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Andrea Gentile

Publications and source records attributed to Andrea Gentile.

15 recordsLinked to original sources
arXiv2026

Quantitative Kröger inequalities for Neumann eigenvalues of convex domains

Refining the sharp upper bounds $μ_{k,d}^* $ obtained by Kröger (1999) for the $k$-th Neumann eigenvalue of a convex domain $Ω\subset \mathbb{R}^d$, we prove the following inequalities: for any $k\in \mathbb{N}$ there exists a constant $C(k,d) >0$ such that $$D_Ω^2 μ_k(Ω) \leq μ_{k,d}^* - C(k,d) a_2(Ω)^2/D_Ω^2$$ where $D_Ω$ is the diameter of $Ω$ and $a_2(Ω)$ is the second largest semiaxis of the John ellipsoid of $Ω$. In the planar case, for $k=1$ we also give an explicit value of the constant $C(1,2)$.

math.AP↗
arXiv2025

Some optimal control and shape optimisation problems for bulk-surface cooperative systems

The goal of this paper is to address some optimal control and shape optimisation problems arising from bulk-surface cooperative systems. The basic model under consideration is the following: letting $Ω$ be a fixed domain, we assume that a population (with density $u$) lives inside $Ω$ and can access some resources $f$, while a second population (with density $v$) lives on the boundary $\partial Ω$ and can access other resources $g$. These two populations are coupled in a cooperative manner by a constant exchange rate at the boundary, leading to a non-standard PDE system that has already been studied in previous works by Bogosel, Giletti and Tellini, for its connection with road-field models. Building on the considerations of the aforementioned previous works, we have two main objectives here: first, investigate the question of optimal resources distribution inside the domain $Ω$ and on the surface $\partial Ω$, i.e. how to spread resources in order to guarantee an optimal survival of the two species. We establish rigid Talenti inequalities and comparison results when $Ω$ is a ball, extending in particular the results of J. J. Langford on symmetrisation for Neumann and Robin problems. Second, when the resources distribution $f$ and $g$ are constant, we provide a partial analysis of the natural shape optimisation problem: which shape $Ω$ maximises the survival rate of the two species? Namely, we show that in certain regimes there can be no optimal shape and, by computing second-order shape derivatives, we investigate the local optimality of the ball.

math.AP↗
arXiv2023

On the gradient rearrangement of functions

In this paper, we introduce a symmetrization technique for the gradient of a $\BV$ function, which separates its absolutely continuous part from its singular part (sum of the jump and the Cantorian part). In particular, we prove an $\text{\emph{L}}^{\text{1}}$ comparison between the function and its symmetrized. Furthermore, we apply this result to obtain Saint-Venant type inequalities for some geometric functionals.

math.AP↗
arXiv2023

On the symmetric rearrangement of the gradient of a Sobolev function

In this paper, we generalize a classical comparison result for solutions to Hamilton-Jacobi equations with Dirichlet boundary conditions, to solutions to Hamilton-Jacobi equations with non-zero boundary trace. As a consequence, we prove the isoperimetric inequality for the torsional rigidity (with Robin boundary conditions) and for other functionals involving such boundary conditions.

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arXiv2023

Higher regularity for weak solutions to degenerate parabolic problems

In this paper, we study the regularity of weak solutions to the following strongly degenerate parabolic equation \begin{equation*} u_t-÷\left(\left(\left|Du\right|-1\right)_+^{p-1}\frac{Du}{\left|Du\right|}\right)=f\qquad\mbox{ in }Ω_T, \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $p\geq2$ and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. We prove the higher differentiability of a nonlinear function of the spatial gradient of the weak solutions, assuming only that $f\in L^{2}_{\loc}\left(Ω_T\right)$. This allows us to establish the higher integrability of the spatial gradient under the same minimal requirement on the datum $f$.

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arXiv2022

Estimates for Robin $p$-Laplacian eigenvalues of convex sets with prescribed perimeter

In this paper, we prove an upper bound for the first Robin eigenvalue of the $p$-Laplacian with a positive boundary parameter and a quantitative version of the reverse Faber-Krahn type inequality for the first Robin eigenvalue of the $p$-Laplacian with negative boundary parameter, among convex sets with prescribed perimeter. The proofs are based on a comparison argument obtained by means of inner sets, introduced by Payne, Weimberger and Polya.

math.AP↗
arXiv2022

Higher differentiability results for solutions to a class of non-homogeneouns elliptic problems under sub-quadratic growth conditions

We prove a sharp higher differentiability result for local minimizers of functionals of the form $$\mathcal{F}\left(w,Ω\right)=\int_Ω\left[ F\left(x,Dw(x)\right)-f(x)\cdot w(x)\right]dx$$ with non-autonomous integrand $F(x,ξ)$ which is convex with respect to the gradient variable, under $p$-growth conditions, with $1<p<2$. The main novelty here is that the results are obtained assuming that the partial map $x\mapsto D_ξF(x,ξ)$ has weak derivatives in some Lebesgue space $L^q$ and the datum $f$ is assumed to belong to a suitable Lebesgue space $L^r$. We also prove that it is possible to weaken the assumption on the datum $f$ and on the map $x\mapsto D_ξF(x,ξ)$, if the minimizers are assumed to be a priori bounded.

