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Francesca Prinari

Publications and source records attributed to Francesca Prinari.

17 recordsLinked to original sources
arXiv2026

Sharp Makai-type inequalities for the best Poincar\'e-Sobolev constants

Given a bounded convex open set $\Omega\subseteq \mathbb R^N$, we prove that the Poincar\'e-Sobolev constants $\lambda_{p,q}(\Omega)$ can be bounded from below by the $p$-power of the ratio between the perimeter of $\Omega$ and a suitable power of its volume, with an optimal constant which is explicitly given. This generalises an old result for torsional rigidity due to Makai when $N=2$. The proof relies on new geometric optimal bounds for the Lebesgue norms of the distance function from the boundary which are of independent interest. These results allow us to give a complete picture of the sharp inequalities for $\lambda_{p,q}(\Omega)$ in terms of suitable powers of perimeter, inradius and volume of $\Omega$.

math.AP↗
arXiv2026

Stochastic homogenization of nonconvex unbounded integral functionals with generalized Orlicz growth

We consider the homogenization of random integral functionals which are possibly unbounded, that is, the domain of the integrand is not the whole space and may depend on the space-variable. In the vectorial case, we develop a complete stochastic homogenization theory for nonconvex unbounded functionals with convex growth of generalized Orlicz-type, under a standard set of assumptions in the field, in particular a coercivity condition of order $p^->1$, and an upper bound of order $p^+<\infty$. The limit energy is defined in a possibly anisotropic Musielak-Orlicz space, for which approximation results with smooth functions are provided. The proof is based on the localization method of $\Gamma$-convergence and a careful use of truncation arguments.

math.OC↗
arXiv2026

Low eigenvalues of the $p-$Laplacian in general open sets

We consider the minmax Ljusternik-Schnirelmann levels of the constrained $p-$Dirichlet integral, on a general open set of the Euclidean space. We show that, whenever one of these levels lies below the threshold given by the $L^p$ Poincar\'e constant ``at infinity'', it actually defines an eigenvalue of the Dirichlet $p-$Laplacian. We also prove an exponential decay at infinity for the relevant eigenfunctions: this can be seen as a \v{S}nol-Simon--type estimate for the nonlinear case. Finally, we exhibit some peculiar examples of unbounded open sets to which our main result applies.

math.AP↗
arXiv2025

Extremals for sharp Poincar\'e-Sobolev inequalities: periodically perforated sets and beyond

We consider periodically perforated unbounded open sets and prove existence of extremals for the relevant sharp Poincar\'e-Sobolev embedding constant. The existence result holds no matter the shape or the regularity of the hole: it is sufficient that the latter is a compact set with positive capacity. We also show how to apply the main result in order to get a similar existence statement, for sets which are periodic in some directions and bounded in all the others.

math.AP↗
arXiv2025

Extremals for Poincar\'e-Sobolev sharp constants in Steiner symmetric sets

We prove existence of minimizers for the sharp Poincar\'e-Sobolev constant in general Steiner symmetric sets, in the subcritical and superhomogeneous regime. The sets considered are not necessarily bounded, thus the relevant embeddings may suffer from a lack of compactness. We prove existence by means of an elementary compactness method. We also prove an exponential decay at infinity for minimizers, showing that in the case of Steiner symmetric sets the relevant estimates only depend on the underlying geometry. Finally, we illustrate the optimality of the existence result, by means of some examples.

math.AP↗
arXiv2024

On fractional Hardy-type inequalities in general open sets

We show that, when $sp>N$, the sharp Hardy constant $\mathfrak{h}_{s,p}$ of the punctured space $\mathbb R^N\setminus\{0\}$ in the Sobolev-Slobodecki\u{\i} space provides an optimal lower bound for the Hardy constant $\mathfrak{h}_{s,p}(\Omega)$ of an open $\Omega\subsetneq \mathbb R^N$. The proof exploits the characterization of Hardy's inequality in the fractional setting in terms of positive local weak supersolutions of the relevant Euler-Lagrange equation and relies on the construction of suitable supersolutions by means of the distance function from the boundary of $\Omega$. Moreover, we compute the limit of $\mathfrak{h}_{s,p}$ as $s\nearrow 1$, as well as the limit when $p \nearrow \infty$. Finally, we apply our results to establish a lower bound for the non-local eigenvalue $\lambda_{s,p}(\Omega)$ in terms of $\mathfrak{h}_{s,p}$ when $sp>N$, which, in turn, gives an improved Cheeger inequality whose constant does not vanish as $p\nearrow \infty$.

