Enea Parini
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View article: Van der Waerden type theorem for amenable groups and FC-groups
Van der Waerden type theorem for amenable groups and FC-groups Open
We prove that for a discrete, countable, and amenable group $G$, if the direct product $G^2=G \times G$ is finitely colored then $\{ g \in G : \text{exists } (x,y) \in G^2 \text{ such that } \{ (x,y),(xg,y),(xg,yg)\} \text{ is monochromati…
View article: Uniqueness of least energy solutions to the fractional Lane-Emden equation in the ball
Uniqueness of least energy solutions to the fractional Lane-Emden equation in the ball Open
We prove uniqueness of least-energy solutions to the fractional Lane-Emden equation, under homogeneous Dirichlet exterior conditions, when the underlying domain is a ball $B \subset \mathbb{R}^N$. The equation is characterized by a superli…
View article: Optimization of the anisotropic Cheeger constant with respect to the anisotropy
Optimization of the anisotropic Cheeger constant with respect to the anisotropy Open
Given an open, bounded set $\Omega $ in $\mathbb {R}^N$ , we consider the minimization of the anisotropic Cheeger constant $h_K(\Omega )$ with respect to the anisotropy K , under a volume constraint on the associated unit ball. In the plan…
View article: Optimization of the anisotropic Cheeger constant with respect to the anisotropy
Optimization of the anisotropic Cheeger constant with respect to the anisotropy Open
Given an open, bounded set $Ω$ in $\mathbb{R}^N$, we consider the minimization of the anisotropic Cheeger constant $h_K(Ω)$ with respect to the anisotropy $K$, under a volume constraint on the associated unit ball. In the planar case, unde…
View article: Compactness and dichotomy in nonlocal shape optimization
Compactness and dichotomy in nonlocal shape optimization Open
We prove a general result about the behaviour of minimizing sequences for nonlocal shape functionals satisfying suitable structural assumptions. Typical examples include functions of the eigenvalues of the fractional Laplacian under homoge…
View article: Nodal Solutions for Sublinear-Type Problems with Dirichlet Boundary Conditions
Nodal Solutions for Sublinear-Type Problems with Dirichlet Boundary Conditions Open
We consider nonlinear 2nd-order elliptic problems of the type $$\begin{align*} & -\Delta u=f(u)\ \textrm{in}\ \Omega, \qquad u=0\ \textrm{on}\ \partial \Omega, \end{align*}$$where $\Omega $ is an open $C^{1,1}$–domain in ${{\mathbb{R}}}^N$…
View article: Nodal Solutions for sublinear-type problems with Dirichlet boundary\n conditions
Nodal Solutions for sublinear-type problems with Dirichlet boundary\n conditions Open
We consider nonlinear second order elliptic problems of the type \\[ -\\Delta\nu=f(u) \\text{ in } \\Omega, \\qquad u=0 \\text{ on } \\partial \\Omega, \\] where\n$\\Omega$ is an open $C^{1,1}$-domain in $\\mathbb{R}^N$, $N\\geq 2$, under …
View article: Reverse Faber-Krahn inequality for a truncated laplacian operator
Reverse Faber-Krahn inequality for a truncated laplacian operator Open
In this paper we prove a reverse Faber-Krahn inequality for the principal eigenvalue $μ_1(Ω)$ of the fully nonlinear eigenvalue problem \[ \label{eq} \left\{\begin{array}{r c l l} -λ_N(D^2 u) & = & μu & \text{in }Ω, \\ u & = & 0 & \text{on…
View article: On the higher Cheeger problem
On the higher Cheeger problem Open
We develop the notion of higher Cheeger constants for a measurable set $\\Omega \\subset \\mathbb{R}^N$. By the $k$-th Cheeger constant we mean the value \\[h_k(\\Omega) = \\inf \\max \\{h_1(E_1), \\dots, h_1(E_k)\\},\\] where the infimum …
View article: The Eigenvalue Problem for the $\infty$-Bilaplacian
The Eigenvalue Problem for the $\infty$-Bilaplacian Open
We consider the problem of finding and describing minimisers of the Rayleigh quotient \[ Λ_\infty \, :=\, \inf_{u\in \mathcal{W}^{2,\infty}(Ω)\setminus\{0\} }\frac{\|Δu\|_{L^\infty(Ω)}}{\|u\|_{L^\infty(Ω)}}, \] where $Ω\subseteq \mathbb{R}…
View article: Stability of variational eigenvalues for the fractional $p-$Laplacian
Stability of variational eigenvalues for the fractional $p-$Laplacian Open
By virtue of Γ-convergence arguments, we investigate the stability of variational eigenvalues associated with a given topological index for the fractional p-Laplacian operator, in the singular limit as the nonlocal operator converges to th…
View article: Stability of variational eigenvalues for the fractional p-Laplacian
Stability of variational eigenvalues for the fractional p-Laplacian Open
By virtue of $Γ-$convergence arguments, we investigate the stability of variational eigenvalues associated with a given topological index for the fractional $p$-Laplacian operator, in the singular limit as the nonlocal operator converges t…
View article: Reverse Cheeger inequality for planar convex sets
Reverse Cheeger inequality for planar convex sets Open
We prove the sharp inequality \[ J(Ω) := \frac{λ_1(Ω)}{h_1(Ω)^2} < \frac{π^2}{4},\] where $Ω$ is any planar, convex set, $λ_1(Ω)$ is the first eigenvalue of the Laplacian under Dirichlet boundary conditions, and $h_1(Ω)$ is the Cheeger con…