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Chiara Leone

Publications and source records attributed to Chiara Leone.

15 recordsLinked to original sources
arXiv2026

Quasiconvexity and Lower Semicontinuity in the Riemannian Setting

We introduce a notion of quasiconvexity for integrands defined on the bundle of linear maps between the tangent spaces of two smooth, connected and complete Riemannian manifolds $(M,g)$ and $(N,h)$, naturally extending the classical Euclidean notion. For continuous integrands, we prove that this Riemannian quasiconvexity is equivalent to the sequential lower semicontinuity of the associated functional $$ F(u,Ω)=\int_Ωf(du)\,dμ_g $$ with respect to the weak$^*$ convergence in $W^{1,\infty}(Ω,N)$, in the intrinsic sense adopted in the paper, for every bounded open subset $Ω\subseteq M$. We then establish the Riemannian analogues of the Acerbi--Fusco theorems for Carathéodory integrands satisfying the corresponding growth conditions, for every $1\leqslant p\leqslant+\infty$, with respect to the weak and weak$^*$ notions of convergence considered here.

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arXiv2026

Nonlocal-to-local variational limits of energies on partitions

We study, through a $Γ$-convergence procedure, the asymptotic behaviour of a general class of nonlocal energies defined on functions taking values in a finite set $Y=\{y_1,\dots,y_N\}\subset\mathbb R^m$. These energies are written in terms of double integrals accounting for pairwise interactions in an open set $Ω$, weighted by a kernel $K_\varepsilon$. If the latter is controlled from above and below by suitable non-negative radial kernels with unit mass concentrating at the origin, then, up to subsequences, the $Γ$-limit has domain $BV(Ω;Y)$ and is given by a local surface energy. We show that the locality of the $Γ$-limit is not guaranteed if the concentration of mass fails. For a class of reference energies, we obtain a more explicit representation of the limit energy, also for non-radial kernels, recovering the limits of isotropic and anisotropic fractional perimeters. We also prove a homogenization result for periodic kernels.

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arXiv2026

Quasiconvexity in the Riemannian setting

We introduce a notion of quasiconvexity for continuous functions $f$ defined on the vector bundle of linear maps between the tangent spaces of a smooth Riemannian manifold $(M,g)$ and $\mathbb{R}^m$, naturally generalizing the classical Euclidean definition. We prove that this condition characterizes the sequential lower semicontinuity of the associated integral functional \[ F(u, Ω) = \int_Ω f(du) \, dμ\] with respect to the weak$^*$ topology of $W^{1,\infty}(Ω, \mathbb{R}^m)$, for every bounded open subset $Ω\subseteq M$.

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arXiv2025

Gradient regularity for double-phase orthotropic functionals

We prove higher integrability for local minimizers of the double-phase orthotropic functional \[ \sum_{i=1}^{n}\int_Ω\left(\left|u_{x_i}\right|^p+a(x)\left| u_{x_i}\right|^q\right)dx \] when the weight function $a \geq0$ is assumed to be $α$-Hölder continuous, while the exponents $p, q$ are such that $2 \leq p \leq q$ and $\frac{q}{p} < 1 + \fracα{n}$. Under natural Sobolev regularity of~$a$, we further obtain explicit Lipschitz regularity estimates for local minimizers.

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arXiv2024

Variational analysis of nonlocal Dirichlet problems in periodically perforated domains

In this paper we consider a family of non local functionals of convolution-type depending on a small parameter $\varepsilon>0$ and $Γ$-converging to local functionals defined on Sobolev spaces as $\varepsilon\to 0$. We study the asymptotic behaviour of the functionals when the order parameter is subject to Dirichlet conditions on a periodically perforated domains, given by a periodic array of small balls of radius $r_δ$ centered on a $δ$--periodic lattice, being $δ> 0$ an additional small parameter and $r_δ=o(δ)$. We highlight differences and analogies with the local case, according to the interplay between the three scales $\varepsilon$, $δ$ and $r_δ$. A fundamental tool in our analysis turns out to be a non local variant of the classical Gagliardo-Nirenberg-Sobolev inequality in Sobolev spaces which may be of independent interest and useful for other applications.

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arXiv2024

Strong existence for free-discontinuity problems with non-standard growth

An Ahlfors-type regularity result for free-discontinuity energies defined on the space $SBV^φ$ of special functions of bounded variation with $φ$-growth, where $φ$ is a generalized Orlicz function, is proved. Our analysis expands on the regularity theory for minimizers of a class of free-discontinuity problems in the non-standard growth case.

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arXiv2023

Regularity of minimizers for free-discontinuity problems with $p(\cdot)$-growth

A regularity result for free-discontinuity energies defined on the space $SBV^{p(\cdot)}$ of special functions of bounded variation with variable exponent is proved, under the assumption of a log-Hölder continuity for the variable exponent $p(x)$. Our analysis expand on the regularity theory for minimizers of a class of free-discontinuity problems in the nonstandard growth case. This may be seen as a follow-up of the paper Fusco, Mingione and Trombetti (2001), dealing with a constant exponent.

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arXiv2022

Singular orthotropic functionals with nonstandard growth conditions

We pursue the study of a model convex functional with orthotropic structure and nonstandard growth conditions, this time focusing on the sub-quadratic case. We prove that bounded local minimizers are locally Lipschitz. No restriction on the ratio between the highest and the lowest growth rates are needed. The result holds also in presence of a non-autonomous lower order term, under sharp integrability assumptions. Finally, we prove higher differentiability of bounded local minimizers, as well.

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arXiv2022

$\mathcal{A}$-caloric approximation and partial regularity for parabolic systems with Orlicz growth

We prove a new $\mathcal{A}$-caloric approximation lemma compatible with an Orlicz setting. With this result, we establish a partial regularity result for parabolic systems of the type $$ u_{t}- {\rm div} \,a(Du)=0. $$ Here the growth of $a$ is bounded by the derivative of an $N$-function $φ$. The primary assumption for $φ$ is that $tφ''(t)$ and $φ'(t)$ are uniformly comparable on $(0,\infty)$.

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arXiv2021

Partial regularity result for non-autonomous elliptic systems with general growth

In this paper we prove a Hölder partial regularity result for weak solutions $u:Ω\to \mathbb{R}^N$, $N\geq 2$, to non-autonomous elliptic systems with general growth of the type: \begin{equation*} -\rm{div}\, a(x, u, Du)= b(x, u, Du) \quad \mbox{ in } Ω. \end{equation*} The crucial point is that the operator $a$ satisfies very weak regularity properties and a general growth, while the inhomogeneity $b$ has a controllable growth.

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arXiv2018

On the Lipschitz character of orthotropic $p-$harmonic functions

We prove that local weak solutions of the orthotropic $p-$harmonic equation are locally Lipschitz, for every $p\ge 2$ and in every dimension. More generally, the result holds true for more degenerate equations with orthotropic structure, with right-hand sides in suitable Sobolev spaces.

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