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Francesco Della Pietra

Publications and source records attributed to Francesco Della Pietra.

At least 37 records · Page 2Linked to original sources
arXiv2019

Saturation phenomena for some classes of nonlinear nonlocal eigenvalue problems

Let us consider the following minimum problem \[ λ_α(p,r)= \min_{\substack{u\in W_{0}^{1,p}(-1,1)\\ u\not\equiv0}}\dfrac{\displaystyle\int_{-1}^{1}|u'|^{p}dx+α\left|\int_{-1}^{1}|u|^{r-1}u\, dx\right|^{\frac pr}}{\displaystyle\int_{-1}^{1}|u|^{p}dx}, \] where $α\in\mathbb R$, $p\ge 2$ and $\frac p2 \le r \le p$. We show that there exists a critical value $α_C=α_C (p,r)$ such that the minimizers have constant sign up to $α=α_{C}$ and then they are odd when $α>α_{C}$.

math.AP↗
arXiv2018

Motion of level sets by inverse anisotropic mean curvature

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.

math.AP↗
arXiv2017

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

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.

math.AP↗
arXiv2017

On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators

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 $p\to+\infty$.

math.AP↗
arXiv2016

A saturation phenomenon for a nonlinear nonlocal eigenvalue problem

Given $1\le q \le 2$ and $α\in\mathbb R$, we study the properties of the solutions of the minimum problem \[ λ(α,q)=\min\left\{\dfrac{\displaystyle\int_{-1}^{1}|u'|^{2}dx+α\left|\int_{-1}^{1}|u|^{q-1}u\, dx\right|^{\frac2q}}{\displaystyle\int_{-1}^{1}|u|^{2}dx}, u\in H_{0}^{1}(-1,1),\,u\not\equiv 0\right\}. \] In particular, depending on $α$ and $q$, we show that the minimizers have constant sign up to a critical value of $α=α_{q}$, and when $α>α_{q}$ the minimizers are odd.

math.AP↗
arXiv2015

Anisotropic Hardy inequalities

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.

math.AP↗
arXiv2015

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 $u\in W^{2,2}_{\textrm{loc}}(Ω)$ of the nonlinear elliptic problem \begin{equation} \tag{P} \label{abstr} \left\{ \begin{array}{ll} -Δ_{H}u+H(\nabla u)^{q}+λu=f&\text{in }Ω,\\ u\rightarrow +\infty &\text{on }\partialΩ, \end{array} \right. \end{equation} where $H$ is a suitable norm of $\mathbb R^{n}$, $Ω$ is a bounded domain, $Δ_{H}$ is the Finsler Laplacian, $1 0$ and $f$ is a suitable function in $L^{\infty}_{\textrm{loc}}$. Furthermore, we are interested in the behavior of the solutions when $λ\rightarrow 0^{+}$, studying the so-called ergodic problem associated to \eqref{abstr}. A key role in order to study the ergodic problem will be played by local gradient estimates for \eqref{abstr}.

math.AP↗
arXiv2014

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.

math.AP↗
arXiv2014

Some remarks on a shape optimization problem

Given $Ω$ bounded open set of $\mathbb R^{n}$ and $α\in \mathbb R$, let us consider \[ μ(Ω,α)=\min_{\substack{v\in W_{0}^{1,2}(Ω)\\v\not\equiv 0}} \frac{\displaystyle\int_Ω |\nabla v|^{2}dx+α\left|\displaystyle\int_Ω|v|v\,dx \right|}{\displaystyle\int_Ω |v|^{2}dx}. \] We study some properties of $μ(Ω,α)$ and of its minimizers, and, depending on $α$, we determine the set $Ω_α$ among those of fixed measure such that $μ(Ω_α,α)$ is the smallest possible.

math.AP↗
arXiv2014

Sharp estimates and existence for anisotropic elliptic problems with general growth in the gradient

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.

math.AP↗
arXiv2013

Symmetry breaking in a constrained Cheeger type isoperimetric inequality

We study the optimal constant in a Sobolev inequality for BV functions with zero mean value and vanishing outside a bounded open set. We are interested in finding the best possible embedding constant in terms of the measure of the domain alone. We set up an optimal shape problem and we completely characterize the behavior of optimal domains.

math.OC↗