arXiv subjects

Matteo Muratori

Publications and source records attributed to Matteo Muratori.

At least 19 recordsLinked to original sources

arXiv2025

On the subcritical Lane-Emden equation on Riemannian models with polynomial volume growth

We focus on the problems of existence and non-existence of positive solutions for the Sobolev-subcritical Lane-Emden equation on certain Riemannian manifolds (mainly models) with asymptotically negative curvature, which, from the viewpoint of the volume growth of geodesic balls, can be regarded as intermediate settings between the Euclidean and the hyperbolic spaces. A number of interesting phenomena arise: the subcritical regime naturally divides into three further ranges, characterized by existence phenomena (slightly subcritical), non-existence phenomena (strongly subcritical), and by a mixed behavior where existence and non-existence strongly depend on additional assumptions on the manifold (intermediate). In the intermediate regime, we further show that the radial homogeneous Dirichlet problem in geodesics balls may admit multiple positive solutions, thereby revealing substantial differences with respect to both the Euclidean and the hyperbolic settings.

math.AP↗

arXiv2025

ToonOut: Fine-tuned Background-Removal for Anime Characters

While state-of-the-art background removal models excel at realistic imagery, they frequently underperform in specialized domains such as anime-style content, where complex features like hair and transparency present unique challenges. To address this limitation, we collected and annotated a custom dataset of 1,228 high-quality anime images of characters and objects, and fine-tuned the open-sourced BiRefNet model on this dataset. This resulted in marked improvements in background removal accuracy for anime-style images, increasing from 95.3% to 99.5% for our newly introduced Pixel Accuracy metric. We are open-sourcing the code, the fine-tuned model weights, as well as the dataset at: https://github.com/MatteoKartoon/BiRefNet.

cs.CV↗

arXiv2025

Concentration comparison for nonlinear diffusion on model manifolds and P\'olya-Szeg\H{o} inequality

We investigate the validity of the mass concentration comparison for a class of nonlinear diffusion equations posed on Riemannian manifolds $ \mathbb{M}^n $ that are spherically symmetric, that is, model manifolds. The concentration comparison states that the solution of a certain diffusion equation that takes the radially decreasing (Schwarz) rearrangement $ u_0^\star $ as its initial datum is more concentrated than the original solution starting from $u_0$. This is known to hold in $\mathbb{R}^n$ as a consequence of the celebrated P\'olya-Szeg\H{o} inequality, which asserts that the $ L^2 $ norm of the gradient of a function $f$ (belonging to an appropriate Sobolev space) is always larger than the $ L^2 $ norm of the gradient of its radially decreasing rearrangement $f^\star$. However, if $ \mathbb{M}^n $ is a general model manifold, it is not for granted that the P\'olya-Szeg\H{o} inequality holds; in fact, we will provide a simple condition involving the scalar curvature of $\mathbb{M}^n $ under which such an inequality actually fails. The main result we prove states that, given any continuous, nondecreasing, and nontrivial function $ \phi: [0,+\infty) \to [0,+\infty) $, the filtration equation $ \partial_t u = \Delta \phi(u) $ satisfies the concentration comparison in $ \mathbb{M}^n \times (0,+\infty) $ if and only if $ \mathbb{M}^n $ supports the P\'olya-Szeg\H{o} inequality. In particular, the validity of such a comparison for the heat equation is sufficient to guarantee that the same holds for all filtration equations. Moreover, we prove that if $ \mathbb{M}^n $ supports a centered isoperimetric inequality then the P\'olya-Szeg\H{o} inequality, and thus the concentration comparison, holds. This allows us to include important examples such as the hyperbolic space and the sphere.

math.AP↗

arXiv2025

A porous medium equation with rough weights: sharp Widder theory

We establish an optimal \emph{Widder theory} for a weighted porous medium equation with rough and inhomogeneous density that may be singular at a point and tends to zero at spatial infinity. Specifically, for this equation, we identify a class $X$ of initial measure data that give rise to very weak solutions, we show that non-negative very weak solutions necessarily admit an initial trace in $X$ at time $t=0$, and we prove that any two non-negative solutions having the same initial trace are equal. The corresponding theory for the classical (unweighted) equation was established by exploiting various properties that are not available in our weighted setting, such as the continuity of solutions, the explicit scale invariance of the equation, Aleksandrov's reflection principle, and the Aronson--B\'enilan inequality. Therefore, to complete the Widder theory, we must devise several proofs by means of entirely new methods. We also establish an optimal quantitative \emph{a priori} smoothing estimate for unsigned local solutions without resorting to scale invariance, which seems to be new in this form even for the classical porous medium equation. Finally, we show that non-negative very weak solutions are always locally bounded, and in particular that they have locally finite energy.

math.AP↗

arXiv2024

Moduli of continuity and absolute continuity: any relation?

