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Alberto G. Setti

Publications and source records attributed to Alberto G. Setti.

At least 19 recordsLinked to original sources
arXiv2026

On the accretivity and m-accretivity of Laplacians and porous medium-type operators on graphs

We study the accretivity and m-accretivity of Laplacian and porous medium-type operators on weighted graphs. In particular, we give several conditions that imply these properties for maximal operators and investigate when these operators agree with various restrictions. For porous medium-type operators on $\ell^1$ and for Laplacians on $\ell^p$ for $p \in [1,\infty)$, we show that there always exists a dense subset of the domain on which the maximal operator is m-accretive. As a consequence, we establish that accretivity, m-accretivity and injectivity of the shifted operator are all equivalent for these maximal operators. Under additional conditions on the graph, we then prove that the maximal operators are m-accretive on the entire domain, not just a dense subset. We also investigate minimal operators and show that they are m-accretive if and only if the minimal and maximal operators agree and the maximal operator is accretive. We then give some conditions that imply this agreement. Furthermore, for the minimal Laplacian on $\ell^p$, we show that accretivity and m-accretivity are not equivalent. For the $\ell^2$ case, we give connections to Markov uniqueness and essential self-adjointness. For the $\ell^\infty$ case, we establish the equivalence of stochastic completeness at infinity, m-accretivity for the maximal Laplacian on $\ell^\infty$, and m-accretivity of the minimal Laplacian on $\ell^1$.

math.FA↗
arXiv2026

Nonlinear parabolic characterizations of stochastic completeness at infinity on weighted graphs

We prove a nonlinear parabolic characterization of stochastic completeness at infinity for weighted graphs. For the filtration equation \[ (\partial_t + ΔΦ)u =0 \] where $Δ$ is the non-negative formal graph Laplacian and $Φu =ϕ\circ u$ with $ϕ\colon \R\to\R$ nonconstant, continuous and increasing, stochastic completeness at infinity is equivalent to uniqueness of bounded pointwise solutions for every bounded initial datum. For \(Φ=\id\), this recovers the classical heat equation characterization of stochastic completeness at infinity, and of stochastic completeness when the killing term is trivial, i.e., when \(κ=0\). If stochastic completeness at infinity fails, then every bounded initial datum admits infinitely many bounded pointwise solutions of the filtration equation. Admissible nonlinearities include the signed porous medium and fast diffusion powers $ϕ(s)=s|s|^{m-1}$ for all $m>0$, as well as many others. Stochastic completeness at infinity is further characterized by a generalized mass balance: the total mass of a positive pointwise solution at time $t$, augmented by the mass $\int_0^t\sum_{x}κ(x)ϕ(u(s,x)) \dd s$ dissipated by the killing term $κ$, equals the initial mass. This balance holds for every bounded positive solution on graphs of finite measure and for bounded finite-mass data on graphs of arbitrary measure under the sharp condition $\limsup_{r\to0^+}ϕ(r)/r<\infty$. It also extends to positive pointwise solutions in $\ell^1$ that are bounded on every positive time interval. When the killing term is trivial, stochastic completeness at infinity reduces to stochastic completeness and generalized balance to conservation of mass.

math.PR↗
arXiv2026

The generalized porous medium equation on graphs: well-posedness, extinction, and mass conservation

We study the Cauchy problem for the generalized porous medium equation on infinite weighted graphs. For a general nonlinearity, we establish Dirichlet comparison and weak maximum principles on finite subgraphs and, through an exhaustion argument, construct minimal and maximal pointwise solutions for arbitrary $\ell^\infty$ initial data, controlled by explicit, possibly time-dependent, barriers. For the porous nonlinearity $ϕ(s)=s|s|^{m-1}$, assuming a $ν$-Sobolev inequality with $ν>2$, we derive quantitative energy estimates for $\ell^1$-mild solutions. These yield finite-time extinction in the fast diffusion range $0 2/ν$. Interestingly, we recover the Euclidean critical exponent for several model graphs. Finally, we prove an exact generalized mass balance for nonnegative $\ell^1$-mild solutions on graphs that are stochastically complete at infinity, allowing for an arbitrary killing term. The same balance is established for suitable classical and bounded pointwise solutions. In the absence of killing, these reduce to conservation of mass.

math.AP↗
arXiv2026

Semilinear Diffusion Equations on Infinite Graphs: The Dissipative and Lipschitz Cases

