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Giorgio Stefani

Publications and source records attributed to Giorgio Stefani.

At least 37 records · Page 2Linked to original sources
arXiv2023

Existence and uniqueness theorems for some semi-linear equations on locally finite graphs

We study some semi-linear equations for the $(m,p)$-Laplacian operator on locally finite weighted graphs. We prove existence of weak solutions for all $m\in\mathbb{N}$ and $p\in(1,+\infty)$ via a variational method already known in the literature by exploiting the continuity properties of the energy functionals involved. When $m=1$, we also establish a uniqueness result in the spirit of the Brezis-Strauss Theorem. We finally provide some applications of our main results by dealing with some Yamabe-type and Kazdan-Warner-type equations on locally finite weighted graphs.

math.AP↗
arXiv2023

On the convex components of a set in $\mathbb{R}^n$

We prove a lower bound on the number of the convex components of a compact set with non-empty interior in $\mathbb{R}^n$ for all $n\ge2$. Our result generalizes and improves the inequalities previously obtained in M. Carozza, F. Giannetti, F. Leonetti and A. Passarelli di Napoli, "Convex components", in Communications in Contemporary Mathematics, Vol. 21, No. 06, 1850036 (2019) and in M. La Civita and F. Leonetti, "Convex components of a set and the measure of its boundary", Atti. Sem. Mat. Fis. Univ. Modena Reggio Emilia 56 (2008-2009) 71-78.

math.MG↗
arXiv2023

Full double Hölder regularity of the pressure in bounded domains

We consider Hölder continuous weak solutions $u\in C^γ(Ω)$, $u\cdot n|_{\partial Ω}=0$, of the incompressible Euler equations on a bounded and simply connected domain $Ω\subset\mathbb{R}^d$. If $Ω$ is of class $C^{2,1}$ then the corresponding pressure satisfies $p\in C^{2γ}_*(Ω)$ in the case $γ\in (0,\frac{1}{2}]$, where $C^{2γ}_*$ is the Hölder-Zygmund space, which coincides with the usual Hölder space for $γ<\frac12$. This result, together with our previous one in [11] covering the case $γ\in(\frac12,1)$, yields the full double regularity of the pressure on bounded and sufficiently regular domains. The interior regularity comes from the corresponding $C^{2γ}_*$ estimate for the pressure on the whole space $\mathbb{R}^d$, which in particular extends and improves the known double regularity results (in the absence of a boundary) in the borderline case $γ=\frac{1}{2}$. The boundary regularity features the use of local normal geodesic coordinates, pseudodifferential calculus and a fine Littlewood-Paley analysis of the modified equation in the new coordinate system. We also discuss the relation between different notions of weak solutions, a step which plays a major role in our approach.

math.AP↗
arXiv2023

Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries

We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure, the Bakry-Émery inequality for the corresponding sub-Laplacian implies the existence of enough Killing vector fields on the tangent cone to force the latter to be Euclidean at each point, yielding the failure of the curvature-dimension condition in full generality. Our approach does not apply to non-strictly-positive measures. In fact, we prove that the weighted Grushin plane does not satisfy any curvature-dimension condition, but, nevertheless, does admit an a.e. pointwise version of the Bakry-Émery inequality. As recently observed by Pan and Montgomery, one half of the weighted Grushin plane satisfies the RCD(0,N) condition, yielding a counterexample to gluing theorems in the RCD setting.

math.DG↗
arXiv2023

Non-local $BV$ functions and a denoising model with $L^1$ fidelity

We study a general total variation denoising model with weighted $L^1$ fidelity, where the regularizing term is a non-local variation induced by a suitable (non-integrable) kernel $K$, and the approximation term is given by the $L^1$ norm with respect to a non-singular measure with positively lower-bounded $L^\infty$ density. We provide a detailed analysis of the space of non-local $BV$ functions with finite total $K$-variation, with special emphasis on compactness, Lusin-type estimates, Sobolev embeddings and isoperimetric and monotonicity properties of the $K$-variation and the associated $K$-perimeter. Finally, we deal with the theory of Cheeger sets in this non-local setting and we apply it to the study of the fidelity in our model.

