On the optimization of the first weighted eigenvalue
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866918041993871360 |
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| author | Biswas, Nirjan Das, Ujjal Ghosh, Mrityunjoy |
| author_facet | Biswas, Nirjan Das, Ujjal Ghosh, Mrityunjoy |
| contents | For $N\geq 2$, a bounded smooth domain $Ω$ in $\mathbb{R}^N$, and $g_0, V_0 \in L^1_{loc}(Ω)$, we study the optimization of the first eigenvalue for the following weighted eigenvalue problem: \begin{align*}
-Δ_p ϕ+ V |ϕ|^{p-2}ϕ= λg |ϕ|^{p-2}ϕ\text{ in } Ω, \quad ϕ=0 \text{ on } \partial Ω, \end{align*} where $g$ and $V$ vary over the rearrangement classes of $g_0$ and $V_0$, respectively. We prove the existence of a minimizing pair $(\underline{g},\underline{V})$ and a maximizing pair $(\overline{g},\overline{V})$ for $g_0$ and $V_0$ lying in certain Lebesgue spaces. We obtain various qualitative properties such as polarization invariance, Steiner symmetry of the minimizers as well as the associated eigenfunctions for the case $p=2$. For annular domains, we prove that the minimizers and the corresponding eigenfunctions possess the foliated Schwarz symmetry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_05543 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the optimization of the first weighted eigenvalue Biswas, Nirjan Das, Ujjal Ghosh, Mrityunjoy Analysis of PDEs Mathematical Physics Functional Analysis Optimization and Control 35B06, 49J30, 35P15, 35Q93, 49R05 For $N\geq 2$, a bounded smooth domain $Ω$ in $\mathbb{R}^N$, and $g_0, V_0 \in L^1_{loc}(Ω)$, we study the optimization of the first eigenvalue for the following weighted eigenvalue problem: \begin{align*} -Δ_p ϕ+ V |ϕ|^{p-2}ϕ= λg |ϕ|^{p-2}ϕ\text{ in } Ω, \quad ϕ=0 \text{ on } \partial Ω, \end{align*} where $g$ and $V$ vary over the rearrangement classes of $g_0$ and $V_0$, respectively. We prove the existence of a minimizing pair $(\underline{g},\underline{V})$ and a maximizing pair $(\overline{g},\overline{V})$ for $g_0$ and $V_0$ lying in certain Lebesgue spaces. We obtain various qualitative properties such as polarization invariance, Steiner symmetry of the minimizers as well as the associated eigenfunctions for the case $p=2$. For annular domains, we prove that the minimizers and the corresponding eigenfunctions possess the foliated Schwarz symmetry. |
| title | On the optimization of the first weighted eigenvalue |
| topic | Analysis of PDEs Mathematical Physics Functional Analysis Optimization and Control 35B06, 49J30, 35P15, 35Q93, 49R05 |
| url | https://arxiv.org/abs/2109.05543 |