math.AP↗
arXiv2023

On Morrey's inequality in Sobolev-Slobodecki\u{\i} spaces

We study the sharp constant in the Morrey inequality for fractional Sobolev-Slobodecki\u{\i} spaces on the whole $\mathbb{R}^N$. By generalizing a recent work by Hynd and Seuffert, we prove existence of extremals, together with some regularity estimates. We also analyze the sharp asymptotic behaviour of this constant as we reach the borderline case $s\,p=N$, where the inequality fails. This can be done by means of a new elementary proof of the Morrey inequality, which combines: a local fractional Poincar\'e inequality for punctured balls, the definition of capacity of a point and Hardy's inequality for the punctured space. Finally, we compute the limit of the sharp Morrey constant for $s\nearrow 1$, as well as its limit for $p\nearrow \infty$. We obtain convergence of extremals, as well.

math.AP↗
arXiv2023

On the sharp Makai inequality

On a convex bounded open set, we prove that Poincar\'e-Sobolev constants for functions vanishing at the boundary can be bounded from below in terms of the norm of the distance function in a suitable Lebesgue space. This generalizes a result shown, in the planar case, by E. Makai, for the torsional rigidity. In addition, we compare the sharp Makai constants obtained in the class of convex sets with the optimal constants defined in other classes of open sets. Finally, an alternative proof of the Hersch-Protter inequality for convex sets is given.

math.OC↗
arXiv2023

Asymptotic analysis of thin structures with point dependent energy growth

$3d-2d$ dimensional reduction for hyperelastic thin films modeled through energies with point dependent growth, assuming that the sample is clamped on the lateral boundary, is performed in the framework of $\Gamma$-convergence. Integral representation results, with a more regular lagrangian related to the original energy density, are provided for the lower dimensional limiting energy, in different contexts.

math.AP↗
arXiv2023

Sobolev embeddings and distance functions

On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space $\mathcal{D}^{1,p}_0$ into $L^q$ and the summability properties of the distance function. We prove that in the superconformal case (i.e. when $p$ is larger than the dimension) these two facts are equivalent, while in the subconformal and conformal cases (i.e. when $p$ is less than or equal to the dimension) we construct counterexamples to this equivalence. In turn, our analysis permits to study the asymptotic behaviour of the positive solution of the Lane-Emden equation for the $p-$Laplacian with sub-homogeneous right-hand side, as the exponent $p$ diverges to $\infty$. The case of first eigenfunctions of the $p-$Laplacian is included, as well. As particular cases of our analysis, we retrieve some well-known convergence results, under optimal assumptions on the open sets. We also give some new geometric estimates for generalized principal frequencies.

math.AP↗
arXiv2021

On a class of Cheeger inequalities

We study a general version of the Cheeger inequality by considering the shape functional $\mathcal{F}_{p,q}(\Omega)=\lambda_p^{1/p}(\Omega)/\lambda_q(\Omega)^{1/q}$. The infimum and the supremum of $\mathcal{F}_{p,q}$ are studied in the class of all domains $\Omega$ of $\mathbb{R}^d$ and in the subclass of convex domains. In the latter case the issue concerning the existence of an optimal domain for $\mathcal{F}_{p,q}$ is discussed.

math.OC↗
arXiv2021

A comparison principle for the Lane-Emden equation and applications to geometric estimates

We prove a comparison principle for positive supersolutions and subsolutions to the Lane-Emden equation for the $p-$Laplacian, with subhomogeneous power in the right-hand side. The proof uses variational tools and the result applies with no regularity assumptions, both on the set and the functions. We then show that such a comparison principle can be applied to prove: uniqueness of solutions; sharp pointwise estimates for positive solutions in convex sets; localization estimates for maximum points and sharp geometric estimates for generalized principal frequencies in convex sets.

