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

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

On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators

Let $\Omega$ be a bounded open set of $\mathbb R^{n}$, $n\ge 2$. In this paper we mainly study some properties of the second Dirichlet eigenvalue $\lambda_{2}(p,\Omega)$ of the anisotropic $p$-Laplacian $$-\mathcal Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_\xi (\nabla u)\right),$$ where $F$ is a suitable smooth norm of $\mathbb R^{n}$ and $p\in]1,+\infty[$. We provide a lower bound of $\lambda_{2}(p,\Omega)$ among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as $p\to+\infty$

Symmetrization with respect to the anisotropic perimeter and applications

In this paper we introduce a new type of symmetrization, which preserves the anisotropic perimeter of the level sets of a suitable concave smooth function, in order to prove sharp comparison results for solutions of a class of homogeneous Dirichlet fully nonlinear elliptic problems of second order and for suitable anisotropic Hessian integrals