475 research outputs found
Relative isoperimetric inequality in the plane: the anisotropic case
We prove a relative isoperimetric inequality in the plane, when the perimeter
is defined with respect to a convex, positively homogeneous function of degree
one, and characterize the minimizers
A saturation phenomenon for a nonlinear nonlocal eigenvalue problem
Given and , we study the properties of the
solutions of the minimum problem In particular, depending on
and , we show that the minimizers have constant sign up to a
critical value of , and when the
minimizers are odd
Stability results for some fully nonlinear eigenvalue estimates
In this paper, we give some stability estimates for the Faber-Krahn
inequality relative to the eigenvalues of Hessian operatorsComment: 18 pages, the proofs of Lemma 4.3 and Theorem 4.1 were clarifie
Faber-Krahn inequality for anisotropic eigenvalue problems with Robin boundary conditions
In this paper we study the main properties of the first eigenvalue and its
eigenfunctions of a class of highly nonlinear elliptic operators in a bounded
Lipschitz domain, assuming a Robin boundary condition. Moreover, we prove a
Faber-Krahn inequality
An optimal bound for nonlinear eigenvalues and torsional rigidity on domains with holes
In this paper we prove an optimal upper bound for the first eigenvalue of a
Robin-Neumann boundary value problem for the p-Laplacian operator in domains
with convex holes. An analogous estimate is obtained for the corresponding
torsional rigidity problem
On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators
Let be a bounded open set of , . In this
paper we mainly study some properties of the second Dirichlet eigenvalue
of the anisotropic -Laplacian where
is a suitable smooth norm of and . We
provide a lower bound of among bounded open sets of
given measure, showing the validity of a Hong-Krahn-Szego type inequality.
Furthermore, we investigate the limit problem as
Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators
The aim of this paper is to obtain optimal estimates for the first Robin
eigenvalue of the anisotropic -Laplace operator, namely: \begin{equation*}
\lambda_1(\beta,\Omega)=\min_{\psi\in W^{1,p}(\Omega)\setminus\{0\} }
\frac{\displaystyle\int_\Omega F(\nabla \psi)^p dx +\beta
\displaystyle\int_{\partial\Omega}|\psi|^pF(\nu_{\Omega}) d\mathcal H^{N-1}
}{\displaystyle\int_\Omega|\psi|^p dx}, \end{equation*} where
, is a bounded, mean convex domain in , is its Euclidean outward normal, is a real
number, and is a sufficiently smooth norm on . The
estimates we found are in terms of the first eigenvalue of a one-dimensional
nonlinear problem, which depends on and on geometrical quantities
associated to . More precisely, we prove a lower bound of
in the case , and a upper bound in the case . As a
consequence, we prove, for , a lower bound for
in terms of the anisotropic inradius of
and, for , an upper bound of in terms of
.Comment: 24 page
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