arXiv subjects

Riccardo Durastanti

Publications and source records attributed to Riccardo Durastanti.

13 recordsLinked to original sources

arXiv2026

Scaling limits for Moran processes on metric strategy spaces: an Eulerian derivation of pure replicator and Fleming--Viot measure-valued PDEs

We study the large-population limit of a discrete-time Moran process featuring multiple strategies, drawn from a possibly infinite strategy space $\mathcal{V}$ (a metric space), under both weak and strong selection. In the Eulerian density formulation, the limiting dynamics depend critically on the relative scaling of population size, mutation rate, and selection intensity. Depending on these scalings, the limit behavior is governed either by a purely deterministic replicator-type continuity equation or by a diffusion-enhanced PDE. In the latter case, the diffusion operator recovers the classical Fleming--Viot operator and the Kimura equation as special instances. Methodologically, we derive the limit by reinterpreting the discrete process in Eulerian coordinates, constructing interpolating curves that satisfy an approximate PDE, and establishing convergence via a compactness argument in a suitable topology on the space of probability measures over probabilities over $\mathcal{V}$.

math.AP↗

arXiv2026

Advancing fronts for the thin-film equation with null slip and repulsive potentials: the case of partial wetting

For negative values of the spreading coefficient (that is, in the so-called ``partial wetting'' regime), we prove that the thin-film equation with zero slip and repulsive potentials $P$ of the form $P(h)\approx h^{1-m}$ as $h\to 0$, $m>1$, admits for any positive speed a one-parameter family of travelling-wave solutions with a contact line and (as in standard slippage models) a logarithmically-corrected linear behaviour as $h\to +\infty$. These waves have locally finite rate of dissipation for any $m>1$ and locally finite energy for any $m\in (1,3)$. The result thus confirms that mildly repulsive potentials effectively resolve the no-slip paradox. The family is parametrized by a thermodynamically consistent contact-line condition which reduces to the classical fixed microscopic contact-angle one if $P\equiv 0$.

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arXiv2025

Mean field first order optimality condition under low regularity of controls

We show that mean field optimal controls satisfy a first order optimality condition (at a.e. time) without any a priori requirement on their spatial regularity. This principle is obtained by a careful limit procedure of the Pontryagin maximum principle for finite particle systems. In particular, our result applies to the case of mean field selective optimal control problems for multipopulation and replicator dynamics.

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arXiv2025

A Pontryagin Maximum Principle for agent-based models with convex state space

We derive a first order optimality condition for a class of agent-based systems, as well as for their mean-field counterpart. A relevant difficulty of our analysis is that the state equation is formulated on possibly infinite-dimensional convex subsets of Banach spaces. This is a typical feature of many problems in multi-population dynamics, where a convex set of probability measures may account for the population, the degree of influence or the strategy attached to each agent. Due to the lack of a linear structure and of local compactness, the usual tools of needle variations and linearisation procedures used to derive Pontryagin type conditions have to be generalised to the setting at hand. This is done by considering suitable notions of differentials and by a careful inspection of the underlying functional structures.

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arXiv2024

Nonlinear asymptotic mean value characterizations of holomorphic functions

Starting from a characterization of holomorphic functions in terms of a suitable mean value property, we build some nonlinear asymptotic characterizations for complex-valued solutions of certain nonlinear systems, which have to do with the classical Cauchy-Riemann equations. From these asymptotic characterizations, we derive suitable asymptotic mean value properties, which are used to construct appropriate vectorial dynamical programming principles. The aim is to construct approximation schemes for the so-called contact solutions, recently introduced by N. Katzourakis, of the nonlinear systems here considered.

math.AP↗

arXiv2023

The Dirichlet problem for possibly singular elliptic equations with degenerate coercivity

We deal with existence, uniqueness and regularity of nonnegative solutions to a Dirichlet problem for equations as \begin{equation*} \displaystyle -\operatorname{div}\left(\frac{|\nabla u|^{p-2}\nabla u}{(1+u)^{θ(p-1)}}\right) = h(u)f \quad \text{in }Ω, \end{equation*} where $Ω$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge 2$), $p>1$, $θ\ge 0$, $f\geq 0$ belongs to a suitable Lebesgue space and $h$ is a continuous, nonnegative function which may blow up at zero and it is bounded at infinity.

math.AP↗

arXiv2023

Thin-film equations with singular potentials: an alternative solution to the contact-line paradox

In the regime of lubrication approximation, we look at spreading phenomena under the action of singular potentials of the form $P(h)\approx h^{1-m}$ as $h\to 0^+$ with $m>1$, modeling repulsion between the liquid-gas interface and the substrate. We assume zero slippage at the contact line. Based on formal analysis arguments, we report that for any $m>1$ and any value of the speed (both positive and negative) there exists a three-parameter, hence generic, family of fronts (i.e., traveling-wave solutions with a contact line). A two-parameter family of advancing "linear-log" fronts also exists, having a logarithmically corrected linear behaviour in the liquid bulk. All these fronts have finite rate of dissipation, indicating that singular potentials stand as an alternative solution to the contact-line paradox. In agreement with steady states, fronts have microscopic contact angle equal to $π/2$ for all $m>1$ and finite energy for all $m<3$. We also propose a selection criterion for the fronts, based on thermodynamically consistent contact-line conditions modeling friction at the contact line. So as contact-angle conditions do in the case of slippage models, this criterion selects a unique (up to translation) linear-log front for each positive speed. Numerical evidence suggests that, fixed the speed and the frictional coefficient, its shape depends on the spreading coefficient, with steeper fronts in partial wetting and a more prominent precursor region in dry complete wetting.

