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

Marian Bocea; Mihai Mihăilescu; Denisa Stancu-Dumitru

doi:10.5802/crmath.348

Équations aux dérivées partielles

The Monotonicity of the Principal Frequency of the Anisotropic

p

-Laplacian

Marian Bocea ¹ ; Mihai Mihăilescu ^2,³ ; Denisa Stancu-Dumitru 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, Volume 360 (2022), 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 :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Marian Bocea and Mihai Mih\u{a}ilescu and Denisa Stancu-Dumitru},
     title = {The {Monotonicity} of the {Principal} {Frequency} of the {Anisotropic} $p${-Laplacian}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {993--1000},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2022},
     doi = {10.5802/crmath.348},
     language = {en},
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Marian Bocea; Mihai Mihăilescu; Denisa Stancu-Dumitru. The Monotonicity of the Principal Frequency of the Anisotropic $p$-Laplacian. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 993-1000. doi : 10.5802/crmath.348. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.348/

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Maria Farcaseanu; Mihai Mihăilescu; Denisa Stancu-Dumitru A minimization problem related to the principal frequency of the p -Bilaplacian with coupled Dirichlet–Neumann boundary conditions, Electronic Journal of Qualitative Theory of Differential Equations (2023) no. 51, p. 1 | DOI:10.14232/ejqtde.2023.1.51
Mihai Mihailescu; Denisa Stancu-Dumitru "Monotonicity with respect to p of the best constants associated with Sobolev immersions of type $W_{0}^{1, p} (Ω) ↪ L^{q} (Ω)$ when $q \in {1, p, \infty}$ ", Studia Universitatis Babes-Bolyai Matematica, Volume 68 (2023) no. 1, p. 109 | DOI:10.24193/subbmath.2023.1.08

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