We construct a monotone, continuous, but not absolutely continuous function whose minimal modulus of continuity is absolutely continuous. In particular, we establish that there is no equivalence between the absolute continuity of a function and the absolute continuity of its modulus of continuity, in contrast with a well-known property of Lipschitz functions.

math.CA↗

arXiv2024

An inhomogeneous porous medium equation with non-integrable data: asymptotics

We investigate the asymptotic behavior as $t\to+\infty$ of solutions to a weighted porous medium equation in $ \mathbb{R}^N $, whose weight $\rho(x)$ behaves at spatial infinity like $ |x|^{-\gamma} $ with subcritical power, namely $ \gamma \in [0,2) $. Inspired by some results by Alikakos-Rostamian and Kamin-Ughi from the 1980s on the unweighted problem, we focus on solutions whose initial data $u_0(x)$ are not globally integrable with respect to the weight and behave at infinity like $ |x|^{-\alpha} $, for $\alpha\in(0,N-\gamma)$. In the special case $ \rho(x)=|x|^{-\gamma} $ and $ u_0(x)=|x|^{-\alpha} $ we show that self-similar solutions of Barenblatt type, i.e. reminiscent of the usual source-type solutions, still exist, although they are no longer compactly supported. Moreover, they exhibit a transition phenomenon which is new even for the unweighted equation. We prove that such self-similar solutions are attractors for the original problem, and convergence takes place globally in suitable weighted $ L^p $ spaces for $p\in[1,\infty)$ and even globally in $L^\infty$ under some mild additional regularity assumptions on the weight. Among the fundamental tools that we exploit, it is worth mentioning a global smoothing effect for non-integrable data.

math.AP↗

arXiv2023

The Lane-Emden system on Cartan-Hadamard manifolds: asymptotics and rigidity of radial solutions

We investigate existence and qualitative properties of globally defined and positive radial solutions of the Lane-Emden system, posed on a Cartan-Hadamard model manifold $ \mathbb{M}^n $. We prove that, for critical or supercritical exponents, there exists at least a one-parameter family of such solutions. Depending on the stochastic completeness or incompleteness of $ \mathbb{M}^n $, we show that the existence region stays one dimensional in the former case, whereas it becomes two dimensional in the latter. Then, we study the asymptotics at infinity of solutions, which again exhibit a dichotomous behavior between the stochastically complete (where both components are forced to vanish) and incomplete cases. Finally, we prove a rigidity result for finite-energy solutions, showing that they exist if and only if $ \mathbb{M}^n $ is isometric to $ \mathbb{R}^n $.

math.AP↗

arXiv2023

A general nonlinear characterization of stochastic incompleteness

Stochastic incompleteness of a Riemannian manifold $M$ amounts to the nonconservation of probability for the heat semigroup on $M$. We show that this property is equivalent to the existence of nonnegative, nontrivial, bounded (sub)solutions to $\Delta W=\psi(W)$ for one, hence all, general nonlinearity $\psi$ which is only required to be continuous, nondecreasing, with $\psi(0)=0$ and $\psi>0$ in $(0,+\infty)$. Similar statements hold for unsigned (sub)solutions. We also prove that stochastic incompleteness is equivalent to the nonuniqueness of bounded solutions to the nonlinear parabolic equation $\partial_t u =\Delta\phi(u)$ with bounded initial data for one, hence all, general nonlinearity $\phi$ which is only required to be continuous, nondecreasing and nonconstant. Such a generality allows us to deal with equations of both fast-diffusion and porous-medium type, as well as with the one-phase and two-phase classical Stefan problems, which seem to have never been investigated in the manifold setting.

math.AP↗

arXiv2022

An inhomogeneous porous medium equation with large data: well-posedness

We study solutions of a Euclidean weighted porous medium equation when the weight behaves at spacial infinity like $|x|^{-\gamma}$, for $\gamma\in [0,2)$, and is allowed to be singular at the origin. In particular we show local-in-time existence and uniqueness for a class of large initial data which includes as "endpoints" those growing at a rate of $ |x|^{(2-\gamma)/(m-1)}$, in a weighted $L^1$-average sense. We also identify global-existence and blow-up classes, whose respective forms strongly support the claim that such a growth rate is optimal, at least for positive solutions. As a crucial step in our existence proof we establish a local smoothing effect for large data without resorting to the classical Aronson-B\'{e}nilan inequality and using the B\'{e}nilan-Crandall inequality instead, which may be of independent interest since the latter holds in much more general settings.

math.AP↗

arXiv2022

The porous medium equation with large data on Cartan-Hadamard manifolds under general curvature bounds

