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
Gianpaolo Piscitelli
In corso di stampa
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 $eta$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.