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Francesco Della Pietra

Publications and source records attributed to Francesco Della Pietra.

At least 19 recordsLinked to original sources
arXiv2026

An optimal inequality between nonlinear eigenvalues in convex sets

Let $λ_p(Ω)$ be the first Dirichlet eigenvalue of the $p$-Laplacian in a bounded convex open set $Ω\subset\mathbb R^N$, $N\ge2$. We prove that $p\mapstoλ_p(Ω)^{1/p}/π_p$ is strictly decreasing, where $π_p^p$ is the first eigenvalue on the unit interval. Consequently, for $1<q<p<\infty$, the sharp upper bound for $λ_p(Ω)^{1/p}/λ_q(Ω)^{1/q}$ is $π_p/π_q$. This bound is not attained and is approached by slabs, resolving the conjecture on the supremum formulated in the plane by Parini (J. Convex Anal. 2017). The proof combines log-concavity of the first eigenfunction with a vector field constructed from the one-dimensional eigenfunctions. The endpoint limits recover the Hersch--Protter and reverse Cheeger (or Buser) inequalities.

math.AP↗
arXiv2026

Optimal insulation and concentration breaking for nonlinear Robin boundary value problems

We consider an optimal insulation problem for a bounded domain in $\mathbb{R}^N$ driven by the $p$-Laplace operator ($p>1$). We model the convective heat transfer between the body and the environment, which corresponds, before insulation, to a nonlinear Robin boundary value problem. Assuming the body is surrounded by a thin layer of insulating material of size $\varepsilon^{\frac{1}{p-1}}$, we compute the $Γ$-limit of the governing energy functional as $\varepsilon \to 0^+$. Furthermore, we study the optimization of the heat content among all possible distributions of the insulating material with a fixed total mass. Finally, we highlight a concentration breaking phenomenon. Under a suitable non-degeneracy condition, if the boundary of the domain is connected or the external temperature profile is constant, the optimal insulating layer fails to cover the entire boundary whenever the total mass is sufficiently small. This is shown to be optimal: an explicit example provides that a disconnected boundary can trigger an anomalous double-phase transition, causing the insulation to fracture again even at intermediate mass regimes.

math.AP↗
arXiv2026

Asymptotic behavior of the first Robin eigenvalue of nonlinear operators

Let $Ω$ be a bounded Lipschitz domain of $\mathbb R^N$, $N\geq 2$. In this paper, we study the asymptotic behavior of the first Robin eigenvalue of the $p$-Laplace operator as $β$ goes to $0$ and as $β$ goes to $+\infty$, deriving sharp asymptotic expansions of the eigenvalue; the expansion in the Dirichlet limit $β\to+\infty$ is obtained under the additional assumption that $\partialΩ$ is of class $C^{1,1}$.

math.AP↗
arXiv2026

Maximization of the efficiency of the first Dirichlet eigenfunction and improved eigenvalue inequalities

We study the efficiency of the first Dirichlet eigenfunction $u$ on bounded convex domains $Ω\subset \mathbb{R}^N$, defined as the ratio between the mean value of $u$ on $Ω$ and its maximum value. By exploiting improved log-concavity estimates, we establish new sharp lower bounds for the first eigenvalue $λ_1$ and upper bounds for the efficiency in terms of the geometry of the domain, refining classical inequalities by Payne, Stakgold, and Hersch. Furthermore, we investigate the asymptotic behavior of the efficiency for elongating planar convex domains, making use of 1D limit profiles and Schr{ö}dinger operators with convex potentials. As a main consequence of our analysis, we prove that among all planar convex domains the Payne-Stakgold upper bound is not optimal, and that there exists a maximizer of the efficiency.

math.AP↗
arXiv2025

Weighted Robin eigenvalue problems and nonlinear elliptic equations with general growth in the gradient

