Francesco Della Pietra

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.

Blow-up solutions for some nonlinear elliptic equations involving a Finsler-Laplacian

In this paper we prove existence results and asymptotic behavior for strong solutions

Stability results for some fully nonlinear eigenvalue estimates

u\in W^{2,2}_{\textrm{loc}}(\Omega)

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

of the nonlinear elliptic problem \begin{equation} \tag{P} \label{abstr} \left\{ \begin{array}{ll} -\Delta_{H}u+H(\nabla u)^{q}+\lambda u=f&\text{in }\Omega,\\ u\rightarrow +\infty &\text{on }\partial\Omega, \end{array} \right. \end{equation} where

Anisotropic Hardy inequalities

H

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

is a suitable norm of

Anisotropic elliptic problems involving Hardy-type potentials

\mathbb R^{n}