The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic $p$-Laplace operator, namely: egin{equation*} lambda_1(eta,Omega)= min_{psiin W^{1,p}(Omega)setminus{0} } rac{displaystyleint_Omega F( abla psi)^p dx +eta dsint_{deOmega}|psi|^pF( u_{Omega}) dcH^{N-1} }{displaystyleint_Omega|psi|^p dx}, end{equation*} where $pin]1,+infty[$, $Omega$ is a bounded, mean convex domain in $R^{N}$, $ u_{Omega}$ is its Euclidean outward normal, $eta$ is a real number, and $F$ is a sufficiently smooth norm on $R^{N}$. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on $eta$ and on geometrical quantities associated to $Omega$. More precisely, we prove a lower bound of $lambda_{1}$ in the case $eta>0$, and a upper bound in the case $eta<0$. As a consequence, we prove, for $eta>0$, a lower bound for $lambda_{1}(eta,Omega)$ in terms of the anisotropic inradius of $Omega$ and, for $eta<0$, an upper bound of $lambda_{1}(eta,Omega)$ in terms of $eta$.

Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators / DELLA PIETRA, Francesco; Piscitelli, Gianpaolo. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 386:(2024), pp. 269-293. [10.1016/j.jde.2023.12.039]

Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators

Francesco Della Pietra
;
Gianpaolo Piscitelli
2024

Abstract

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic $p$-Laplace operator, namely: egin{equation*} lambda_1(eta,Omega)= min_{psiin W^{1,p}(Omega)setminus{0} } rac{displaystyleint_Omega F( abla psi)^p dx +eta dsint_{deOmega}|psi|^pF( u_{Omega}) dcH^{N-1} }{displaystyleint_Omega|psi|^p dx}, end{equation*} where $pin]1,+infty[$, $Omega$ is a bounded, mean convex domain in $R^{N}$, $ u_{Omega}$ is its Euclidean outward normal, $eta$ is a real number, and $F$ is a sufficiently smooth norm on $R^{N}$. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on $eta$ and on geometrical quantities associated to $Omega$. More precisely, we prove a lower bound of $lambda_{1}$ in the case $eta>0$, and a upper bound in the case $eta<0$. As a consequence, we prove, for $eta>0$, a lower bound for $lambda_{1}(eta,Omega)$ in terms of the anisotropic inradius of $Omega$ and, for $eta<0$, an upper bound of $lambda_{1}(eta,Omega)$ in terms of $eta$.
2024
Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators / DELLA PIETRA, Francesco; Piscitelli, Gianpaolo. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 386:(2024), pp. 269-293. [10.1016/j.jde.2023.12.039]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11588/881837
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