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Edoardo Mainini

Publications and source records attributed to Edoardo Mainini.

At least 19 recordsLinked to original sources
arXiv2026

Smoothing effect and uniqueness for aggregation diffusion models

We consider aggregation-diffusion models for a density of mass in $\mathbb R^d$, $d\ge 2$, where the diffusion can be either linear or of porous medium type, while the aggregation effect is governed by the attractive Newtonian or Bessel potential. It is well known that the dynamics can be interpreted as Wasserstein gradient flow of the natural free energy of the system, and that, under suitable assumptions on the diffusion exponent and the mass of the initial datum, solutions exist globally in time. In these regimes, we perform the analysis of the discrete variational approach by means of the JKO scheme. We establish a sharp $L^\infty$ smoothing effect, which proves to be of same rate as that of the porous medium equation. Thanks to this estimate, we obtain uniqueness of global gradient flow solutions for initial data of finite energy and, in the diffusion dominated regime, for measure data having finite second moment. We prove the energy dissipation equality and characterize the gradient flow in terms of suitable evolution variational inequalities. In the fair competition regime, we show uniform extinction of solutions for large time under smallness conditions on the mass.

math.AP↗
arXiv2026

On fractional critical problems with multi-polar Hardy potentials

We investigate the existence of positive solutions to fractional equations presenting a double criticality: a multi-polar Hardy-type potential and a Sobolev critical nonlinearity. The nonlocal nature of the operator and the absence of explicit ground states for the single-pole equation stand as major difficulties. We overcome these obstacles by passing to an extended formulation of the problem and by establishing sharp asymptotic estimates for the solutions in the case of a single pole. Then, through a concentration-compactness argument, we show that the existence of minimizers is dictated by the magnitude of the masses and the mutual distances between the corresponding poles.

math.AP↗
arXiv2025

Equilibria of aggregation-diffusion models with nonlinear potentials

We consider an evolution model with nonlinear diffusion of porous medium type in competition with a nonlocal drift term favoring mass aggregation. The distinguishing trait of the model is the choice of a nonlinear $(s,p)$ Riesz potential for describing the overall aggregation effect. We investigate radial stationary states of the dynamics, showing their relation with extremals of suitable Hardy-Littlewood-Sobolev inequalities. In the case that aggregation does not dominate over diffusion, radial stationary states also relate to global minimizers of a homogeneous free energy functional featuring the $(s,p)$ energy associated to the nonlinear potential. In the limit as the fractional parameter $s$ tends to zero, the nonlocal interaction term becomes a backward diffusion and we describe the asymptotic behavior of the stationary states.

math.AP↗
arXiv2023

Linearization of finite elasticity with surface tension

We propose models in nonlinear elasticity for nonsimple materials that include surface energy terms. Additionally, we also discuss living surface loads on the boundary. We establish corresponding linearized models and show their relationship to the original ones by means of $\Gamma$-convergence.

math.AP↗
arXiv2023

On the weighted inertia-energy approach to forced wave equations

We show convergence of minimizers of weighted inertia-energy functionals to solutions of initial value problems for a class of nonlinear wave equations. The result is given for the nonhomogeneous case under a natural growth assumption on the forcing term.

math.AP↗
arXiv2023

Newton's second law as limit of variational problems

We show that the solution of the Cauchy problem for the classical ode $m \mathbf y''=\mathbf f$ can be obtained as limit of minimizers of exponentially weighted convex variational integrals. This complements the known results about weighted inertia-energy approach to Lagrangian mechanics and hyperbolic equations.

math.AP↗
arXiv2022

Nonlinear aggregation-diffusion equations with Riesz potentials

We consider an aggregation-diffusion model, where the diffusion is nonlinear of porous medium type and the aggregation is governed by the Riesz potential of order s. The addition of a quadratic diffusion term produces a more precise competition with the aggregation term for small s, as they have the same scaling if s=0. We prove existence and uniqueness of stationary states and we characterize their asymptotic behavior as s goes to zero. Moreover, we prove existence of gradient flow solutions to the evolution problem by applying the JKO scheme.

math.AP↗
arXiv2022

A new approach to posterior contraction rates via Wasserstein dynamics

This paper presents a new approach to the classical problem of quantifying posterior contraction rates (PCRs) in Bayesian statistics. Our approach relies on Wasserstein distance, and it leads to two main contributions which improve on the existing literature of PCRs. The first contribution exploits the dynamic formulation of Wasserstein distance, for short referred to as Wasserstein dynamics, in order to establish PCRs under dominated Bayesian statistical models. As a novelty with respect to existing approaches to PCRs, Wasserstein dynamics allows us to circumvent the use of sieves in both stating and proving PCRs, and it sets forth a natural connection between PCRs and three well-known classical problems in statistics and probability theory: the speed of mean Glivenko-Cantelli convergence, the estimation of weighted Poincaré-Wirtinger constants and Sanov large deviation principle for Wasserstein distance. The second contribution combines the use of Wasserstein distance with a suitable sieve construction to establish PCRs under full Bayesian nonparametric models. As a novelty with respect to existing literature of PCRs, our second result provides with the first treatment of PCRs under non-dominated Bayesian models. Applications of our results are presented for some classical Bayesian statistical models, e.g., regular parametric models, infinite-dimensional exponential families, linear regression in infinite dimension and nonparametric models under Dirichlet process priors.