We prove an existence result for Robin boundary value problems modeled on \[ \begin{cases} Δu + |\nabla u|^2 + λf(x) = 0 & \text{in } Ω \\ \frac{\partial u}{\partial ν} + βu = 0 & \text{on } \partialΩ\end{cases} \] where $Ω$ is a bounded, sufficiently smooth open set in $\mathbb R^N$, $f(x)$ belongs to the Marcinkiewicz space $M^{\frac N2}$ and {$β>0$}, under a smallness assumption on the datum $λ$. In order to study such problem, we will show several properties of the weighted, singular Robin eigenvalue problem \[ λ_{1,f,γ}(Ω)= \inf_{ψ\in H^{1},\;\int_Ωfψ^{2}=1}\left\{\int_Ω|\nabla ψ|^{2}dx+γ\int_{\partialΩ}ψ^{2}\right\}. \]

math.AP↗
arXiv2024

Some remarks on optimal insulation with Robin boundary conditions

We consider an optimal insulation problem of a given domain in $\mathbb R^N$. We study a model of heat trasfer determined by convection; this corresponds, before insulation, to a Robin boundary value problem. We deal with a prototype which involves the first eigenvalue of an elliptic differential operator. Such optimization problem, if the convection heat transfer coefficient is sufficiently large and the total amount of insulation is small enough, presents a symmetry breaking.

math.AP↗
arXiv2024

Existence and regularity results for a class of non-uniformly elliptic Robin problems

In this paper, we study the existence and the summability of solutions to a Robin boundary value problem whose prototype is the following: $$ \begin{cases} -\text{div}(b(|u|)\nabla u)=f &\text{in }Ω,\\[.2cm] \displaystyle\frac{\partial u}{\partial ν}+βu=0 &\text{on }\partialΩ\end{cases} $$ where $Ω$ is a bounded Lipschitz domain in $\mathbb R^N$, $N>2$, $β>0$, $b(s)$ is a positive function which may vanish at infinity and $f$ belongs to a suitable Lebesgue space. The presence of such a function $b$ in the principal part of the operator prevents it from being uniformly elliptic when $u$ is large.

math.AP↗
arXiv2024

Finite element approximation of the Hardy constant

We consider finite element approximations to the optimal constant for the Hardy inequality with exponent $p=2$ in bounded domains of dimension $n=1$ or $n \geq 3$. For finite element spaces of piecewise linear and continuous functions on a mesh of size $h$, we prove that the approximate Hardy constant converges to the optimal Hardy constant at a rate proportional to $1/| \log h |^2$. This result holds in dimension $n=1$, in any dimension $n \geq 3$ if the domain is the unit ball and the finite element discretization exploits the rotational symmetry of the problem, and in dimension $n=3$ for general finite element discretizations of the unit ball. In the first two cases, our estimates show excellent quantitative agreement with values of the discrete Hardy constant obtained computationally.

math.NA↗
arXiv2023

On a nonlinear Robin problem with an absorption term on the boundary and $L^1$ data

We deal with existence and uniqueness of nonnegative solutions to \begin{equation*} \left\{ \begin{array}{l} -Δu = f(x) \text{ in }Ω, \frac{\partial u}{\partial ν} + λ(x) u = \frac{g(x)}{u^η} \text{ on } \partialΩ, \end{array} \right. \end{equation*} where $η\ge 0$ and $f,λ$ and $g$ are nonnegative integrable functions. The set $Ω\subset\mathbb{R}^N (N> 2)$ is open and bounded with smooth boundary and $ν$ denotes its unit outward normal vector. More generally, we handle equations driven by monotone operators of $p$-Laplacian type jointly with nonlinear boundary conditions. We prove existence of an entropy solution and check that this solution is unique under natural assumptions. Among other features, we study the regularizing effect given to the solution by both the absorption and the nonlinear boundary term.