math.FA↗
arXiv2023

Existence and stability of weak solutions of the Vlasov--Poisson system in localized Yudovich spaces

We consider the Vlasov--Poisson system both in the repulsive (electrostatic potential) and in the attractive (gravitational potential) cases. In our first main theorem, we prove the uniqueness and the quantitative stability of Lagrangian solutions $f=f(t,x,v)$ whose associated spatial density $ρ_f=ρ_f(t,x)$ is potentially unbounded but belongs to suitable uniformly-localized Yudovich spaces. This requirement imposes a condition of slow growth on the function $p \mapsto \|ρ_f(t,\cdot)\|_{L^p}$ uniformly in time. Previous works by Loeper, Miot and Holding--Miot have addressed the cases of bounded spatial density, i.e., $\|ρ_f(t,\cdot)\|_{L^p} \lesssim 1$, and spatial density such that $\|ρ_f(t,\cdot)\|_{L^p} \sim p^{1/α}$ for $α\in[1,+\infty)$. Our approach is Lagrangian and relies on an explicit estimate of the modulus of continuity of the electric field and on a second-order Osgood lemma. It also allows for iterated-logarithmic perturbations of the linear growth condition. In our second main theorem, we complement the aforementioned result by constructing solutions whose spatial density sharply satisfies such iterated-logarithmic growth. Our approach relies on real-variable techniques and extends the strategy developed for the Euler equations by the first and fourth-named authors. It also allows for the treatment of more general equations that share the same structure as the Vlasov--Poisson system. Notably, the uniqueness result and the stability estimates hold for both the classical and the relativistic Vlasov--Poisson systems.

math.AP↗
arXiv2023

Weak-strong uniqueness and vanishing viscosity for incompressible Euler equations in exponential spaces

In the class of admissible weak solutions, we prove a weak-strong uniqueness result for the incompressible Euler equations assuming that the symmetric part of the gradient belongs to $L^1_{\rm loc}([0,+\infty);L^{\rm exp}(\mathbb{R}^d;\mathbb{R}^{d\times d}))$, where $L^{\rm exp}$ denotes the Orlicz space of exponentially integrable functions. Moreover, under the same assumptions on the limit solution to the Euler system, we obtain the convergence of vanishing-viscosity Leray--Hopf weak solutions of the Navier--Stokes equations.

math.AP↗
arXiv2023

An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces

We revisit Yudovich's well-posedness result for the $2$-dimensional Euler equations for an inviscid incompressible fluid on either a sufficiently regular (not necessarily bounded) open set $Ω\subset\mathbb{R}^2$ or on the torus $Ω=\mathbb{T}^2$. We construct global-in-time weak solutions with vorticity in $L^1\cap L^p_{\mathrm{ul}}$ and in $L^1\cap Y^Θ_{\mathrm{ul}}$, where $L^p_{\mathrm{ul}}$ and $Y^Θ_{\mathrm{ul}}$ are suitable uniformly-localized versions of the Lebesgue space $L^p$ and of the Yudovich space $Y^Θ$ respectively, with no condition at infinity for the growth function $Θ$. We also provide an explicit modulus of continuity for the velocity depending on the growth function $Θ$. We prove uniqueness of weak solutions in $L^1\cap Y^Θ_{\mathrm{ul}}$ under the assumption that $Θ$ grows moderately at infinity. In contrast to Yudovich's energy method, we employ a Lagrangian strategy to show uniqueness. Our entire argument relies on elementary real-variable techniques, with no use of either Sobolev spaces, Calderón-Zygmund theory or Littlewood-Paley decomposition, and actually applies not only to the Biot-Savart law, but also to more general operators whose kernels obey some natural structural assumptions.

