Symmetry and asymmetry of minimizers of a class of noncoercive functionals

For $N\geq 2$ , a bounded smooth domain $\Omega$ in $\mathbb {R}^{N}$ , and $g_0,\, V_0 \in L^{1}_{loc}(\Omega )$ , we study the optimization of the first eigenvalue for the following weighted eigenvalue problem: \[ -\Delta_p \phi + V |\phi|^{p-2}\phi = \lambda g |\phi|^{p-2}\phi \text{ in } \Omega, \quad \phi=0 \text{ on } \partial \Omega, \] 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.

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“…Foliated Schwarz symmetry. First, we define the foliated Schwarz symmetrization of a function on radial domains following [12].…”

Section: Steiner Symmetrymentioning

confidence: 99%

On the optimization of the first weighted eigenvalue

Biswas¹,

Das²,

Ghosh³

2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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A saturation phenomenon for a nonlinear nonlocal eigenvalue problem

Pietra

Piscitelli

2016

Nonlinear Differ. Equ. Appl.

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On the optimization of the first weighted eigenvalue

Biswas¹,

Das²,

Ghosh³

2021

Preprint

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For 𝑁 ≥ 2, a bounded smooth domain Ω in R 𝑁 , and 𝑔0, 𝑉0 ∈ 𝐿 1 𝑙𝑜𝑐 (Ω), we study the optimization of the first eigenvalue for the following weighted eigenvalue problem:where 𝑔 and 𝑉 vary over the rearrangement classes of 𝑔0 and 𝑉0, respectively. We prove the existence of a minimizing pair (𝑔, 𝑉 ) and a maximizing pair (𝑔, 𝑉 ) for 𝑔0 and 𝑉0 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.

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Symmetry and asymmetry of minimizers of a class of noncoercive functionals

Cited by 3 publications

References 20 publications

On the optimization of the first weighted eigenvalue

On the optimization of the first weighted eigenvalue

A saturation phenomenon for a nonlinear nonlocal eigenvalue problem

On the optimization of the first weighted eigenvalue

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