arXiv subjects
Publications and source records attributed to Emanuele Cristoforoni.
We variationally characterize the bottom of the spectrum of the buckling problem in an infinite cylinder $A\times\mathbb{R}^{n-d}$, where $A$ is an open bounded subset of $\mathbb{R}^d$, and compute it explicitly when $A$ is a ball.
We consider the first eigenvalue, $λ_β(Ω,A)$, of a two-phase eigenvalue problem for the Laplacian with Robin boundary conditions, where the two phases are characterised by different ellipticity constants. We characterise the conditions under which a ball $B_R$ is a local minimum under a volume constraint for the minimisation problem $A\mapstoλ_β(B_r,A)$, in terms of the principal Neumann eigenvalue of the fixed inner ball $B_r$.
We investigate the spectral properties of a differential elliptic operator on $H^1(\barΩ\cup Σ)$, where $Ω$ is a smooth domain surrounded by a layer $Σ$. The thickness of the layer is given by $\varepsilon h$, where $h$ is a positive function defined on the boundary $\partial Ω$ and $\varepsilon$ is the ellipticity constant of the operator in $Σ$. We prove that, in the limit for $\varepsilon$ going to $0$, the spectrum converges to the spectrum of a differential elliptic operator in $H^1(Ω)$, and we investigate a first-order asymptotic development.
In the present survey, we consider the classical reinforcement problem for elliptic boundary value problems originally studied by Sanchez-Palencia in 1969. We focus on the seminar papers by Brezis, Caffarelli, & Friedman, and by Acerbi & Buttazzo, and discuss the related optimisation problems proposed by Friedman and by Buttazzo.
A celebrated inequality by Payne relates the first eigenvalue of the Dirichlet Laplacian to the first eigenvalue of the buckling problem. Motivated by the goal of establishing a quantitative version of this inequality, we show that Payne's original estimate - which is not sharp - can in fact be improved. Our result provides a refined spectral bound and opens the way to further investigations into quantitative enhancements of classical inequalities in spectral theory.
We investigate the relationship between the Neumann and Steklov principal eigenvalues emerging from the study of collapsing convex domains in $\mathbb{R}^2$. Such a relationship allows us to give a partial proof of a conjecture concerning estimates of the ratio of the former to the latter: we show that thinning triangles maximize the ratio among convex thinning sets, while thinning rectangles minimize the ratio among convex thinning with some symmetry property.
For every given $β<0$, we study the problem of maximizing the first Robin eigenvalue of the Laplacian $λ_β(Ω)$ among convex (not necessarily smooth) sets $Ω\subset\mathbb{S}^{n}$ with fixed perimeter. In particular, denoting by $σ_n$ the perimeter of the $n$-dimensional hemisphere, we show that for fixed perimeters $P<σ_n$, geodesic balls maximize the eigenvalue. Moreover, we prove a quantitative stability result for this isoperimetric inequality in terms of volume difference between $Ω$ and the ball $D$ of the same perimeter.
We investigate the behavior of the solution to an elliptic diffraction problem in the union of a smooth set $Ω$ and a thin layer $Σ$ locally described by $\varepsilon h$, where $h$ is a positive function defined on the boundary $\partialΩ$, and $\varepsilon$ is the ellipticity constant of the differential operator in the thin layer $Σ$. We study the problem in the limit for $\varepsilon$ going to zero and prove a first-order asymptotic development by $Γ$-convergence of the associated energy functional.
We are interested in the thermal insulation of a bounded open set $Ω$ surrounded by a set whose thickness is locally described by $\varepsilon h$, where $h$ is a non-negative function defined on the boundary $\partialΩ$. We study the problem in the limit for $\varepsilon$ going to zero using a first-order asymptotic development by $Γ$-convergence.
In this paper, we generalize the notion of relative $p$-capacity of $K$ with respect to $Ω$, by replacing the Dirichlet boundary condition with a Robin one. We show that, under volume constraints, our notion of $p$-capacity is minimal when $K$ and $Ω$ are concentric balls. We use the $H$-function and a derearrangement technique.
We study a nonlinear generalization of a free boundary problem that arises in the context of thermal insulation. We consider two open sets $Ω\subseteq A$, and we search for an optimal $A$ in order to minimize a non-linear energy functional, whose minimizers $u$ satisfy the following conditions: $Δ_p u=0$ inside $A\setminusΩ$, $u=1$ in $Ω$, and a nonlinear Robin-like boundary $(p,q)$-condition on the free boundary $\partial A$. We study the variational formulation of the problem in SBV, and we prove that, under suitable conditions on the exponents $p$ and $q$, a minimizer exists and its jump set satisfies uniform density estimates.
We study the thermal insulation of a bounded body $Ω\subset\mathbb{R}^n$, under a prescribed heat source $f>0$, via a bulk layer of insulating material. We consider a model of heat transfer between the insulated body and the environment determined by convection; this corresponds to Robin boundary conditions on the free boundary of the layer. We show that a minimal configuration exists and that it satisfies uniform density estimates.