On the optimization of the first weighted eigenvalue
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 6 / December 2023
- Published online by Cambridge University Press:
- 12 September 2022, pp. 1777-1804
- Print publication:
- December 2023
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- Article
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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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