Équations aux dérivées partielles

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

Bocea, Marian¹ ; Mihăilescu, Mihai^2, 3  ; Stancu-Dumitru, Denisa⁴

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 $\Omega \subset {ℝ}^D$ and $H:{ℝ}^D\to [0,\infty )$ a convex, even, and $1$-homogeneous function of class $C^{3,\alpha }({ℝ}^D\setminus \left\{0\right\})$ for which the Hessian matrix $D^2\left(H^p\right)$ is positive definite in ${ℝ}^D\setminus \left\{0\right\}$ 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 $\Omega$ with respect to $p\in (1,\infty )$. As an application, we find a new variational characterization for the principal frequency on domains $\Omega$ 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].

Détail

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

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

Bibliographie

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

[2] Belloni, Marino; Kawohl, Bernd; Juutinen, Petri 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] Bocea, Marian; Mihăilescu, Mihai Minimization problems for inhomogeneous Rayleigh quotients, Commun. Contemp. Math., Volume 20 (2018) no. 7, 1750074, 13 pages | MR | Zbl

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

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

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

[7] Fukagai, Noboyoshi; Ito, Masayuki; Narukawa, Kimiaki 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] Huang, Yin X. On the eigenvalues of the $p$-Laplacian with varying $p$, Proc. Am. Math. Soc., Volume 125 (1997) no. 11, pp. 3347-3354 | MR | DOI | Zbl

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

[10] Kajikiya, Ryuji; Tanaka, Mieko; Tanaka, Satoshi 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] Lindqvist, Peter Note on a nonlinear eigenvalue problem, Rocky Mt. J. Math., Volume 23 (1993) no. 1, pp. 281-288 | MR | Zbl

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