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Francesco Solombrino

Publications and source records attributed to Francesco Solombrino.

At least 19 recordsLinked to original sources
arXiv2026

Stochastic homogenization of nonconvex unbounded integral functionals with generalized Orlicz growth

We consider the homogenization of random integral functionals which are possibly unbounded, that is, the domain of the integrand is not the whole space and may depend on the space-variable. In the vectorial case, we develop a complete stochastic homogenization theory for nonconvex unbounded functionals with convex growth of generalized Orlicz-type, under a standard set of assumptions in the field, in particular a coercivity condition of order $p^->1$, and an upper bound of order $p^+<\infty$. The limit energy is defined in a possibly anisotropic Musielak-Orlicz space, for which approximation results with smooth functions are provided. The proof is based on the localization method of $Γ$-convergence and a careful use of truncation arguments.

math.OC↗
arXiv2025

Relations between principal eigenvalue and torsional rigidity with Robin boundary conditions

We consider the torsional rigidity and the principal eigenvalue related to the Laplace operator with Dirichlet and Robin boundary conditions. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in the class of Lipschitz domains. The threshold exponent for the Robin case is explicitly recovered and shown to be strictly smaller than in the Dirichlet one.

math.AP↗
arXiv2025

A general perspective on CBO methods with stochastic rate of information

This paper studies a class of Consensus-Based Optimization (CBO) models featuring an additional stochastic rate of information, modeling the agents' knowledge of the environment and energy landscape. The well-posedness of the stochastic system is proved, together with its finite-particle approximation and the mean-field convergence to a kinetic PDE. Particles are shown to concentrate around the consensus point under mild assumptions on the initial spatial distribution and initial level of knowledge. In particular, the analysis unveils that a positive, however small, initial level of knowledge is enough for convergence to consensus to happen. The framework presented is general enough to include the first instances of CBO proposed in the literature.

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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

From discrete to continuum in the helical XY-model: emergence of chirality transitions in the $S^1$ to $S^2$ limit

We analyze the discrete-to-continuum limit of a frustrated ferromagnetic/anti-ferromagnetic $\mathbb{S}^2$-valued spin system on the lattice $λ_n\mathbb{Z}^2$ as $λ_n\to 0$. For $\mathbb{S}^2$ spin systems close to the Landau-Lifschitz point (where the helimagnetic/ferromagnetic transition occurs), it is well established that for chirality transitions emerge with vanishing energy. Inspired by recent work on the $N$-clock model, we consider a spin model where spins are constrained to $k_n$ copies of $\mathbb{S}^1$ covering $\mathbb{S}^2$ as $n\to\infty$. We identify a critical energy-scaling regime and a threshold for the divergence rate of $k_n\to+\infty$, below which the $Γ$-limit of the discrete energies capture chirality transitions while retaining an $\mathbb{S}^2$-valued energy description in the continuum limit.

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arXiv2024

Optimal control problems driven by nonlinear degenerate Fokker-Planck equations

The well-posedness of a class of optimal control problems is analysed, where the state equation couples a nonlinear degenerate Fokker-Planck equation with a system of Ordinary Differential Equations (ODEs). Such problems naturally arise as mean-field limits of Stochastic Differential models for multipopulation dynamics, where a large number of agents (followers) is steered through parsimonious intervention on a selected class of leaders. The proposed approach combines stability estimates for measure solutions of nonlinear degenerate Fokker-Planck equations with a general framework of assumptions on the cost functional, ensuring compactness and lower semicontinuity properties. The Lie structure of the state equations allows one for considering non-Lipschitz nonlinearities, provided some suitable dissipativity assumptions are considered in addition to non-Euclidean Hölder and sublinearity conditions.

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arXiv2024

Korn and Poincaré-Korn inequalities: A different perspective

We present a concise point of view on the first and the second Korn's inequality for general exponent $p$ and for a class of domains that includes Lipschitz domains. Our argument is conceptually very simple and, for $p = 2$, uses only the classical Riesz representation theorem in Hilbert spaces. Moreover, the argument for the general exponent $1<p<\infty$ remains the same, the only change being invoking now the $q$-Riesz representation theorem (with $q$ the harmonic conjugate of $p$). We also complement the analysis with elementary derivations of Poincaré-Korn inequalities in bounded and unbounded domains, which are essential tools in showing the coercivity of variational problems of elasticity but also propedeutic to the proof of the first Korn inequality.

