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Kumar, K. Ashok; Biswas, Nirjan

Strict monotonicity of the first \(q\)-eigenvalue of the fractional \(p\)-Laplace operator over annuli. (English) Zbl 1532.35328

J. Geom. Anal. 34, No. 3, Paper No. 96, 21 p. (2024).
Summary: 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 \(\lambda^s_{p,q}(B\setminus\overline{B'})\) be the first \(q\)-eigenvalue of the fractional \(p\)-Laplace operator \((-\Delta_p)^s\) in \(B\setminus \overline{B'}\) with the homogeneous nonlocal Dirichlet boundary conditions. We prove that \(\lambda^s_{p,q}(B\setminus \overline{B'})\) 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 \(\lambda_{p,q}^s (\cdot)\) under polarization. This extends some monotonicity results obtained by S. M. Djitte et al. [Calc. Var. Partial Differ. Equ. 60, No. 6, Paper No. 231, 31 p. (2021; Zbl 1476.49051)] in the case of \((-\Delta)^s\) and \(q=1, 2\) to \((-\Delta_p)^s\) 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 \(\mathbb{R}^d\).

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35P15 Estimates of eigenvalues in context of PDEs
35B06 Symmetries, invariants, etc. in context of PDEs
35B51 Comparison principles in context of PDEs
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35R11 Fractional partial differential equations
49Q10 Optimization of shapes other than minimal surfaces
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems

Keywords:

nonlocal operator; polarizations; strong comparison principle; Faber-Krahn inequality; monotonicity of the first eigenvalue

Citations:

Zbl 1476.49051
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Full Text: DOI arXiv

References:

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