On the optimization of the first weighted eigenvalue

Abstract

For $N\ge 2$, a bounded smooth domain $\Omega$ in ${\mathbb{R}}^N$, and $g_0,V_0\in L_{loc}^1(\Omega )$, 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.

22 pages, 1 figure

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