math.AP↗
arXiv2023

On the first Robin eigenvalue of the Finsler $p$-Laplace operator as $p\to 1$

Let $Ω$ be a bounded, connected, sufficiently smooth open set, $p>1$ and $β\in\mathbb R$. In this paper, we study the $Γ$-convergence, as $p\rightarrow 1^+$, of the functional \[ J_p(φ)=\frac{\int_ΩF^p(\nabla φ)dx+β\int_{\partial Ω} |φ|^pF(ν)d\mathcal{H}^{N-1}}{\int_Ω|φ|^pdx} \] where $φ\in W^{1,p}(Ω)\setminus\{0\}$ and $F$ is a sufficientely smooth norm on $\mathbb R^n$. We study the limit of the first eigenvalue $λ_1(Ω,p,β)=\inf_{\substack{φ\in W^{1,p}(Ω)\\ φ\ne 0}}J_p(φ)$, as $p\to 1^+$, that is: \begin{equation*} Λ(Ω,β)=\inf_{\substack{φ\in BV(Ω)\\ φ\not\equiv 0}}\dfrac{|Du|_F(Ω)+\min\{β,1\}\displaystyle \int_{\partial Ω}|φ|F(ν)d\mathcal H^{N-1}}{\displaystyle s\int_Ω|φ|dx}. \end{equation*} Furthermore, for $β>-1$, we obtain an isoperimetric inequality for $Λ(Ω,β)$ depending on $β$. The proof uses an interior approximation result for $BV(Ω)$ functions by $C^\infty(Ω)$ functions in the sense of strict convergence on $\mathbb R^n$ and a trace inequality in $BV$ with respect to the anisotropic total variation.

math.AP↗
arXiv2022

Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic $p$-Laplace operator, namely: \begin{equation*} λ_1(β,Ω)=\min_{ψ\in W^{1,p}(Ω)\setminus\{0\} } \frac{\displaystyle\int_ΩF(\nabla ψ)^p dx +β\displaystyle\int_{\partialΩ}|ψ|^pF(ν_Ω) d\mathcal H^{N-1} }{\displaystyle\int_Ω|ψ|^p dx}, \end{equation*} where $p\in]1,+\infty[$, $Ω$ is a bounded, mean convex domain in $\mathbb R^{N}$, $ν_Ω$ is its Euclidean outward normal, $β$ is a real number, and $F$ is a sufficiently smooth norm on $\mathbb R^{N}$. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on $β$ and on geometrical quantities associated to $Ω$. More precisely, we prove a lower bound of $λ_{1}$ in the case $β>0$, and a upper bound in the case $β<0$. As a consequence, we prove, for $β>0$, a lower bound for $λ_{1}(β,Ω)$ in terms of the anisotropic inradius of $Ω$ and, for $β<0$, an upper bound of $λ_{1}(β,Ω)$ in terms of $β$.

math.AP↗
arXiv2022

Behaviour of solutions to $p$-Laplacian with Robin boundary conditions as $p$ goes to $1$

We study the asymptotic behaviour, as $p\to 1^{+}$, of the solutions of the following inhomogeneous Robin boundary value problem: \begin{equation} \label{pbabstract} \tag{P} \left\{\begin{array}{ll} \displaystyle -Δ_p u_p = f & \text{in }Ω, \displaystyle |\nabla u_p|^{p-2}\nabla u_p\cdot ν+λ|u_p|^{p-2}u_p = g& \text{on } \partialΩ, \end{array}\right. \end{equation} where $Ω$ is a bounded domain in $\mathbb R^{N}$ with sufficiently smooth boundary, $ν$ is its unit outward normal vector and $Δ_p v$ is the $p$-Laplacian operator with $p>1$. The data $f\in L^{N,\infty}(Ω)$ (which denotes the Marcinkiewicz space) and $λ,g$ are bounded functions defined on $\partialΩ$ with $λ\ge0$. We find the threshold below which the family of $p$--solutions goes to 0 and above which this family blows up. As a second interest we deal with the $1$-Laplacian problem formally arising by taking $p\to 1^+$ in \eqref{pbabstract}.

math.AP↗
arXiv2022

On the behaviour of the first eigenvalue of the $p$-Laplacian with Robin boundary conditions as $p$ goes to $1$

In this paper we study the $Γ$-limit, as $p\to 1$, of the functional $$ J_{p}(u)=\frac{\displaystyle\int_Ω|\nabla u|^p + β\int_{ \partial Ω} |u|^p}{\displaystyle \int_Ω|u|^p}, $$ where $Ω$ is a smooth bounded open set in $\mathbb R^{N}$, $p>1$ and $β$ is a real number. Among our results, for $β>-1$, we derive an isoperimetric inequality for \[ Λ(Ω,β)=\inf_{u \in BV(Ω), u\not \equiv 0} \frac{\displaystyle |Du|(Ω) + \min(β,1)\int_{ \partial Ω} |u|}{\displaystyle \int_Ω|u|} \] which is the limit as $p\to 1^{+}$ of $ λ(Ω,p,β)= \displaystyle \min_{u\in W^{1,p}(Ω)} J_{p}(u). $ We show that among all bounded and smooth open sets with given volume, the ball maximizes $Λ(Ω, β)$ when $β\in$ $(-1,0)$ and minimizes $Λ(Ω, β)$ when $β\in[0, \infty)$.