We study a class of semilinear diffusion equations on infinite, connected, weighted graphs, focusing on two types of nonlinearities: monotone decreasing and Lipschitz continuous. Under minimal structural assumptions on the graph, we establish existence, uniqueness, and regularity of mild solutions for initial data in $\ell^p$ spaces, with $1\leq p<\infty$. Our approach relies on time discretization via an implicit Euler scheme and an exhaustion technique using Dirichlet subgraphs. As a by-product, we obtain existence and uniqueness results for a related time-independent equation. Finite-time extinction and positivity for solutions under a specific forcing term are also proved.

math.AP↗
arXiv2026

Characterizations of $p$-Parabolicity on Graphs

We study $p$-energy functionals on infinite locally summable graphs for $p\in (1,\infty)$ and show that many well-known characterizations for a parabolic space are also true in this discrete, non-local and non-linear setting. Among the characterizations are an Ahlfors-type, a Kelvin-Nevanlinna-Royden-type, a Khas'minski\uı-type and a Poincaré-type characterization. We also illustrate some applications and describe examples of graphs which are locally summable but not locally finite. Finally, we study the obstacle problem for the $p$-Laplacian using an approximation procedure by finite graphs in the summable, not necessarily locally finite, case. This is then utilized to give an alternative proof of the Khas'minski\uı-type characterization.

math.FA↗
arXiv2026

Proper solutions of the $1/H$-flow and the Green kernel of the $p$-Laplacian

We show existence and optimal growth estimate for the weak inverse mean curvature flow issuing from a point, on manifolds with certain curvature and isoperimetric conditions. These theorems imply analogous ones for the flow issuing from relatively compact sets. Some of the results are obtained by proving new decay estimates for the Green kernel of the $p$-Laplacian which fix a gap in the literature. Additionally, we address the convergence of renormalized $p$-capacitary potentials to the inverse mean curvature flow with outer obstacle.

math.AP↗
arXiv2025

Hardy--Littlewood maximal operators on certain manifolds with bounded geometry

In this paper we study the $L^p$ boundedness of the centred and the uncentred Hardy--Littlewood maximal operators on certain Riemannian manifolds with bounded geometry. Our results complement those of various authors. We show that, under mild assumptions, $L^p$ estimates for the centred operator are ``stable'' under conformal changes of the metric, and prove sharp~$L^p$ estimates for the centred operator on Riemannian models with pinched negative scalar curvature. Furthermore, we prove that the centred operator is of weak type $(1,1)$ on the connected sum of two space forms with negative curvature, whereas the uncentred operator is, perhaps surprisingly, bounded only on $L^\infty$. We also prove that if two locally doubling geodesic metric measure spaces enjoying the uniform ball size condition are strictly quasi-isometric, then they share the same boundedness properties for both the centred and the uncentred maximal operator. Finally, we discuss some $L^p$ mapping properties for the centred operator on a specific Riemannian surface introduced by Strömberg, providing new interesting results.

math.FA↗
arXiv2024

Neumann cut-offs and essential self-adjointness on complete Riemannian manifolds with boundary

We generalize some fundamental results for noncompact Riemannian manfolds without boundary, that only require completeness and no curvature assumptions, to manifolds with boundary: let $M$ be a smooth Riemannian manifold with boundary $\partial M$ and let $\hat{C}^\infty_c(M)$ denote the space of smooth compactly supported cut-off functions with vanishing normal derivative, Neumann cut-offs. We show, among other things, that under completeness: - $\hat{C}^\infty_c(M)$ is dense in $W^{1,p}(\mathring{M})$ for all $p\in (1,\infty)$; this generalizes a classical result by Aubin [2] for $\partial M=\emptyset$. - $M$ admits a sequence of first order cut-off functions in $\hat{C}^\infty_c(M)$; for $\partial M=\emptyset$ this result can be traced back to Gaffney [7]. - the Laplace-Beltrami operator with domain of definition $\hat{C}^\infty_c(M)$ is essentially self-adjoint; this is a generalization of a classical result by Strichartz [20] for $\partial M=\emptyset$.

math.DG↗
arXiv2023

The $ L^1 $-Liouville property on graphs

In this paper we investigate the $ L^1 $-Liouville property, underlining its connection with stochastic completeness and other structural features of the graph. We give a characterization of the $ L^1 $-Liouville property in terms of the Green function of the graph and use it to prove its equivalence with stochastic completeness on model graphs. Moreover, we show that there exist stochastically incomplete graphs which satisfy the $ L^1 $-Liouville property and prove some comparison theorems for general graphs based on inner-outer curvatures. We also introduce the Dirichlet $L^1$-Liouville property of subgraphs and prove that if a graph has a Dirichlet $L^1$-Liouville subgraph, then it is $L^1$-Liouville itself. As a consequence, we obtain that the $ L^1$-Liouville property is not affected by a finite perturbation of the graph and, just as in the continuous setting, a graph is $ L^1$-Liouville provided that at least one of its ends is Dirichlet $ L^1$-Liouville.