math.AP↗
arXiv2020

$\Gamma$-convergence for power-law functionals with variable exponents

We study the $\Gamma$-convergence of the functionals $F_n(u):= || f(\cdot,u(\cdot),Du(\cdot))||_{p_n(\cdot)}$ and $\mathcal{F}_n(u):= \int_{\Omega} \frac{1}{p_n(x)} f^{p_n(x)}(x,u(x),Du(x))dx$ defined on $X\in \{L^1(\Omega,\mathbb{R}^d), L^\infty(\Omega,\mathbb{R}^d), C(\Omega,\mathbb{R}^d)\}$ (endowed with their usual norms) with effective domain the Sobolev space $W^{1,p_n(\cdot)}(\Omega, \mathbb{R}^d )$. Here $\Omega\subseteq \mathbb{R}^N$ is a bounded open set, $N,d \ge 1$ and the measurable functions $p_n: \overline{\Omega} \rightarrow (1, + \infty) $ satisfy the conditions ${\mathop{\rm ess\: sup }}_{\ \overline \Omega} p_n \le \, \beta \, {\mathop{\rm ess\: inf }}_{\ \overline \Omega} p_n $ for a fixed constant $\beta > 1$ and $ {\mathop{\rm ess\: inf }}_{\ \overline \Omega} p_n \rightarrow + \infty$ as $n \rightarrow + \infty$. We show that when $f(x,u,\cdot)$ is level convex and lower semicontinuous and it satisfies a uniform growth condition from below, then, as $n\to \infty$, the sequences $(F_n)_n$ $\Gamma$-converges in $X$ to the functional $F$ represented as $F(u)= || f(\cdot,u(\cdot),Du(\cdot))||_{\infty}$ on the effective domain $W^{1,\infty}(\Omega, \mathbb{R}^d )$. Moreover we show that the $\Gamma$-$\lim_n \mathcal F_n$ is given by the functional $ \mathcal{F}(u):=\left\{\begin {array}{lll} \!\!\!\!\!\! & 0 & \hbox{if } || f(\cdot,u(\cdot),Du(\cdot)) ||_{\infty}\leq 1,\\ \!\!\!\!\!\! & +\infty & \hbox{otherwise in } X.\\ \end{array}\right. $

math.OC↗
arXiv2019

A relaxation result in the vectorial setting and $L^p$-approximation for $L^\infty$-functionals

We provide relaxation for not lower semicontinuous supremal functionals of the type $W^{1,\infty}(\Omega;\mathbb R^d) \ni u \mapsto\supess_{ x \in \Omega}f(\nabla u(x))$ in the vectorial case, where $\Omega\subset \mathbb R^N$ is a Lipschitz, bounded open set, and $f$ is level convex. The connection with indicator functionals is also enlightened, thus extending previous lower semicontinuity results in that framework. Finally we discuss the $L^p$-approximation of supremal functionals, with non-negative, coercive densities $f=f(x,\xi)$, which are only $\L^N \otimes \B_{d \times N}$-measurable.

math.OC↗
arXiv2018

The role of intrinsic distances in the relaxation of $L^\infty$-functionals

We consider a supremal functional of the form $$F(u)=\mathop{\rm ess\: sup }_{x \in \Omega} f(x,Du(x))$$ where $\Omega\subseteq \mathbf {R}^N$ is a regular bounded open set, $u\in W^{1,\infty}(\Omega)$ and $f$ is a Borel function. Assuming that the intrinsic distances $d^{\lambda}_F(x,y):= \sup \Big\{ u(x) - u(y): \, F(u)\leq \lambda \Big\}$ are locally equivalent to the euclidean one for every $\lambda>\inf_{W^{1,\infty}(\Omega)} F$, we give a description of the sublevel sets of the weak$^*$-lower semicontinuous envelope of $F$ in terms of the sub-level sets of the difference quotient functionals $R_{d^\lambda_F}(u):=\sup_{x\not =y} \frac{u(x)-u(y)}{d^\lambda_F(x,y)}. $ As a consequence we prove that the relaxed functional of positive $1$-homogeneous supremal functionals coincides with $R_{d^1_F}$. Moreover, for a more general supremal functional $F$ (a priori non coercive), we prove that the sublevel sets of its relaxed functionals with respect to the weak$^*$ topology, the weak$^*$ convergence and the uniform convergence are convex. The proof of these results relies both on a deep analysis of the intrinsic distances associated to $F$ and on a careful use of variational tools such as $\Gamma$-convergence.

math.OC↗
arXiv2006

On the Yamabe equation with rough potentials

We study the existence of non--trivial solutions to the Yamabe equation: $$-Δu+ a(x)= μu|u|^\frac4{n-2} \hbox{} μ>0, x\in Ω\subset {\mathbf R}^n \hbox{with} n\geq 4,$$ $$ u(x)=0 \hbox{on} \partial Ω$$ under weak regularity assumptions on the potential $a(x)$. More precisely in dimension $n\geq 5$ we assume that: \begin{enumerate} \item $a(x)$ belongs to the Lorentz space $L^{\frac n2, d}(Ω)$ for some $1\leq d <\infty$, \item $a(x) \leq M<\infty \hbox{a.e.} x\in Ω$, \item the set $\{x\in Ω|a(x)<0\}$ has positive measure, \item there exists $c>0$ such that $$\int_Ω(|\nabla u|^2 + a(x) |u|^2) \hbox{} dx \geq c\int_Ω|\nabla u|^2 \hbox{} dx \hbox{} \forall u\in H^1_0(Ω).$$ \end{enumerate} \noindent In dimension $n=4$ the hypothesis $(2)$ above is replaced by $$a(x)\leq 0 \hbox{} a.e. \hbox{} x\in Ω.$$

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