July/August 2023 Domain variations of the first eigenvalue via a strict Faber-Krahn type inequality
T.V. Anoop, K. Ashok Kumar
Adv. Differential Equations 28(7/8): 537-568 (July/August 2023). DOI: 10.57262/ade028-0708-537

Abstract

For $d\geq 2$ and $\frac{2d+2}{d+2} < p < \infty $, we prove a strict Faber-Krahn type inequality under polarizations for the first eigenvalue $\lambda _1(\Omega )$ of the $p$-Laplace operator on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ with mixed boundary conditions. We apply this inequality to the obstacle problems on domains of the form $\Omega \setminus \mathscr{O}$, where $\mathscr{O}\subset \subset \Omega $ is an obstacle. Under some geometric assumptions on $\Omega $ and $\mathscr{O}$, we prove the strict monotonicity of $\lambda _1 (\Omega \setminus \mathscr{O})$ with respect to certain translations and rotations of $\mathscr{O}$ in $\Omega$.

Citation

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T.V. Anoop. K. Ashok Kumar. "Domain variations of the first eigenvalue via a strict Faber-Krahn type inequality." Adv. Differential Equations 28 (7/8) 537 - 568, July/August 2023. https://doi.org/10.57262/ade028-0708-537

Information

Published: July/August 2023
First available in Project Euclid: 10 April 2023

Digital Object Identifier: 10.57262/ade028-0708-537

Subjects:
Primary: 35J92 , 35M12 , 47J10 , 49Q10

Rights: Copyright © 2023 Khayyam Publishing, Inc.

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Vol.28 • No. 7/8 • July/August 2023
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