The Monotonicity of the Principal Frequency of the Anisotropic $p$-Laplacian

Bocea, Marian; Mihăilescu, Mihai; Stancu-Dumitru, Denisa

doi:10.5802/crmath.348

Équations aux dérivées partielles

The Monotonicity of the Principal Frequency of the Anisotropic

p

-Laplacian

Bocea, Marian ¹ ; Mihăilescu, Mihai ^2,³ ; Stancu-Dumitru, Denisa ⁴

¹ Division of Mathematical Sciences, National Science Foundation, 2415 Eisenhower Avenue, Alexandria, VA 22314 U.S.A.
² Department of Mathematics, University of Craiova, 200585 Craiova, Romania
³ “Gheorghe Mihoc - Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, Romania
⁴ Department of Mathematics and Computer Science, Politehnica University of Bucharest, 060042 Bucharest, Romania

Comptes Rendus. Mathématique, Tome 360 (2022) no. G9, pp. 993-1000.

Résumé

Let $D > 1$ be a fixed integer. Given a smooth bounded, convex domain $Ω \subset ℝ^{D}$ and $H : ℝ^{D} \to [0, \infty)$ a convex, even, and $1$ -homogeneous function of class $C^{3, α} (ℝ^{D} ∖ {0})$ for which the Hessian matrix $D^{2} (H^{p})$ is positive definite in $ℝ^{D} ∖ {0}$ for any $p \in (1, \infty)$ , we study the monotonicity of the principal frequency of the anisotropic $p$ -Laplacian (constructed using the function $H$ ) on $Ω$ with respect to $p \in (1, \infty)$ . As an application, we find a new variational characterization for the principal frequency on domains $Ω$ having a sufficiently small inradius. In the particular case where $H$ is the Euclidean norm in $ℝ^{D}$ , we recover some recent results obtained by the first two authors in [3, 4].

Reçu le : 2021-11-05
Accepté le : 2022-02-26
Publié le : 2022-09-29

DOI : 10.5802/crmath.348

Classification : 35P30, 47J10, 49R05, 49J40, 58C40

Affiliations des auteurs :

Bocea, Marian ¹ ; Mihăilescu, Mihai ^{2, 3} ; Stancu-Dumitru, Denisa ⁴

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Bocea, Marian; Mihăilescu, Mihai; Stancu-Dumitru, Denisa. The Monotonicity of the Principal Frequency of the Anisotropic $p$-Laplacian. Comptes Rendus. Mathématique, Tome 360 (2022) no. G9, pp. 993-1000. doi : 10.5802/crmath.348. http://www.numdam.org/articles/10.5802/crmath.348/

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