math.AP↗
arXiv2023

Lagrangian stability for a system of non-local continuity equations under Osgood condition

We extend known existence and uniqueness results of weak measure solutions for systems of non-local continuity equations beyond the usual Lipschitz regularity. Existence of weak measure solutions holds for uniformly continuous vector fields and convolution kernels, while uniqueness follows from a Lagrangian stability estimate under an additional Osgood condition.

math.AP↗
arXiv2023

Failure of the local chain rule for the fractional variation

We prove that the local version of the chain rule cannot hold for the fractional variation defined in arXiv:1809.08575. In the case $n = 1$, we prove a stronger result, exhibiting a function $f \in BV^α(\mathbb{R})$ such that $|f| \notin BV^α(\mathbb{R})$. The failure of the local chain rule is a consequence of some surprising rigidity properties for non-negative functions with bounded fractional variation which, in turn, are derived from a fractional Hardy inequality localized to half-spaces. Our approach exploits the results of arXiv:2111.13942 and the distributional approach of the previous papers arXiv:1809.08575, arXiv:1910.13419, arXiv:2011.03928, arXiv:2109.15263. As a byproduct, we refine the fractional Hardy inequality obtained in arXiv:1611.07204, arXiv:1806.07588 and we prove a fractional version of the closely related Meyers-Ziemer trace inequality.

math.FA↗
arXiv2023

On Double Hölder Regularity of the Hydrodynamic Pressure in Bounded Domains

We prove that the hydrodynamic pressure $p$ associated to the velocity $u\in C^θ(Ω)$, $θ\in(0,1)$, of an inviscid incompressible fluid in a bounded and simply connected domain $Ω\subset \mathbb R^d$ with $C^{2+}$ boundary satisfies $p\in C^θ(Ω)$ for $θ\leq \frac12$ and $p\in C^{1,2θ-1}(Ω)$ for $θ>\frac12$. Moreover, when $\partial Ω\in C^{3+}$, we prove that an almost double Hölder regularity $p\in C^{2θ-}(Ω)$ holds even for $θ<\frac12$. This extends and improves the recent result of Bardos and Titi obtained in the planar case to every dimension $d\ge2$ and it also doubles the pressure regularity. Differently from Bardos and Titi, we do not introduce a new boundary condition for the pressure, but instead work with the natural one. In the boundary-free case of the $d$-dimensional torus, we show that the double regularity of the pressure can be actually achieved under the weaker assumption that the divergence of the velocity is sufficiently regular, thus not necessarily zero.

math.AP↗
arXiv2022

Leibniz rules and Gauss-Green formulas in distributional fractional spaces

We apply the results established in arXiv:2109.15263 to prove some new fractional Leibniz rules involving $BV^{α,p}$ and $S^{α,p}$ functions, following the distributional approach adopted in the previous works arXiv:1809.08575, arXiv:1910.13419, arXiv:2011.03928. In order to achieve our main results, we revise the elementary properties of the fractional operators involved in the framework of Besov spaces and we rephraze the Kenig-Ponce-Vega Leibniz-type rule in our fractional context. We apply our results to prove the well-posedness of the boundary-value problem for a general $2α$-order fractional elliptic operator in divergence form.

math.FA↗
arXiv2022

The fractional variation and the precise representative of $BV^{α,p}$ functions

We continue the study of the fractional variation following the distributional approach developed in the previous works arXiv:1809.08575, arXiv:1910.13419 and arXiv:2011.03928. We provide a general analysis of the distributional space $BV^{α,p}(\mathbb{R}^n)$ of $L^p$ functions, with $p\in[1,+\infty]$, possessing finite fractional variation of order $α\in(0,1)$. Our two main results deal with the absolute continuity property of the fractional variation with respect to the Hausdorff measure and the existence of the precise representative of a $BV^{α,p}$ function.