math.AP↗
arXiv2021

Efficiency and localisation for the first Dirichlet eigenfunction

Bounds are obtained for the efficiency or mean to peak ratio $E(Ω)$ for the first Dirichlet eigenfunction (positive) for open, connected sets $Ω$ with finite measure in Euclidean space $\R^m$. It is shown that (i) localisation implies vanishing efficiency, (ii) a vanishing upper bound for the efficiency implies localisation, (iii) localisation occurs for the first Dirichlet eigenfunctions for a wide class of elongating bounded, open, convex and planar sets, (iv) if $Ω_n$ is any quadrilateral with perpendicular diagonals of lengths $1$ and $n$ respectively, then the sequence of first Dirichlet eigenfunctions localises, and $E(Ω_n)=O\big(n^{-2/3}\log n\big)$. This disproves some claims in the literature. A key technical tool is the Feynman-Kac formula.

math.SP↗
arXiv2021

Symmetrization with respect to mixed volumes

In this paper we introduce new symmetrization with respect to mixed volume or anisotropic curvature integral, which generalizes the one with respect to quermassintegral due to Talenti and Tso. We show a Pólya-Szego type principle for such symmetrization -- it diminishes the anisotropic Hessian integral for quasi-convex functions. We achieve this by a systematic study of invariants on non-symmetric matrices with real eigenvalues and the higher order anisotropic mean curvatures of level sets, which may be of independent interest. As applications, we establish a comparison principle for anisotropic Hessian equations and sharp anisotropic Sobolev inequalities.

math.AP↗
arXiv2021

An optimal insulation problem

In this paper we consider a minimization problem which arises from thermal insulation. A compact connected set $K$, which represents a conductor of constant temperature, say $1$, is thermally insulated by surrounding it with a layer of thermal insulator, the open set $Ω\setminus K$ with $K\subset\barΩ$. The heat dispersion is then obtained as \[ \inf\left\{ \int_Ω|\nabla φ|^{2}dx +β\int_{\partial^{*}Ω}φ^{2}d\mathcal H^{n-1} ,\;φ\in H^{1}(\mathbb R^{n}), \, φ\ge 1\text{ in } K\right\}, \] for some positive constant $β$. We mostly restrict our analysis to the case of an insulating layer of constant thickness. We let the set $K$ vary, under prescribed geometrical constraints, and we look for the best (or worst) geometry in terms of heat dispersion. We show that under perimeter constraint the disk in two dimensions is the worst one. The same is true for the ball in higher dimension but under different constraints. We finally discuss few open problems.

math.AP↗
arXiv2020

An optimization problem in thermal insulation with Robin boundary conditions

We study thermal insulating of a bounded body $Ω\subset \mathbb{R}^n$. Under a prescribed heat source $f\geq 0$, we consider a model of heat transfer between $Ω$ and the environment determined by convection; this corresponds, before insulation, to Robin boundary conditions. The body is then surrounded by a layer of insulating material of thickness of size $\varepsilon>0$, and whose conductivity is also proportional to $\varepsilon$. This corresponds to the case of a small amount of insulating material, with excellent insulating properties. We then compute the $Γ$-limit of the energy functional $F_\varepsilon$ and prove that this is a functional $F$ whose minimizers still satisfy an elliptic PDEs system with a non uniform Robin boundary condition depending on the distribution of insulating layer around $Ω$. In a second step we study the maximization of heat content (which measures the goodness of the insulation) among all the possible distributions of insulating material with fixed mass, and prove an optimal upper bound in terms of geometric properties. Eventually we prove a conjecture which states that the ball surrounded by a uniform distribution of insulating material maximizes the heat content.

math.AP↗