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Ulisse Stefanelli

Publications and source records attributed to Ulisse Stefanelli.

At least 19 recordsLinked to original sources
arXiv2026

A free boundary problem in accretive growth

We study a free boundary problem inspired by the modelization of accretive growth. The growth process is formulated through a level-set approach, leading to a boundary-value problem for a Hamilton-Jacobi equation within a prescribed constraining set. Existence, variational representability, and regularity of solutions to the growth subproblem are investigated. The full system arises from coupling the growth dynamics with an elliptic equation for the activation field. Existence of solutions to the fully coupled free boundary problems is obtained via an iterative procedure.

math.AP↗
arXiv2026

Evolution variational inequalities with general costs

We extend the theory of gradient flows beyond metric spaces by studying evolution variational inequalities (EVIs) driven by general cost functions $c$, including Bregman and entropic transport divergences. We establish several properties of the resulting flows, including stability and energy identities. Using novel notions of convexity related to costs $c$, we prove that EVI flows are the limit of splitting schemes, providing assumptions for both implicit and explicit iterations.

math.FA↗
arXiv2025

Level sets of solutions to the stationary Hamilton-Jacobi equation are John regular

Let $u$ be the unique nonnegative viscosity solution of the Hamilton-Jacobi equation $H(x,\nabla u)=0$ in the external domain ${\mathbb R}^{ n} \setminus K$ with $u=0$ on $K$. Under general conditions on $H$, we prove that all sublevels of $u$ are John domains. Moreover, if $K$ itself is a John domain, we provide a uniform lower bound on the John constant of all sublevels. We exhibit counterexamples showing that John regularity is sharp in this setting.

math.AP↗
arXiv2025

Crystallization in the Winterbottom shape and sharp fluctuation laws

We address finite crystallization in two dimensions in the presence of a flat crystalline substrate. Particles interact through short-range two- and three-body potentials favoring local square-lattice arrangements. An additional interaction term of relative strength $β>0$ couples the particles and the substrate. Our first main result proves crystallization for all $β>0$, corresponding to the onset of discrete Winterbottom configurations. The proof relies on a stratification technique from [31], characterizing the topology of the bond graph of minimizing configurations. Our second main result concerns fluctuations estimates for $β\in (0,1)$. We obtain bounds on the distance between distinct minimizers with the same number $N$ of particles, showing a sharp scaling law $N^{3/4}$ when $β$ is rational, and $N^{1/3}$ when $β$ is irrational and algebraic. This reveals a genuine substrate-driven effect on fluctuation laws. As a corollary, we derive a discrete-to-continuum convergence of minimizers towards the Winterbottom equilibrium shape in the large-particle limit.

cond-mat.mes-hall↗
arXiv2025

Existence for accreting viscoelastic solids at large strains

By revisiting a model proposed in [45], we address the accretive growth of a viscoelastic solid at large strains. The accreted material is assumed to accumulate at the boundary of the body in an unstressed state. The growth process is driven by the deformation state of the solid. The progressive build-up of incompatible strains in the material is modeled by considering an additional backstrain. The model is regularized by postulating the presence of a fictitious compliant material surrounding the accreting body. We show the existence of solutions to the coupled accretion and viscoelastic equilibrium problem.

math.AP↗
arXiv2025

Global well-posedness for a time-fractional doubly nonlinear equation

We consider a time-fractional parabolic equation of doubly nonlinear type, featuring nonlinear terms both inside and outside the differential operator in time. The main nonlinearities are maximal monotone graphs, without restrictions on the growth. In addition, a Lipschitz continuous perturbation is considered. The existence of global weak solutions is obtained via a regularization and Galerkin approximation method. Uniqueness is also discussed under some additional assumptions.

math.AP↗
arXiv2025

Weak stability by noise for approximations of doubly nonlinear evolution equations

Doubly nonlinear stochastic evolution equations are considered. Upon assuming the additive noise to be rough enough, we prove the existence of probabilistically weak solutions of Friedrichs type and study their uniqueness in law. This entails stability for approximations of stochastic doubly nonlinear equations in a weak probabilistic sense. Such effect is a genuinely stochastic, as doubly nonlinear equations are not even expected to exhibit uniqueness in the deterministic case.

math.PR↗
arXiv2025

Quasistatic nonassociative plasticity at finite strains

We investigate finite-strain elastoplastic evolution in the nonassociative setting. The constitutive material model is formulated in variational terms and coupled with the quasistatic equilibrium system. We introduce measure-valued energetic solutions and prove their existence via a time discretization approach. The existence theory hinges on a suitable regularization of the dissipation term via a space-time mollification. Eventually, we discuss the possibility of solving the problem in the setting of functions, instead of measures.

math.AP↗
arXiv2025

Viscoelasticity and accretive phase-change at finite strains

We investigate the evolution of a two-phase viscoelastic material at finite strains. The phase evolution is assumed to be irreversible: One phase accretes in time in its normal direction, at the expense of the other. Mechanical response depends on the phase. At the same time, growth is influenced by the mechanical state at the boundary of the accreting phase, making the model fully coupled. This setting is inspired by the early stage development of solid tumors, as well as by the swelling of polymer gels. We formulate the evolution problem by coupling the balance of momenta in weak form and the growth dynamics in the viscosity sense. Both a diffused- and a sharp-interface variant of the model are proved to admit solutions and the sharp-interface limit investigated.