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arXiv2024

A fractional approach to strain-gradient plasticity: beyond core-radius of discrete dislocations

We derive a strain-gradient theory for plasticity as the $Γ$-limit of discrete dislocation fractional energies, without the introduction of a core-radius. By using the finite horizon fractional gradient introduced by Bellido, Cueto, and Mora-Corral of 2023, we consider a nonlocal model of semi-discrete dislocations, in which the stored elastic energy is computed via the fractional gradient of order $1-α$. As $α$ goes to $0$, we show that suitably rescaled energies $Γ$-converge to the macroscopic strain-gradient model of Garroni, Leoni, and Ponsiglione of 2010.

math.AP↗
arXiv2024

Strong existence for free-discontinuity problems with non-standard growth

An Ahlfors-type regularity result for free-discontinuity energies defined on the space $SBV^φ$ of special functions of bounded variation with $φ$-growth, where $φ$ is a generalized Orlicz function, is proved. Our analysis expands on the regularity theory for minimizers of a class of free-discontinuity problems in the non-standard growth case.

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arXiv2024

Well-posedness of Kolmogorov-Fokker-Planck equations with unbounded drift

We consider Kolmogorov-Fokker-Planck equations with unbounded drift terms which are only measurable in time and locally Hölder continuous in space. In particular, we extend the parametrix method to this setting and we prove existence and uniqueness of measure solutions to the associated Cauchy problem, as well as the equivalence with the corresponding stochastic formulation.

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arXiv2023

Geometric rigidity for incompatible fields in the multi-well case and an application to strain-gradient plasticity

We derive a quantitative rigidity estimate for a multi-well problem in nonlinear elasticity with dislocations. Precisely, we show that the $L^{1^{*}}$-distance of a possibly incompatible strain field $β$ from a single well is controlled in terms of the $L^{1^{*}}$-distance from a finite set of wells, of ${\rm curl}β$, and of ${\rm div}β$. As a consequence, we derive a strain-gradient plasticity model as $Γ$-limit of a nonlinear finite dislocation model, containing a singular perturbation term accounting for the divergence of the strain field. This can also be seen as a generalization of the result of (Alicandro et al. 2018) to the case of incompatible vector fields.

math.AP↗
arXiv2023

Geometric rigidity on Sobolev spaces with variable exponent and applications

We present extensions of rigidity estimates and of Korn's inequality to the setting of (mixed) variable exponents growth. The proof techniques, based on a classical covering argument, rely on the log-Hölder continuity of the exponent to get uniform regularity estimates on each cell of the cover, and on an extension result à la Nitsche in Sobolev spaces with variable exponents. As an application, by means of $Γ$-convergence we perform a passage from nonlinear to linearized elasticity under variable subquadratic energy growth far from the energy well.

math.AP↗
arXiv2023

Integral representation and $Γ$-convergence for free-discontinuity problems with $p(\cdot)$-growth

An integral representation result for free-discontinuity energies defined on the space $GSBV^{p(\cdot)}$ of generalized special functions of bounded variation with variable exponent is proved, under the assumption of log-Hölder continuity for the variable exponent $p(x)$. Our analysis is based on a variable exponent version of the global method for relaxation devised in Bouchittè, Fonseca, Leoni and Mascarenhas (2002) for a constant exponent. We prove $Γ$-convergence of sequences of energies of the same type, we identify the limit integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions.

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arXiv2023

Inertial Balanced Viscosity (IBV) solutions to infinite-dimensional rate-independent systems

A suitable notion of weak solution to infinite-dimensional rate-independent systems, called Inertial Balanced Viscosity (IBV) solution, is introduced. The key feature of such notion is that the energy dissipated at jump discontinuities takes both into account inertial and viscous effects. Under a general set of assumptions it is shown that IBV solutions arise as vanishing inertia and viscosity limits of second order dynamic evolutions as well as of the corresponding time-incremental approximations. Relevant examples coming from applications, such as Allen-Cahn type evolutions and Kelvin-Voigt models in linearized elasticity, are considered.

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arXiv2023

Regularity of minimizers for free-discontinuity problems with $p(\cdot)$-growth

A regularity result for free-discontinuity energies defined on the space $SBV^{p(\cdot)}$ of special functions of bounded variation with variable exponent is proved, under the assumption of a log-Hölder continuity for the variable exponent $p(x)$. Our analysis expand on the regularity theory for minimizers of a class of free-discontinuity problems in the nonstandard growth case. This may be seen as a follow-up of the paper Fusco, Mingione and Trombetti (2001), dealing with a constant exponent.

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