
| url | https://arxiv.org/abs/2405.06936 |