arXiv subjects
Publications and source records attributed to Nunzia Gavitone.
The paper is devoted to the study of a quantitative Weinstock inequality in higher dimension for the first non trivial Steklov eigenvalue of Laplace operator for convex sets. The key rule is played by a quantitative isoperimetric inequality which involves the boundary momentum, the volume and the perimeter of a convex open set of $\mathbb R^n$, $n \ge 2$.
In this paper we consider the weak formulation of the inverse anisotropic mean curvature flow, in the spirit of Huisken-Ilmanen. By using approximation method involving Finsler-p-Laplacian, we prove the existence and uniqueness of weak solutions.
The paper is devoted to the study of some properties of the first eigenvalue of the anisotropic $p$-Laplace operator with Robin boundary condition involving a function $β$ which in general is not constant. In particular we obtain sharp lower bounds in terms of the measure of the domain and we prove a monotonicity property of the eigenvalue with respect the set inclusion.
In this paper we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue $λ_{F}(p,Ω)$ of the anisotropic $p$-Laplacian, $1<p<+\infty$. Our aim is to enhance how, by means of the $\mathcal P$-function method, it is possible to get several sharp estimates for $λ_{F}(p,Ω)$ in terms of several geometric quantities associated to the domain. The $\mathcal P$-function method is based on a maximum principle for a suitable function involving the eigenfunction and its gradient.
In this paper we study optimal estimates for two functionals involving the anisotropic $p$-torsional rigidity $T_p(Ω)$, $1<p<+\infty$. More precisely, we study $Φ(Ω)=\frac{T_p(Ω)}{|Ω|M(Ω)}$ and $Ψ(Ω)=\frac{T_p(Ω)}{|Ω|[R_{F}(Ω)]^{\frac{p}{p-1}}}$, where $M(Ω)$ is the maximum of the torsion function $u_Ω$ and $R_F(Ω)$ is the anisotropic inradius of $Ω$.
We prove an optimal lower bound for the best constant in a class of weighted anisotropic Poincaré inequalities
Let $Ω$ 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 $λ_{2}(p,Ω)$ of the anisotropic $p$-Laplacian \[ -\mathcal Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_ξ(\nabla u)\right), \] where $F$ is a suitable smooth norm of $\mathbb R^{n}$ and $p\in]1,+\infty[$. We provide a lower bound of $λ_{2}(p,Ω)$ among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as $p\to+\infty$.
We study some Hardy-type inequalities involving a general norm in $R^n$ and an anisotropic distance function to the boundary. The case of the optimality of the constants is also addressed.
We consider a class of quasilinear operators on a bounded domain $Ω\subset \mathbb R^n$ and address the question of optimizing the first eigenvalue with respect to the boundary conditions, which are of the Robin-type. We describe the optimizing boundary conditions and establish upper and lower bounds on the respective maximal and minimal eigenvalue.
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.
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: $ \left\{ \begin{array}{ll} -\mathcal Q_{p}u =[H(Du)]^{q}+f(x) &\text{in }Ω,\\ u=0&\text{on }\partialΩ. \end{array} \right. $ Here $Ω$ is a bounded open set of $\mathbb R^{N}$, $N\ge 2$, $0<p-1<q\le p<N$, and $\mathcal Q_{p}$ is the anisotropic operator $ \mathcal Q_{p} u ={\rm div}\left( [H(Du)]^{p-1}H_ξ(Du) \right)$, where $H$ is a suitable norm of $\mathbb R^{N}$. Moreover, $f$ belongs to an appropriate Marcinkiewicz space.
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.
The study of the optimal constant in an Hessian-type Sobolev inequality leads to a fully nonlinear boundary value problem, overdetermined with non standard boundary conditions. We show that all the solutions have ellipsoidal symmetry. In the proof we use the maximum principle applied to a suitable auxiliary function in conjunction with an entropy estimate from affine curvature flow.
In this paper, we give some stability estimates for the Faber-Krahn inequality relative to the eigenvalues of Hessian operators
We prove a sharp upper bound for the first Dirichlet eigenvalue of a class of nonlinear elliptic operators which includes the p-Laplace and the pseudo-p-Laplace operators. Moreover, we prove a stability result by means of a suitable isoperimetric deficity. Finally, we give a sharp lower bound for the anisotropic p-torsional rigidity.
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.
We prove some upper bounds for the Dirichlet eigenvalues of a class of fully nonlinear elliptic equations, namely the Hessian equations
We give existence and regularity results for solutions of some nonlinear elliptic problems. The equations we deal with are modeled on a problem which involves in its principal part an anisotropic operator, a Hardy-type potential, and a lower-order term depending on the gradient.