arXiv subjects

Diego Berti

Publications and source records attributed to Diego Berti.

16 recordsLinked to original sources

arXiv2026

Short-time behavior and invariant surfaces for caloric functions with non-constant boundary values

We consider a Cauchy-Dirichlet problem for a heat equation with variable coefficients in non-divergence form. The initial values are assumed to be homogeneous, while the Dirichlet boundary values are non-constant. In this setting, we derive an asymptotic formula for the short-time behavior of the solution, which extends the celebrated Varadhan formula. We stress the fact that the boundary values are allowed to vanish or change sign. The formula is obtained by combining the original ideas of Varadhan with those of Evans and Ishii, pertaining to the theory of viscosity solutions, and some further remarks. In passing, we prove the elliptic counterpart of the formula. This concerns the slow-diffusion behavior of the solutions of the resolvent equation. Furthermore, for the case of the classical Laplace operator, we prove asymptotic formulas for the heat content and the mean value of the solution of the resolvent equation on spheres touching the boundary of the domain. These extend to the case of non-constant boundary values, certain formulas previously obtained by the second author and S. Sakaguchi. The formulas involve the principal curvatures at the touching point and the Dirichlet boundary values. We then use our formulas to describe time-invariant surfaces for the heat equation, in presence of non-constant Dirichlet boundary values. We give two symmetry results in case of non-negative Dirichlet boundary values and a non-existence result, when those values change sign.

math.AP↗

arXiv2026

Shock wavefronts for parabolic equations with sign-changing diffusivity

We consider a reaction-diffusion equation in a one-dimensional space, where the diffusion coefficient changes sign from positive to negative and back to positive. The reaction term is bistable, with its interior zero located in the region where the diffusivity is negative. The model does not admit continuous wavefronts, i.e., continuous traveling waves that connect the steady states $0$ and $1$. We prove the existence of a family of shock wavefronts, that is, wavefronts with profiles exhibiting a jump discontinuity. We investigate the properties of these profiles and their propagation speeds. Finally, we apply the results to a recently proposed model describing the movement of a population composed of both isolated and grouped individuals.

math.AP↗

arXiv2026

Expansive solutions and the boundary at infinity for the homogeneous $N$-body problem

We investigate expansive solutions of the $N$-body problem in $\mathbb{R}^d$ ($d\ge2$) driven by homogeneous Newtonian potentials of degree $-α$. We establish the existence of half-entire expansive motions with prescribed initial configuration and asymptotic direction for a wide range of homogeneity exponents $α$. Our approach is variational and relies on the minimization of a suitably renormalized Lagrangian action, allowing us to treat in a unified framework the hyperbolic, parabolic, and hyperbolic-parabolic regimes in the sense of Chazy's classification. Beyond existence, we derive refined asymptotic expansions for all classes of expansive solutions, identifying higher-order correction terms and improving previously known growth estimates, including the classical Newtonian case $α=1$. In particular, for hyperbolic-parabolic solutions, we provide a detailed description of the interplay between linear escape of cluster centers and internal parabolic dynamics, extending the cluster scattering picture to general homogeneous potentials. Finally, we interpret these solutions within the geometric framework of the Jacobi-Maupertuis metric and the weak KAM theory. In this perspective, expansive motions correspond to geodesic rays and calibrating curves for the associated Hamilton-Jacobi equation, yielding a dynamical characterization of the boundary at infinity and a refined description of global viscosity solutions.

math.DS↗

arXiv2025

Symmetric solutions of the $n$-body problem: a numerical study of Floquet multipliers and Morse indices

In this paper, we consider periodic solutions of the $n$-body problem that satisfy symmetry constraints, expressed through invariance under finite group actions. We focus on their stability properties and present algorithms specifically designed for the computation of Floquet multipliers and Morse indices. Numerical results are provided to illustrate our methods in both two and three dimensional configuration spaces, and for different choices on the number of bodies.

math.DS↗

arXiv2025

On the regularity of solutions to the Hamilton-Jacobi equations for the N-body problem

We prove that certain suitably renormalized value functions associated with the $d$-dimensional ($d\geq2$) $N$-body problem corresponding to different limiting shapes of expanding solutions, under the assumption that the center of mass is at the origin, are viscosity solutions of the associated Hamilton-Jacobi equation. We analyze their singularities, defined as the initial configurations for which the minimizer of the associated variational problem is not unique. Moreover, we estimate the size of the closure of the singular set by proving its $\mathcal{H}^{d(N-1)-1}$-rectifiability, and we provide an upper bound on the Hausdorff dimension of the set of regular conjugate points.