We consider very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds, that are assumed to satisfy general curvature bounds and to be stochastically complete. We identify a class of initial data that can grow at infinity at a prescribed rate, which depends on the assumed curvature bounds through an integral function, such that the corresponding solution exists at least on $[0,T]$ for a suitable $T>0$. The maximal existence time $T$ is estimated in terms of a suitable weighted norm of the initial datum. Our results are sharp, in the sense that slower growth rates yield global existence, whereas one can construct data with critical growth for which the corresponding solutions blow up in finite time. Under further assumptions, uniqueness of very weak solutions is also proved, in the same growth class.

math.AP↗

arXiv2021

The Fractional Porous Medium Equation on noncompact Riemannian manifolds

We study nonnegative solutions to the Fractional Porous Medium Equation on a suitable class of connected, noncompact Riemannian manifolds. We provide existence and smoothing estimates for solutions, in an appropriate weak (dual) sense, for data belonging either to the usual $L^1$ space or to a considerably larger weighted space determined in terms of the fractional Green function. The class of manifolds for which the results hold include both the Euclidean and the hyperbolic spaces and even in the Euclidean situation involve a class of data which is larger than previously known one.

math.AP↗

arXiv2021

On the equivalence of Sobolev norms in Malliavin spaces

We investigate the problem of the equivalence of $L^q$-Sobolev norms in Malliavin spaces for $q\in [1,\infty)$, focusing on the graph norm of the $k$-th Malliavin derivative operator and the full Sobolev norm involving all derivatives up to order $k$, where $k$ is any positive integer. The case $q=1$ in the infinite-dimensional setting is challenging, since at such extreme the standard approach involving Meyer's inequalities fails. In this direction, we are able to establish the mentioned equivalence for $q=1$ and $k=2$ relying on a vector-valued Poincar\'e inequality that we prove independently and that turns out to be new at this level of generality, while for $q=1$ and $k>2$ the equivalence issue remains open, even if we obtain some functional estimates of independent interest. With our argument (that also resorts to the Wiener chaos) we are able to recover the case $q\in (1,\infty)$ in the infinite-dimensional setting; the latter is known since the eighties, however our proof is more direct than those existing in the literature, and allows to give explicit bounds on all the multiplying constants involved in the functional inequalities. Finally, we also deal with the finite-dimensional case for all $q\in [1,\infty)$ (where the equivalence, without explicit constants, follows from standard compactness arguments): our proof in such setting is much simpler, relying on Gaussian integration-by-parts formulas and an adaptation of Sobolev inequalities in Euclidean spaces, and it provides again quantitative bounds on the multiplying constants, which however blow up when the dimension diverges to $\infty$ (whence the need for a different approach in the infinite-dimensional setting).

math.FA↗

arXiv2021

Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds

The Cartan-Hadamard conjecture states that, on every $n$-dimensional Cartan-Hadamard manifold $ \mathbb{M}^n $, the isoperimetric inequality holds with Euclidean optimal constant, and any set attaining equality is necessarily isometric to a Euclidean ball. This conjecture was settled, with positive answer, for $n \le 4$. It was also shown that its validity in dimension $n$ ensures that every $p$-Sobolev inequality ($ 1 < p < n $) holds on $ \mathbb{M}^n $ with Euclidean optimal constant. In this paper we address the problem of classifying all Cartan-Hadamard manifolds supporting an optimal function for the Sobolev inequality. We prove that, under the validity of the $n$-dimensional Cartan-Hadamard conjecture, the only such manifold is $ \mathbb{R}^n $, and therefore any optimizer is an Aubin-Talenti profile (up to isometries). In particular, this is the case in dimension $n \le 4$. Optimal functions for the Sobolev inequality are weak solutions to the critical $p$-Laplace equation. Thus, in the second part of the paper, we address the classification of radial solutions (not necessarily optimizers) to such a PDE. Actually, we consider the more general critical or supercritical equation \[ -\Delta_p u = u^q \, , \quad u>0 \, , \qquad \text{on } \mathbb{M}^n \, , \] where $q \ge p^*-1$. We show that if there exists a radial finite-energy solution, then $\mathbb{M}^n$ is necessarily isometric to $\mathbb{R}^n$, $q=p^*-1$ and $u$ is an Aubin-Talenti profile. Furthermore, on model manifolds, we describe the asymptotic behavior of radial solutions not lying in the energy space $\dot{W}^{1,p}(\mathbb{M}^n)$, studying separately the $p$-stochastically complete and incomplete cases.

math.AP↗

arXiv2020

Fast diffusion on noncompact manifolds: well-posedness theory and connections with semilinear elliptic equations

