*Published Paper*

**Inserted:** 8 feb 2022

**Last Updated:** 28 jul 2023

**Journal:** Advances in Differential Equations

**Volume:** 28

**Number:** 7-8

**Pages:** 537-568

**Year:** 2023

**Doi:** https://doi.org/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 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) under the polarizations. We apply this inequality to the obstacle problems on the 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 $.

**Keywords:**
obstacle problems, Faber-Krahn inequality, Polarizations, Zaremba problem for $p$-Laplacian, Strict monotonicity of first eigenvalue

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