math.DS↗

arXiv2025

New bounds on the high Sobolev norms of the 1d NLS solutions

We introduce modified energies that are suitable to get upper bounds on the high Sobolev norms for solutions to the $1$D periodic NLS. Our strategy is rather flexible and allows us to get a new and simpler proof of the bounds obtained by Bourgain in the case of the quintic nonlinearity, as well as its extension to the case of higher order nonlinearities. Our main ingredients are a combination of integration by parts and classical dispersive estimates.

math.AP↗

arXiv2023

Global existence for a 3D Tropical Climate Model with damping and small initial data in $\dot H^{1/2}(\mathbb{R}^3)$

We consider a 3D Tropical Climate Model with damping terms in the equation of the barotropic mode $u$ and in the equation of the first baroclinic mode $v$ of the velocity. The equation for the temperature $θ$ is free from dampings. We prove global existence in time for this system assuming the initial data $(u_0, v_0,θ_0)$ small, in terms of the homogeneous space $\dot H^{1/2}(\mathbb{R}^3)$.

math.AP↗

arXiv2023

The role of convection in the existence of wavefronts for biased movements

We investigate a model, inspired by (Johnston et al., Sci. Rep., 7:42134, 2017), to describe the movement of a biological population which consists of isolated and grouped organisms. We introduce biases in the movements and then obtain a scalar reaction-diffusion equation which includes a convective term as a consequence of the biases. We focus on the case the diffusivity makes the parabolic equation of forward-backward-forward type and the reaction term models a strong Allee effect, with the Allee parameter lying between the two internal zeros of the diffusion. In such a case, the unbiased equation (i.e., without convection) possesses no smooth traveling-wave solutions; on the contrary, in the presence of convection, we show that traveling-wave solutions do exist for some significant choices of the parameters. We also study the sign of their speeds, which provides information on the long term behavior of the population, namely, its survival or extinction.

math.AP↗

arXiv2022

A regularity criterion for a 3D tropical climate model with damping

In this paper we deal with the 3D tropical climate model with damping terms in the equation of the barotropic mode $u$ and in the equation of the first baroclinic mode $v$ of the velocity, and we establish a regularity criterion for this system thanks to which the local smooth solution $(u, v, θ)$ can actually be extended globally in time.

math.AP↗

arXiv2021

Wavefronts for degenerate diffusion-convection reaction equations with sign-changing diffusivity

We consider in this paper a diffusion-convection reaction equation in one space dimension. The main assumptions are about the reaction term, which is monostable, and the diffusivity, which changes sign once or twice; then, we deal with a forward-backward parabolic equation. Our main results concern the existence of globally defined traveling waves, which connect two equilibria and cross both regions where the diffusivity is positive and regions where it is negative. We also investigate the monotony of the profiles and show the appearance of sharp behaviours at the points where the diffusivity degenerates. In particular, if such points are interior points, then the sharp behaviours are new and unusual.

math.AP↗

arXiv2021

Diffusion-convection reaction equations with sign-changing diffusivity and bistable reaction term

We consider a reaction-diffusion equation with a convection term in one space variable, where the diffusion changes sign from the positive to the negative and the reaction term is bistable. We study the existence of wavefront solutions, their uniqueness and regularity. The presence of convection reveals several new features of wavefronts: according to the mutual positions of the diffusivity and reaction, profiles can occur either for a single value of the speed or for a bounded interval of such values; uniqueness (up to shifts) is lost; moreover, plateaus of arbitrary length can appear; profiles can be singular where the diffusion vanishes.

math.AP↗

arXiv2020

Uniqueness and nonuniqueness of fronts for degenerate diffusion-convection reaction equations

We consider a scalar parabolic equation in one spatial dimension. The equation is constituted by a convective term, a reaction term with one or two equilibria, and a positive diffusivity which can however vanish. We prove the existence and several properties of traveling-wave solutions to such an equation. In particular, we provide a sharp estimate for the minimal speed of the profiles and improve previous results about the regularity of wavefronts. Moreover, we show the existence of an infinite number of semi-wavefronts with the same speed.