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arXiv2022

Spreading equilibria under mildly singular potentials: pancakes versus droplets

We study global minimizers of a functional modeling the free energy of thin liquid layers over a solid substrate under the combined effect of surface, gravitational, and intermolecular potentials. When the latter ones have a mild repulsive singularity at short ranges, global minimizers are compactly supported and display a microscopic contact angle of $π/2$. Depending on the form of the potential, the macroscopic shape can either be droplet-like or pancake-like, with a transition profile between the two at zero spreading coefficient. These results generalize, complete, and give mathematical rigor to de Gennes' formal discussion of spreading equilibria. Uniqueness and non-uniqueness phenomena are also discussed.

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arXiv2021

Shape programming of a magnetic elastica

We consider a cantilever beam which possesses a possibly non-uniform permanent magnetization, and whose shape is controlled by an applied magnetic field. We model the beam as a plane elastic curve and we suppose that the magnetic field acts upon the beam by means of a distributed couple that pulls the magnetization towards its direction. Given a list of target shapes, we look for a design of the magnetization profile and for a list of controls such that the shapes assumed by the beam when acted upon by the controls are as close as possible to the targets, in an averaged sense. To this effect, we formulate and solve an optimal design and control problem leading to the minimization of a functional which we study by both direct and indirect methods. In particular, we prove that minimizers exist, solve the associated Lagrange-multiplier formulation (besides non-generic cases), and are unique at least for sufficiently low intensities of the controlling magnetic fields. To achieve the latter result, we use two nested fixed-point arguments relying on the Lagrange-multiplier formulation of the problem, a method which also suggests a numerical scheme. Various relevant open question are also discussed.

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arXiv2019

Comparison principle for elliptic equations with mixed singular nonlinearities

We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by \begin{equation*} \begin{cases} \displaystyle -Δ_p u= \frac{f}{u^γ} + g u^q & \mbox{in $Ω$,} \\ u = 0 & \mbox{on $\partialΩ$,} \end{cases} \end{equation*} where $Ω$ is an open bounded subset of $\mathbb{R}^N$, $Δ_p u:=\text{div}(|\nabla u|^{p-2}\nabla u)$ is the usual $p$-Laplacian operator, $γ\geq 0$ and $0\leq q\leq p-1$; $f$ and $g$ are nonnegative functions belonging to suitable Lebesgue spaces.

math.AP↗

arXiv2019

Asymptotic behavior and existence of solutions for singular elliptic equations

We study the asymptotic behavior, as $γ$ tends to infinity, of solutions for the homogeneous Dirichlet problem associated to singular semilinear elliptic equations whose model is $$ -Δu=\frac{f(x)}{u^γ}\,\text{ in }Ω, $$ where $Ω$ is an open, bounded subset of $\RN$ and $f$ is a bounded function. We deal with the existence of a limit equation under two different assumptions on $f$: either strictly positive on every compactly contained subset of $Ω$ or only nonnegative. Through this study we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated to $$ -Δv + \frac{|\nabla v|^2}{v} = f\,\text{ in }Ω. $$

math.AP↗

arXiv2019

Regularizing effect for some p-Laplacian systems

We study existence and regularity of weak solutions for the following $p$-Laplacian system \begin{cases} -Δ_p u+Aφ^{θ+1}|u|^{r-2}u=f, \ &u\in W_0^{1,p}(Ω),\\-Δ_p φ=|u|^rφ^θ, \ &φ\in W_0^{1,p}(Ω), \end{cases} where $Ω$ is an open bounded subset of $\mathbb{R}^N$ $(N\geq 2)$, $Δ_p v :=\operatorname{div}(|\nabla v|^{p-2}\nabla v)$ is the $p$-Laplacian operator, for $1 0$, $r>1$, $0\leqθ<p-1$ and $f$ belongs to a suitable Lebesgue space. In particular, we show how the coupling between the equations in the system gives rise to a regularizing effect producing the existence of finite energy solutions.

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arXiv2017

Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data

We study existence and uniqueness of solutions to a nonlinear elliptic boundary value problem with a general, and possibly singular, lower order term, whose model is $$\begin{cases} -Δ_p u = H(u)μ& \text{in}\ Ω,\\ u>0 &\text{in}\ Ω,\\ u=0 &\text{on}\ \partialΩ. \end{cases}$$ Here $Ω$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge2$), $Δ_p u:= \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ ($1<p<N$) is the $p$-laplacian operator, $μ$ is a nonnegative bounded Radon measure on $Ω$ and $H(s)$ is a continuous, positive and finite function outside the origin which grows at most as $s^{-γ}$, with $γ\ge0$, near zero.

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