math.ST↗
arXiv2022

Strong posterior contraction rates via Wasserstein dynamics

In Bayesian statistics, posterior contraction rates (PCRs) quantify the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of a true model, in a suitable way, as the sample size goes to infinity. In this paper, we develop a new approach to PCRs, with respect to strong norm distances on parameter spaces of functions. Critical to our approach is the combination of a local Lipschitz-continuity for the posterior distribution with a dynamic formulation of the Wasserstein distance, which allows to set forth an interesting connection between PCRs and some classical problems arising in mathematical analysis, probability and statistics, e.g., Laplace methods for approximating integrals, Sanov's large deviation principles in the Wasserstein distance, rates of convergence of mean Glivenko-Cantelli theorems, and estimates of weighted Poincar\'e-Wirtinger constants. We first present a theorem on PCRs for a model in the regular infinite-dimensional exponential family, which exploits sufficient statistics of the model, and then extend such a theorem to a general dominated model. These results rely on the development of novel techniques to evaluate Laplace integrals and weighted Poincar\'e-Wirtinger constants in infinite-dimension, which are of independent interest. The proposed approach is applied to the regular parametric model, the multinomial model, the finite-dimensional and the infinite-dimensional logistic-Gaussian model and the infinite-dimensional linear regression. In general, our approach leads to optimal PCRs in finite-dimensional models, whereas for infinite-dimensional models it is shown explicitly how the prior distribution affect PCRs.

math.ST↗
arXiv2022

Wasserstein posterior contraction rates in non-dominated Bayesian nonparametric models

Posterior contractions rates (PCRs) strengthen the notion of Bayesian consistency, quantifying the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of the true model, with probability tending to 1 or almost surely, as the sample size goes to infinity. Under the Bayesian nonparametric framework, a common assumption in the study of PCRs is that the model is dominated for the observations; that is, it is assumed that the posterior can be written through the Bayes formula. In this paper, we consider the problem of establishing PCRs in Bayesian nonparametric models where the posterior distribution is not available through the Bayes formula, and hence models that are non-dominated for the observations. By means of the Wasserstein distance and a suitable sieve construction, our main result establishes PCRs in Bayesian nonparametric models where the posterior is available through a more general disintegration than the Bayes formula. To the best of our knowledge, this is the first general approach to provide PCRs in non-dominated Bayesian nonparametric models, and it relies on minimal modeling assumptions and on a suitable continuity assumption for the posterior distribution. Some refinements of our result are presented under additional assumptions on the prior distribution, and applications are given with respect to the Dirichlet process prior and the normalized extended Gamma process prior.

math.ST↗
arXiv2022

Asymptotic behavior of constrained local minimizers in finite elasticity

We provide an approximation result for the pure traction problem of linearized elasticity in terms of local minimizers of finite elasticity, under the constraint of vanishing average curl for admissible deformation maps. When suitable rotations are included in the constraint, the limit is shown to be the linear elastic equilibrium associated to rotated loads.

math.AP↗
arXiv2020

Sharp conditions for the linearization of finite elasticity

We consider the topic of linearization of finite elasticity for pure traction problems. We characterize the variational limit for the approximating sequence of rescaled nonlinear elastic energies. We show that the limiting minimal value can be strictly lower than the minimal value of the standard linear elastic energy if a strict compatibility condition for external loads does not hold. The results are provided for both the compressible and the incompressible case.

math.AP↗
arXiv2020

Uniqueness of entire ground states for the fractional plasma problem

We establish uniqueness of vanishing radially decreasing entire solutions, which we call ground states, to some semilinear fractional elliptic equations. In particular, we treat the fractional plasma equation and the supercritical power nonlinearity. As an application, we deduce uniqueness of radial steady states for nonlocal aggregation-diffusion equations of Keller-Segel type, even in the regime that is dominated by aggregation.

math.AP↗
arXiv2020

Lipschitz continuity of probability kernels in the optimal transport framework

In Bayesian statistics, a continuity property of the posterior distribution with respect to the observable variable is crucial as it expresses well-posedness, i.e., stability with respect to errors in the measurement of data. Essentially, this requires to analyze the continuity of a probability kernel or, equivalently, of a conditional probability distribution with respect to the conditioning variable. Here, we give general conditions for the Lipschitz continuity of probability kernels with respect to metric structures arising within the optimal transport framework, such as the Wasserstein metric. For dominated probability kernels over finite-dimensional spaces, we show Lipschitz continuity results with a Lipschitz constant enjoying explicit bounds in terms of Fisher-information functionals and weighted Poincar\'e constants. We also provide results for kernels with moving support, for infinite-dimensional spaces and for non dominated kernels. We show applications to several problems in Bayesian statistics, such as approximation of posterior distributions by mixtures and posterior consistency.

math.PR↗