math.AP↗

arXiv2020

Small diffusion and short-time asymptotics for Pucci operators

This paper presents asymptotic formulas in the case of the following two problems for the {\it Pucci's extremal operators} $\mathcal{M}^\pm$. It is considered the solution $u^\varepsilon(x)$ of $-\varepsilon^2 \mathcal{M}^\pm\left(\nabla ^2 u^\varepsilon\right)+u^\varepsilon=0$ in $Ω$ such that $u^\varepsilon=1$ on $Γ$. Here, $Ω\subset \mathbb{R}^N$ is a domain (not necessarily bounded) and $Γ$ is its boundary. It is also considered $v(x,t)$ the solution of $v_t - \mathcal{M}^\pm\left(\nabla^2 v\right)=0$ in $Ω\times (0,\infty)$, $v=1$ on $Γ\times(0,\infty)$ and $v=0$ on $Ω\times \{0\}$. In the spirit of their previous works, the authors establish the profiles as $\varepsilon$ or $t\to 0^+$ of the values of $u^\varepsilon(x)$ and $v(x,t)$ as well as of those of their $q$-means on balls touching $Γ$. The results represent a further step in the extensions of those obtained by Varadhan and by Magnanini-Sakaguchi in the linear regime.

math.AP↗

arXiv2019

Asymptotic analysis of solutions related to the game-theoretic p-laplacian

We consider the (viscosity) solution $u(x,t)$ of the nonlinear evolution equation $u_t-Δ^G_p u=0$ in a (not necessarily bounded) domain $Ω$, such that $u=0$ in $Ω$ at time $t=0$ and $u=1$ on the boundary of $Ω$ at all times. Here, $Δ_p^G$ is the game-theoretic $p$-laplacian, a $1$-homogeneous version of the standard $p$-laplacian. Also, we consider the (viscosity) solution $u^\varepsilon$ of the nonlinear elliptic equation $\varepsilon^2Δ_p^G u^\varepsilon= u^\varepsilon$ in $Ω$, satisfying $u^\varepsilon=1$ on its boundary. In this thesis, we establish asymptotic formulas for small positive values of $t$ and $\varepsilon$ involving both the values of $u(x,t)$ and $u^\varepsilon(x)$ and their $q$-means on balls touching the boundary. In the spirit of S.~R.~S.~Varadhan's work, we associate appropriate rescalings of the values of $u(x,t)$ and $u^\varepsilon(x)$ to the distance of $x$ to the boundary of $Ω$. We also provide accurate uniform estimates of the rate of approximation in these formulas, highlighting the dependence on both the parameter $p$ and the regularity of the domain. The uniform estimates are new results also in the linear case. Also, we connect the asymptotic behavior of $q$-means on balls touching the boundary to a suitable function of principal curvatures. These results generalize and extend formulas for the heat content, obtained by R. Magnanini and S. Sakaguchi for $p=q=2$. Finally, we give a few applications of the asymptotic formulas to geometric and symmetry results. In particular, we characterize time-invariant level surfaces of $u(x,t)$ (or $\varepsilon$-invariant level surfaces of $u^\varepsilon(x)$) as spheres and hyperplanes.

math.AP↗

arXiv2018

Short time behaviour for game-theoretic $p$-caloric functions

We consider the solution of $u_t-Δ^G_p u=0$ in a (not necessarily bounded) domain, satisfying $u=0$ initially and $u=1$ on the boundary at all times. Here, $Δ^G_p u$ is the game-theoretic or normalized $p$-laplacian. We derive new precise asymptotic formulas for short times, that generalize the work of S. R. S. Varadhan for large deviations and that of the second author and S. Sakaguchi for the heat content of a ball touching the boundary. We also compute the short-time behavior of the $q$-mean of $u$ on such a ball. Applications to time-invariant level surfaces of $u$ are then derived.

math.AP↗

arXiv2018

Asymptotics for the resolvent equation associated to the game-theoretic $p$-laplacian

We consider the (viscosity) solution $u^\varepsilon$ of the elliptic equation $\varepsilon^2Δ_p^G u= u$ in a domain (not necessarily bounded), satisfying $u=1$ on its boundary. Here, $Δ_p^G$ is the {\it game-theoretic or normalized $p$-laplacian}. We derive asymptotic formulas for $\varepsilon\to 0^+$ involving the values of $u^\varepsilon$, in the spirit of Varadhan's work \cite{Va}, and its $q$-mean on balls touching the boundary, thus generalizing that obtained in \cite{MS-AM} for $p=q=2$. As in a related parabolic problem, investigated in \cite{BM}, we link the relevant asymptotic behavior to the geometry of the domain.

math.AP↗