math.FA↗
arXiv2022

A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II

We continue the study of the space $BV^α(\mathbb R^n)$ of functions with bounded fractional variation in $\mathbb R^n$ and of the distributional fractional Sobolev space $S^{α,p}(\mathbb R^n)$, with $p\in [1,+\infty]$ and $α\in(0,1)$, considered in the previous works arXiv:1809.08575 and arXiv:1910.13419. We first define the space $BV^0(\mathbb R^n)$ and establish the identifications $BV^0(\mathbb R^n)=H^1(\mathbb R^n)$ and $S^{α,p}(\mathbb R^n)=L^{α,p}(\mathbb R^n)$, where $H^1(\mathbb R^n)$ and $L^{α,p}(\mathbb R^n)$ are the (real) Hardy space and the Bessel potential space, respectively. We then prove that the fractional gradient $\nabla^α$ strongly converges to the Riesz transform as $α\to0^+$ for $H^1\cap W^{α,1}$ and $S^{α,p}$ functions. We also study the convergence of the $L^1$-norm of the $α$-rescaled fractional gradient of $W^{α,1}$ functions. To achieve the strong limiting behavior of $\nabla^α$ as $α\to0^+$, we prove some new fractional interpolation inequalities which are stable with respect to the interpolating parameter.

math.FA↗
arXiv2019

A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up

We introduce the new space $BV^α(\mathbb{R}^n)$ of functions with bounded fractional variation in $\mathbb{R}^n$ of order $α\in (0, 1)$ via a new distributional approach exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature. In analogy with the classical $BV$ theory, we give a new notion of set $E$ of (locally) finite fractional Caccioppoli $α$-perimeter and we define its fractional reduced boundary $\mathscr{F}^α E$. We are able to show that $W^{α,1}(\mathbb{R}^n)\subset BV^α(\mathbb{R}^n)$ continuously and, similarly, that sets with (locally) finite standard fractional $α$-perimeter have (locally) finite fractional Caccioppoli $α$-perimeter, so that our theory provides a natural extension of the known fractional framework. Our main result partially extends De Giorgi's Blow-up Theorem to sets of locally finite fractional Caccioppoli $α$-perimeter, proving existence of blow-ups and giving a first characterisation of these (possibly non-unique) limit sets.

math.FA↗
arXiv2019

Heat and entropy flows in Carnot groups

We prove the correspondence between the solutions of the sub-elliptic heat equation in a Carnot group $\mathbb{G}$ and the gradient flows of the relative entropy functional in the Wasserstein space of probability measures on $\mathbb{G}$. Our result completely answers a question left open in a previous paper by N. Juillet, where the same correspondence was proved for $\mathbb{G}=\mathbb{H}^n$, the $n$-dimensional Heisenberg group.

math.DG↗
arXiv2019

Symmetric double bubbles in the Grushin plane

We address the double bubble problem for the anisotropic Grushin perimeter $P_α$, $α\geq 0$, and the Lebesgue measure in $\mathbb R^2$, in the case of two equal volumes. We assume that the contact interface between the bubbles lays on either the vertical or the horizontal axis. Since no regularity theory is available in this setting, in both cases we first prove existence of minimizers via the direct method by symmetrization arguments and then characterize them in terms of the given area by first variation techniques. Angles at which minimal boundaries intersect satisfy the standard 120-degree rule up to a suitable change of coordinates. While for $α=0$ the Grushin perimeter reduces to the Euclidean one and both minimizers coincide with the symmetric double bubble found in [Foisy et al., Pacific J. Math. (1993)], for $α=1$ vertical interface minimizers have Grushin perimeter strictly greater than horizontal interface minimizers. As the latter ones are obtained by translating and dilating the Grushin isoperimetric set found in [Monti Morbidelli, J. Geom. Anal. (2004)], we conjecture that they solve the double bubble problem with no assumptions on the contact interface.

math.OC↗