Nunzia Gavitone

Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators

In this paper we prove a sharp upper bound for the first Dirichlet eigenvalue of a class of nonlinear elliptic operators which includes the operator , that is the p-Laplacian, and , namely the pseudo-p-Laplacian. Moreover we prove a stability result by means of a suitable isoperimetric deficit. Finally, we give a sharp lower bound for the anisotropic p-torsional rigidity.

Anisotropic elliptic equations with general growth in the gradient and Hardy-type potentials

Abstract In this paper we give existence and regularity results for the solutions of problems whose prototype is { − Q v = β ( | v | ) H ( D v ) q + λ H o ( x ) p | v | p − 2 v + f ( x ) in Ω , v = 0 on ∂ Ω , with Ω bounded domain of R N , N ⩾ 2 , 0 p − 1 q ⩽ p N , β is a nonnegative continuous function and λ ⩾ 0 . Moreover, H is a general norm of R N , H o is its polar and Q v : = ∑ i = 1 N ∂ ∂ x i ( H ( D v ) p − 1 H ξ i ( D v ) ) .

Symmetrization for Neumann Anisotropic Problems and Related Questions

Abstract In this paper we prove some comparison results for Neumann elliptic problems whose model involves the anisotropic Laplacian ΔHu = div (H(Du)Hξ(Du)), where H is a positively homogenous convex function. Finally, we find a Poincaré inequality in the anisotropic setting.

Upper bounds for the eigenvalues of Hessian equations

In this paper, we prove some upper bounds for the Dirichlet eigenvalues of a class of fully nonlinear elliptic equations, namely the Hessian equations.

Stability results for some fully nonlinear eigenvalue estimates

In this paper, we give some stability estimates for the Faber–Krahn inequality relative to the eigenvalue λk(Ω) of the Hessian operator Sk, 1 ≤ k ≤ n, in a reasonable bounded domain Ω. Roughly speaking, we prove that if λk(Ω) is near to λk(B), where B is a ball which preserves an appropriate measure of Ω, then, in a suitable sense, Ω is close to B.

Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle

Abstract In this paper, we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue λ F ⁢ ( p , Ω ) {\lambda_{F}(p,\Omega)} of the anisotropic p-Laplacian, 1 < p < + ∞ {1<p<+\infty} . Our aim is to enhance, by means of the 𝒫 {\mathcal{P}} -function method, how it is possible to get several sharp estimates for λ F ⁢ ( p , Ω ) {\lambda_{F}(p,\Omega)} in terms of several geometric quantities associated to the domain. The 𝒫 {\mathcal{P}} -function method is based on a maximum principle for a suitable function involving the eigenfunction and its gradient.

Anisotropic Hardy inequalities

We study some Hardy-type inequalities involving a general norm in

Sharp Estimates and Existence for Anisotropic Elliptic Problems with General Growth in the Gradient

R^n

Anisotropic elliptic problems involving Hardy-type potentials

and an anisotropic distance function to the boundary. The case of the optimality of the constants is also addressed.

Faber-Krahn Inequality for Anisotropic Eigenvalue Problems with Robin Boundary Conditions

In this paper, we prove sharp estimates and existence results for anisotropic nonlinear elliptic problems with lower order terms depending on the gradient. Our prototype is: