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},
     volume = {360},
     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/

Bibliographie
Cité par

[1] Marino Belloni; Vincenzo Ferone; Bernd Kawohl Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys., Volume 54 (2003) no. 5, pp. 771-783 | DOI | MR | Zbl

[2] Marino Belloni; Bernd Kawohl; Petri Juutinen The $p$ -Laplace eigenvalue problem as $p \to \infty$ in a Finsler metric, J. Eur. Math. Soc., Volume 8 (2006) no. 1, pp. 123-138 | DOI | MR | Zbl

[3] Marian Bocea; Mihai Mihăilescu Minimization problems for inhomogeneous Rayleigh quotients, Commun. Contemp. Math., Volume 20 (2018) no. 7, 1750074, 13 pages | MR | Zbl

[4] Marian Bocea; Mihai Mihăilescu On the monotonicity of the principal frequency of the $p$ -Laplacian, Adv. Calc. Var., Volume 14 (2021) no. 1, pp. 147-152 | DOI | MR | Zbl

[5] Francesco Della Pietra; Giuseppina di Blasio; Nunzia Gavitone Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle, Adv. Nonlinear Anal., Volume 9 (2020), pp. 278-291 | DOI | MR | Zbl

[6] Francesco Della Pietra; Nunzia Gavitone; Gianpolo Piscitelli On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operator, Bull. Sci. Math., Volume 155 (2019), pp. 10-32 | DOI | MR | Zbl

[7] Noboyoshi Fukagai; Masayuki Ito; Kimiaki Narukawa Limit as $p \to \infty$ of $p$ -Laplace eigenvalue problems and $L^{\infty}$ -inequality of Poincaré type, Differ. Integral Equ., Volume 12 (1999) no. 2, pp. 183-206 | Zbl

[8] Yin X. Huang On the eigenvalues of the $p$ -Laplacian with varying $p$ , Proc. Am. Math. Soc., Volume 125 (1997) no. 11, pp. 3347-3354 | DOI | MR | Zbl

[9] Petri Juutinen; Peter Lindqvist; Juan J. Manfredi The $\infty$ -eigenvalue problem, Arch. Ration. Mech. Anal., Volume 148 (1999) no. 2, pp. 89-105 | DOI | Zbl

[10] Ryuji Kajikiya; Mieko Tanaka; Satoshi Tanaka Bifurcation of positive solutions for the one-dimensional $(p; q)$ -Laplace equation, Electron. J. Differ. Equ., Volume 2017 (2017), 107, 37 pages | MR | Zbl

[11] Peter Lindqvist Note on a nonlinear eigenvalue problem, Rocky Mt. J. Math., Volume 23 (1993) no. 1, pp. 281-288 | MR | Zbl

[12] Peter Lindqvist On non-linear Rayleigh quotients, Potential Anal., Volume 2 (1993) no. 3, pp. 199-218 | DOI | Zbl

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