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arXiv2022

Regularity results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions

We establish some higher differentiability results for solution to non-autonomous obstacle problems of the form \begin{equation*} \min \left\{\int_Ωf\left(x, Dv(x)\right)dx\,:\, v\in \mathcal{K}_ψ(Ω)\right\}, \end{equation*} where the function $f$ satisfies $p-$growth conditions with respect to the gradient variable, for $1<p<2$, and $\mathcal{K}_ψ(Ω)$ is the class of admissible functions. Here we show that, if the obstacle $ψ$ is bounded, then a Sobolev regularity assumption on the gradient of the obstacle $ψ$ transfers to the gradient of the solution, provided the partial map $x\mapsto D_ξf(x,ξ)$ belongs to a Sobolev space, $W^{1, p+2}$. The novelty here is that we deal with subquadratic growth conditions with respect to the gradient variable, i.e. $f(x, ξ)\approx a(x)|ξ|^p$ with $1<p<2,$ and where the map $a$ belongs to a Sobolev space.

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arXiv2021

Regularity results for bounded solutions to obstacle problems with non-standard growth conditions

In this paper we consider a class of obstacle problems of the type %\begin{equation*} %\int_Ω\left \, \dx\ge0\qquad\forall %φ\in W^{1,q}(Ω) \quad {\mathrm{s.t.}} \quad φ\ge ψ%\end{equation*} \begin{equation*} \min \left\{\int_Ωf(x, Dv)\, \dx\,:\, v\in \mathcal{K}_ψ(Ω)\right\} \end{equation*} where $ψ$ is the obstacle, $\mathcal{K}_ψ(Ω)=\{v\in u_0+W^{1, p}_{0}(Ω, \R): v\geψ\text{ a.e. in }Ω\}$, with $u_0 \in W^{1,p}(Ω)$ a fixed boundary datum, the class of the admissible functions and the integrand $f(x, Dv)$ satisfies non standard $(p,q)$-growth conditions. \\ We prove higher differentiability results for bounded solutions of the obstacle problem under dimension-free conditions on the gap between the growth and the ellipticity exponents. Moreover, also the Sobolev assumption on the partial map $x\mapsto A(x, ξ)$ is independent of the dimension $n$ and this, in some cases, allows us to manage coefficients in a Sobolev class below the critical one $W^{1,n}$.

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arXiv2021

Comparison results for solutions to p-Laplace equations with Robin boundary conditions

In the last decades comparison results of Talenti type for Elliptic Problems with Dirichlet boundary conditions have been widely investigated. In this paper, we generalize the results obtained in arXiv:1909.11950 to the case of p-Laplace operator with Robin boundary conditions. The point-wise comparison, obtained in arXiv:1909.11950 only in the planar case, holds true in any dimension if p is sufficiently small.

math.AP↗
arXiv2020

Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions

We establish some higher differentiability results of integer and fractional order for solution to non-autonomous obstacle problems of the form \begin{equation*} \min \left\{\int_Ωf(x, Dv(x))\,:\, v\in \mathcal{K}_ψ(Ω)\right\}, \end{equation*} where the function $f$ satisfies $p-$growth conditions with respect to the gradient variable, for $1<p<2$, and $\mathcal{K}_ψ(Ω)$ is the class of admissible functions $v\in u_0+W^{1, p}_0(Ω)$ such that $v\geψ$ a. e. in $Ω$, where $u_0\in W^{1,p}(Ω)$ is a fixed boundary datum. Here we show that a Sobolev or Besov-Lipschitz regularity assumption on the gradient of the obstacle $ψ$ transfers to the gradient of the solution, provided the partial map $x\mapsto D_ξf(x,ξ)$ belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with subquadratic growth conditions with respect to the gradient variable, i. e. $f(x, ξ)\approx a(x)|ξ|^p$ with $1<p<2,$ and where the map $a$ belongs to a Sobolev or Besov-Lipschitz space.

math.AP↗
arXiv2019

Regularity results for solutions to obstacle problems with Sobolev coeffcients

We establish the higher differentiability of solutions to a class of obstacle problems for integral functionals where the convex integrand f satisfies p-growth conditions with respect to the gradient variable. We derive that the higher differentiability property of the weak solution v is related to the regularity of the assigned , under a suitable Sobolev assumption on the partial map that measures the oscillation of f with respect to the x variable. The main novelty is that such assumption is independent of the dimension n and that, in the case p<=n-2, improves previous known results.

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arXiv2019

Regularity for minimizers of a class of non-autonomous functionals with sub-quadratic growth

We consider a class of integral functionals with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space W^{1;q}. We prove a result of higer differentiability for the minimizers. We also infer a result of Lipschitz regularity of minimizers if q > n, and a result of higher integrability for the gradient if q = n. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.

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arXiv2018

Regularity for minimizers of non-autonomous non-quadratic functionals in the case 1 < p < 2: an a priori estimate

We prove an a priori estimate for the second derivatives of local minimizers of integral functionals of calculus of variation with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.

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