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On the optimization of the first weighted eigenvalue

Part of: Spectral theory and eigenvalue problems Variational methods for eigenvalues of operators Equations of mathematical physics and other areas of application Partial differential equations Existence theories

Published online by Cambridge University Press:  12 September 2022

Nirjan Biswas Open the ORCID record for Nirjan Biswas [Opens in a new window] ,
Ujjal Das Open the ORCID record for Ujjal Das [Opens in a new window]  and
Mrityunjoy Ghosh
Nirjan Biswas
Affiliation:
Tata Institute of Fundamental Research, Centre For Applicable Mathematics, Post Bag No 6503, Sharada Nagar, Bangalore, 560065, India (nirjan22@tifrbng.res.in)
Ujjal Das
Affiliation:
Department of Mathematics, Technion - Israel Institute of Technology, Haifa, 32000, Israel (ujjal.rupam.das@gmail.com)
Mrityunjoy Ghosh
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600036, India (ghoshmrityunjoy22@gmail.com)
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Abstract

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.

Keywords

Optimization of the principal eigenvaluepolarization invarianceSchwarz symmetrySteiner symmetryfoliated Schwarz symmetry

MSC classification

Primary: 35B06: Symmetries, invariants, etc. 49J30: Optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 35P15: Estimation of eigenvalues, upper and lower bounds 35Q93: PDEs in connection with control and optimization 49R05: Variational methods for eigenvalues of operators (should also be assigned at least one other classification number in Section 49)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 6 , December 2023 , pp. 1777 - 1804
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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