On the optimization of the first weighted eigenvalue

Abstract

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.