Strict monotonicity of the first $q$-eigenvalue of the fractional $p$-Laplace operator over annuli
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911006702174208 |
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| author | Kumar, K Ashok Biswas, Nirjan |
| author_facet | Kumar, K Ashok Biswas, Nirjan |
| contents | Let $B, B'\subset \mathbb{R}^d$ with $d\geq 2$ be two balls such that $B'\subset \subset B$ and the position of $B'$ is varied within $B$. For $p\in (1, \infty ),$ $s\in (0,1)$, and $q \in [1, p^*_s)$ with $p^*_s=\frac{dp}{d-sp}$ if $sp < d$ and $p^*_s=\infty $ if $sp \geq d$, let $λ^s_{p,q}(B\setminus \overline{B'})$ be the first $q$-eigenvalue of the fractional $p$-Laplace operator $(-Δ_p)^s$ in $B\setminus \overline{B'}$ with the homogeneous nonlocal Dirichlet boundary conditions. We prove that $λ^s_{p,q}(B\setminus \overline{B'})$ strictly decreases as the inner ball $B'$ moves towards the outer boundary $\partial B$. To obtain this strict monotonicity, we establish a strict Faber-Krahn type inequality for $λ_{p,q}^s(\cdot )$ under polarization. This extends some monotonicity results obtained by Djitte-Fall-Weth (Calc. Var. Partial Differential Equations, 60:231, 2021) in the case of $(-Δ)^s$ and $q=1, 2$ to $(-Δ_p)^s$ and $q\in [1, p^*_s).$ Additionally, we provide the strict monotonicity results for the general domains that are difference of Steiner symmetric or foliated Schwarz symmetric sets in $\mathbb{R}^d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_16672 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Strict monotonicity of the first $q$-eigenvalue of the fractional $p$-Laplace operator over annuli Kumar, K Ashok Biswas, Nirjan Analysis of PDEs Mathematical Physics Optimization and Control 2020: Primary 35R11, 49Q10, Secondary 35B51, 35B06, 47J10 Let $B, B'\subset \mathbb{R}^d$ with $d\geq 2$ be two balls such that $B'\subset \subset B$ and the position of $B'$ is varied within $B$. For $p\in (1, \infty ),$ $s\in (0,1)$, and $q \in [1, p^*_s)$ with $p^*_s=\frac{dp}{d-sp}$ if $sp < d$ and $p^*_s=\infty $ if $sp \geq d$, let $λ^s_{p,q}(B\setminus \overline{B'})$ be the first $q$-eigenvalue of the fractional $p$-Laplace operator $(-Δ_p)^s$ in $B\setminus \overline{B'}$ with the homogeneous nonlocal Dirichlet boundary conditions. We prove that $λ^s_{p,q}(B\setminus \overline{B'})$ strictly decreases as the inner ball $B'$ moves towards the outer boundary $\partial B$. To obtain this strict monotonicity, we establish a strict Faber-Krahn type inequality for $λ_{p,q}^s(\cdot )$ under polarization. This extends some monotonicity results obtained by Djitte-Fall-Weth (Calc. Var. Partial Differential Equations, 60:231, 2021) in the case of $(-Δ)^s$ and $q=1, 2$ to $(-Δ_p)^s$ and $q\in [1, p^*_s).$ Additionally, we provide the strict monotonicity results for the general domains that are difference of Steiner symmetric or foliated Schwarz symmetric sets in $\mathbb{R}^d$. |
| title | Strict monotonicity of the first $q$-eigenvalue of the fractional $p$-Laplace operator over annuli |
| topic | Analysis of PDEs Mathematical Physics Optimization and Control 2020: Primary 35R11, 49Q10, Secondary 35B51, 35B06, 47J10 |
| url | https://arxiv.org/abs/2305.16672 |