We investigate the well-posedness of the fast diffusion equation (FDE) in a wide class of noncompact Riemannian manifolds. Existence and uniqueness of solutions for globally integrable initial data was established in [5]. However, in the Euclidean space, it is known from Herrero and Pierre [20] that the Cauchy problem associated with the FDE is well posed for initial data that are merely in $ L^1_{\mathrm{loc}} $. We establish here that such data still give rise to global solutions on general Riemannian manifolds. If, in addition, the radial Ricci curvature satisfies a suitable pointwise bound from below (possibly diverging to $-\infty$ at spatial infinity), we prove that also uniqueness holds, for the same type of data, in the class of strong solutions. Besides, under the further assumption that the initial datum is in $L^2_{\mathrm{loc}}$ and nonnegative, a minimal solution is shown to exist, and we are able to establish uniqueness of purely (nonnegative) distributional solutions, which to our knowledge was not known before even in the Euclidean space. The required curvature bound is in fact sharp, since on model manifolds it turns out to be equivalent to stochastic completeness, and it was shown in [13] that uniqueness for the FDE fails even in the class of bounded solutions on manifolds that are not stochastically complete. Qualitatively this amounts to asking that the curvature diverges at most quadratically at infinity. A crucial ingredient of the uniqueness result is the proof of nonexistence of distributional subsolutions to certain semilinear elliptic equations with power nonlinearities, of independent interest.

math.AP↗

arXiv2020

Nonexistence of radial optimal functions for the Sobolev inequality on Cartan-Hadamard manifolds

It is well known that the Euclidean Sobolev inequality holds on any Cartan-Hadamard manifold of dimension $ n\ge 3 $, i.e. any complete, simply connected Riemannian manifold with nonpositive sectional curvature. As a byproduct of the Cartan-Hadamard conjecture, a longstanding problem in the mathematical literature settled only very recently in a breakthrough paper by Ghomi and Spruck, we can now assert that the optimal constant is also Euclidean, namely it coincides with the one achieved in the Euclidean space $ \mathbb{R}^n $ by the Aubin-Talenti functions. One may ask whether there exist at all optimal functions on a generic Cartan-Hadamard manifold $ \mathbb{M}^n $. What we prove here, with ad hoc arguments that do not take advantage of the validity of the Cartan-Hadamard conjecture, is that this is false at least for functions that are radially symmetric with respect to the geodesic distance from a fixed pole. More precisely, we show that if the optimum in the Sobolev inequality is achieved by some radial function, then $\mathbb{M}^n $ must be isometric to $ \mathbb{R}^n $.

math.AP↗

arXiv2019

Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below

Given a complete, connected Riemannian manifold $ \mathbb{M}^n $ with Ricci curvature bounded from below, we discuss the stability of the solutions of a porous medium-type equation with respect to the 2-Wasserstein distance. We produce (sharp) stability estimates under negative curvature bounds, which to some extent generalize well-known results by Sturm and Otto-Westdickenberg. The strategy of the proof mainly relies on a quantitative $L^1-L^\infty$ smoothing property of the equation considered, combined with the Hamiltonian approach developed by Ambrosio, Mondino and Savar\'e in a metric-measure setting.

math.AP↗

arXiv2019

Uniqueness of very weak solutions for a fractional filtration equation

We prove existence and uniqueness of distributional, bounded, nonnegative solutions to a fractional filtration equation in ${\mathbb R}^d$. With regards to uniqueness, it was shown even for more general equations in [19] that if two bounded solutions $u,w$ of (1.1) satisfy $u-w\in L^1({\mathbb R}^d\times(0,T))$, then $u=w$. We obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions, provided they are nonnegative. Indeed, we show that a minimal solution exists and that any other solution must coincide with it. As a consequence, distributional solutions have locally-finite energy.

math.AP↗

arXiv2018

Gradient flows and Evolution Variational Inequalities in metric spaces. I: structural properties

This is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space $(X,\mathsf{d})$ that can be characterized by Evolution Variational Inequalities. We present new results concerning the structural properties of solutions to the $\mathrm{EVI}$ formulation, such as contraction, regularity, asymptotic expansion, precise energy identity, stability, asymptotic behaviour and their link with the geodesic convexity of the driving functional. Under the crucial assumption of the existence of an $\mathrm{EVI}$ gradient flow, we will also prove two main results: the equivalence with the De Giorgi variational characterization of curves of maximal slope and the convergence of the Minimizing Movement-JKO scheme to the $\mathrm{EVI}$ gradient flow, with an explicit and uniform error estimate of order $1/2$ with respect to the step size, independent of any geometric hypothesis (such as upper or lower curvature bounds) on $\mathsf{d}$. In order to avoid any compactness assumption, we will also introduce a suitable relaxation of the Minimizing Movement algorithm obtained by the Ekeland variational principle, and we will prove its uniform convergence as well.

math.FA↗