math.DG↗
arXiv2020

Inner-Outer Curvatures, Ricci-Ollivier Curvature and Volume Growth of Graphs

We are concerned with the study of different notions of curvature on graphs. We show that if a graph has stronger inner-outer curvature growth than a model graph, then it has faster volume growth too. We also study the relationhips of volume growth with other kind of curvatures, such as the Ollivier-Ricci curvature.

math.DG↗
arXiv2020

Qualitative properties of bounded subsolutions of nonlinear PDEs

We study decay and compact support properties of positive and bounded solutions of $Δ_{p} u \geq Λ(u)$ on the exterior of a compact set of a complete manifold with rotationally symmetry. In the same setting, we also give a new characterization of stochastic completeness for the $p$-Laplacian in terms of a global $W^{1,p}$-regularity of such solutions. One of the tools we use is a nonlinear version of the Feller property which we investigate on general Riemannian manifolds and which we establish under integral Ricci curvature conditions.

math.AP↗
arXiv2018

Height estimates for Killing graphs

The paper aims at proving global height estimates for Killing graphs defined over a complete manifold with nonempty boundary. To this end, we first point out how the geometric analysis on a Killing graph is naturally related to a weighted manifold structure, where the weight is defined in terms of the length of the Killing vector field. According to this viewpoint, we introduce some potential theory on weighted manifolds with boundary and we prove a weighted volume estimate for intrinsic balls on the Killing graph. Finally, using these tools, we provide the desired estimate for the weighted height in the assumption that the Killing graph has constant weighted mean curvature and the weighted geometry of the ambient space is suitably controlled.

math.DG↗
arXiv2017

Dirichlet parabolicity and $L^1$-Liouville property under localized geometric conditions

We shed a new light on the $L^1$-Liouville property for positive, superharmonic functions by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space. We also present examples in any dimension showing that the $L^1$-Liouville property is strictly weaker than the stochastic completeness of the manifold. The main tool in our investigations is represented by the potential theory of a manifold with boundary subject to Dirichlet boundary conditions. The paper incorporates, under a unifying viewpoint, some old and new aspects of the theory, with a special emphasis on global maximum principles and on the role of the Dirichlet Green's kernel.

math.DG↗
arXiv2016

Laplacian cut-offs, porous and fast diffusion on manifolds and other applications

We construct exhaustion and cut-off functions with controlled gradient and Laplacian on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the distance from a fixed reference point, without any assumptions on the topology or the injectivity radius. Along the way we prove a generalization of the Li-Yau gradient estimate which is of independent interest. We then apply our cut-offs to the study of the fast and porous media diffusion, of $L^q$-properties of the gradient and of the self-adjointness of Schroedinger-type operators.

math.DG↗
arXiv2015

Curvature estimates for submanifolds immersed into horoballs and horocylinders

We prove mean curvature estimates and a Jorge-Koutroufiotis type theorem for submanifolds confined into either a horocylinder of N X L or a horoball of N, where N is a Cartan-Hadamard manifold with pinched curvature. Thus, these submanifolds behave in many respects like submanifolds immersed into compact balls and into cylinders over compact balls. The proofs rely on the Hessian comparison theorem for the Busemann function.

math.DG↗
arXiv2013

The connectivity at infinity of a manifold and $L^{q,p}$-Sobolev inequalities

The purpose of this paper is to give a self-contained proof that a complete manifold with more than one end never supports an $L^{q,p}$-Sobolev inequality ($2 \leq p$, $q\leq p^{*}$), provided the negative part of its Ricci tensor is small (in a suitable spectral sense). In the route, we discuss potential theoretic properties of the ends of a manifold enjoying an $L^{q,p}$-Sobolev inequality.

math.DG↗
arXiv2013

Global maximum principles and divergence theorems on complete manifolds with boundary

In this paper we extend to non-compact Riemannian manifolds with boundary the use of two important tools in the geometric analysis of compact spaces, namely, the weak maximum principle for subharmonic functions and the integration by parts. The first one is a new form of the classical Ahlfors maximum principle whereas the second one is a version for manifolds with boundary of the so called Kelvin-Nevanlinna-Royden criterion of parabolicity. In fact, we will show that the validity of non-compact versions of these tools serve as a characterization of the Neumann parabolicity of the space. The motivation underlying this study is to obtain new information on the geometry of graphs with prescribed mean curvature inside a Riemannian product of the type $N\times\mathbb{R}$. In this direction two kind of results will be presented: height estimates for constant mean curvature graphs parametrized over unbounded domains in a complete manifold and slice type results for graphs whose superlevel sets have finite volume.

math.DG↗