math.AP↗
arXiv2024

Long-Time behavior of the tangential surface Navier-Stokes equation

We investigate the initial-value problem for the incompressible tangential Navier-Stokes equation with variable viscosity on a given two-dimensional surface without boundary. Existence of global weak and strong solutions under inhomogeneous forcing is proved by a fixed-point and continuation argument. Continuous dependence on data, backward uniqueness, and instantaneous regularization are also discussed. Depending on the effect of the inhomogeneous forcing on the dissipative and the nondissipative components of the system, we investigate the long-time behavior of solutions. We prove the existence and properties of the $σ$-global attractor, in the case of bounded trajectories, and of the so-called unbounded attractor, for unbounded trajectories.

math.AP↗
arXiv2024

A model of gravitational differentiation of compressible self-gravitating planets

We present a dynamic model for inhomogeneous viscoelastic media at finite strains. The model features a Kelvin-Voigt rheology, and includes a self-generated gravitational field in the actual evolving configuration. In particular, a fully Eulerian approach is adopted. We specialize the model to viscoelastic (barotropic) fluids and prove existence and a certain regularity of global weak solutions by a Faedo-Galerkin semi-discretization technique. Then, an extension to multi-component chemically reacting viscoelastic fluids based on a phenomenological approach by Eckart and Prigogine, is advanced and studied. The model is inspired by planetary geophysics. In particular, it describes gravitational differentiation of inhomogeneous planets and moons, possibly undergoing volumetric phase transitions.

math.AP↗
arXiv2024

Modelling of planetary accretion and core-mantle structure formation

We advance a thermodynamically consistent model of self-gravitational accretion and differentiation in planets. The system is modeled in actual variables as a compressible thermoviscoelastic fluid in a fixed, sufficiently large domain. The supply of material to the accreting and differentiating system is described as a bulk source of mass, volume, impulse, and energy localized in some border region of the domain. Mass, momentum, and energy conservation, along with constitutive relations, result in an extended compressible Navier-Stokes-Fourier-Poisson system. After studying some single-component setting, we consider a two-component situation, where metals and silicates mix and differentiate under gravity, eventually forming a core-mantle structure. The energetics of the models are elucidated. Moreover, we prove that the models are stable, in that self-gravitational collapse is excluded. Eventually, we comment on the prospects of devising a rigorous mathematical approximation and existence theory.

math.AP↗
arXiv2024

The Weighted Inertia-Energy-Dissipation Principle

The Weighted Inertia-Energy-Dissipation (WIDE) principle is a global variational approach to nonlinear evolution equations of parabolic and hyperbolic type. The minimization of the parameter-dependent WIDE functional on trajectories delivers an elliptic-in-time regularization. By taking the limit in the parameter, one recovers a solution to the given differential problem. This survey is intended to provide a comprehensive account of the available results on the WIDE variational approach. The basic concepts are illustrated in the simplest finite-dimensional case, and the existing literature, both theoretical and applied, is systematically reviewed.

math.AP↗
arXiv2024

Weighted Energy-Dissipation approach to semilinear gradient flows with state-dependent dissipation

We investigate the Weighted Energy-Dissipation variational approach to semilinear gradient flows with state-dependent dissipation. A family of parameter-dependent functionals defined over entire trajectories is introduced and proved to admit global minimizers. These global minimizers correspond to solutions of elliptic-in-time regularizations of the limiting causal problem. By passing to the limit in the parameter we prove that such global minimizers converge, up to subsequences, to a solution of the gradient flow.

math.AP↗
arXiv2024

Optimal control of gradient flows via the Weighted Energy-Dissipation method

We consider a general optimal control problem in the setting of gradient flows. Two approximations of the problem are presented, both relying on the variational reformulation of gradient-flow dynamics via the Weighted-Energy-Dissipation variational approach. This consists in the minimization of global-in-time functionals over trajectories, combined with a limit passage. We show that the original nonpenalized problem and the two successive approximations admits solutions. Moreover, resorting to a $Γ$-convergence analysis we show that penalised optimal controls converge to nonpenalized one as the approximation is removed.

math.OC↗
arXiv2024

An existence result for accretive growth in elastic solids

We investigate a model for the accretive growth of an elastic solid. The reference configuration of the body is accreted in its normal direction, with space- and deformation-dependent accretion rate. The time-dependent reference configuration is identified via the level sets of the unique viscosity solution of a suitable generalized eikonal equation. After proving the global-in-time well-posedness of the quasistatic equilibrium under prescribed growth, we prove the existence of a local-in-time solution for the coupled equilibrium-growth problem, where both mechanical displacement and time-evolving set are unknown. A distinctive challenge is the limited regularity of the growing body, which calls for proving a new uniform Korn inequality.

math.AP↗