- Preprints

## Browse

### Recent Submissions

Item Homogenisation of vectorial free-discontinuity functionals with cohesive type surface terms(2024-09-12) Dal Maso, Gianni; Donati, DavideShow more The results on Γ-limits of sequences of free-discontinuity functionals with bounded cohesive surface terms are extended to the case of vector-valued functions. In this framework, we prove an integral representation result for the Γ-limit, which is then used to study deterministic and stochastic homogenisation problems for this type of functionals.Show more Item On the Symmetry TFT of Yang–Mills–Chern–Simons theory(2024-04-04) Riccardo Argurio, Francesco Benini, Matteo Bertolini, Giovanni Galati, Pierluigi NiroShow more Three-dimensional Yang–Mills–Chern–Simons theory has the peculiar property that its one-form symmetry defects have non-trivial braiding, namely they are charged under the same symmetry they generate, which is then anomalous. This poses a few puzzles in describing the corresponding Symmetry TFT in a four-dimensional bulk. First, the braiding between lines at the boundary seems to be ill-defined when such lines are pulled into the bulk. Second, the Symmetry TFT appears to be too trivial to allow for topological boundary conditions encoding all the different global variants. We show that both of these puzzles can be solved by including endable (tubular) surfaces in the lass of bulk topological operators one has to consider. In this way, we are able to reproduce all global variants of the theory, with their symmetries and their anomalies. We check the validity ofour proposal also against a top-down holographic realization of the same class of theories.Show more Item A closure theorem for gAMMA-convergence and H-convergence with applications to non-periodic homogenization(2024-02-29) Braides, Andrea; Dal Maso, Gianni; Le Bris, ClaudeShow more In this work we examine the stability of some classes of integrals, and in particular with respect to homogenization. The prototypical case is the homogenization of quadratic energies with periodic coe cients perturbed by a term vanishing at in nity, which has been recently examined in the framework of elliptic PDE.We use localization techniques and higher-integrability Meyers-type results to provide a closure theorem by gamma-convergence within a large class of integral functionals. From such result we derive stability theorems in homogenization which comprise the case of perturbations with zero average on the whole space. The results are also extended to the stochastic case, and specialized to the G-convergence of operators corresponding to quadratic forms. A corresponding analysis is also carried on for non-symmetric operators using the localization properties of H-convergence. Finally, we treat the case of perforated domains with Neumann boundary condition, and their stability.Show more Item EXISTENCE AND BLOW-UP FOR NON-AUTONOMOUS SCALAR CONSERVATION LAWS WITH VISCOSITY(2023-11-23) Bianchini, Stefano; Leccese, Giacomo MariaShow more We consider a question posed in [1], namely the blow-up of the PDE ut + (b(t, x)u1+k)x = uxx when b is uniformly bounded, Lipschitz and k = 2. We give a complete answer to the behavior of solutions when b belongs to the Lorentz spaces b ∈ Lp,∞, p ∈ (2,∞], or bx ∈ Lp,∞, p ∈ (1,∞].Show more Item Homogenisation problems for free discontinuity functionals with bounded cohesive surface terms(2023-07-11) Dal Maso, Gianni; Toader, Rodica; mathematicsShow more We study stochastic homogenisation problems for free discontinuity func- tionals under a new assumption on the surface terms, motivated by cohesive fracture models. The results are obtained using a characterization of the limit functional by means of the asymptotic behaviour of suitable minimum problems on cubes with very simple boundary conditions. An important role is played by the subadditive ergodic theorem.Show more Item Discrete approximation of nonlocal-gradient energies(2023-01-22) Braides, Andrea; Causin, Andrea; Solci, Margherita; mathematicsShow more We study a discrete approximation of functionals depending on nonlocal gradients. The discretized functionals are proved to be coercive in classical Sobolev spaces. The key ingredient in the proof is a formulation in terms of circulant Toeplitz matrices.Show more Item Another look at elliptic homogenization(2023-06-21) Braides, Andrea; Cosma Brusca, Giuseppe; Donati, Davide; mathematicsShow more We consider the limit of sequences of normalized (s, 2)-Gagliardo seminorms with an oscillating coefficient as s → 1. In a seminal paper by Bourgain, Brezis and Mironescu (subsequently extended by Ponce) it is proven that if the coefficient is constant then this sequence Γ-converges to a multiple of the Dirichlet integral. Here we prove that, if we denote by ε the scale of the oscillations and we assume that 1−s << ε2, this sequence converges to the homogenized functional formally obtained by separating the effects of s and ε; that is, by the homogenization as ε → 0 of the Dirichlet integral with oscillating coefficient obtained by formally letting s → 1 first.Show more Item Validity and failure of the integral representation of Γ-limits of convex non-local functionals(2023-05-09) Braides, Andrea; Dal Maso, Gianni; mathematicsShow more We prove an integral-representation result for limits of non-local quadratic forms on H1 0 pΩq, with Ω a bounded open subset of Rd, extending the representation on C8c pΩq given by the Beurling-Deny formula in the theory of Dirichlet forms. We give a counterexample showing that a corresponding representation may not hold if we consider analogous functionals in W1,p0 pΩq, with p ‰ 2 and 1 ă p ď d.Show more Item Γ-CONVERGENCE AND INTEGRAL REPRESENTATION FOR A CLASS OF FREE DISCONTINUITY FUNCTIONALS(2023-05-05) Dal Maso, Gianni; Toader, Rodica; mathematicsShow more We study the Γ -limits of sequences of free discontinuity functionals with linear growth, assuming that the surface energy density is bounded. We determine the relevant properties of the Γ -limit, which lead to an integral representation result by means of integrands obtained by solving some auxiliary minimum problems on small cubes.Show more Item Parker Bound and Monopole Pair Production from Primordial Magnetic Fields(2023-04-11) Kobayashi, Takeshi; Perri, Daniele; physicsShow more We present new bounds on the cosmic abundance of magnetic monopoles based on the survival of primordial magnetic fields during the reheating and radiation-dominated epochs. The new bounds can be stronger than the conventional Parker bound from galactic magnetic fields, as well as bounds from direct searches. We also apply our bounds to monopoles produced by the primordial magnetic fields themselves through the Schwinger effect, and derive additional conditions for the survival of the primordial fields.Show more Item UNIVERSALITY IN THE 2D QUASI-PERIODIC ISING MODEL AND HARRIS-LUCK IRRELEVANCE(2023-04-04) Gallone, Matteo; Mastropietro, Vieri; mathematicsShow more We prove that in the 2d Ising Model with a weak bidimensional quasi-periodic disorder in the interaction, the critical behavior is the same as in the non-disordered case, that is the critical exponents are identical and no logarithmic corrections are present. The result establishes the validity of the prediction based on the Harris-Luck criterion and it provides the first rigorous proof of universality in the Ising model in presence of quasi-periodic disorder. The proof combines Renormalization Group approaches with direct methods used to deal with small divisors in KAM theory.Show more Item Gamma-convergence of quadratic functionals perturbed by bounded linear functionals(2022-12-14) Dal Maso, Gianni; Donati, Davide; mathematicsShow more We study the asymptotic behavior of solutions to elliptic equations of the form (div(Akruk) = fk in ;uk = wk on @; where Rn is a bounded open set, wk is weakly converging in H1(), fk is weakly converging in H1(), and Ak is a sequence square matrices satisfying some uniform ellipticity and boundedness conditions, and H-converging in . In particular, we characterize the weak limits of the solutions uk and of their momenta Akruk . When Ak is symmetric and wk = w = 0, we characterize the limits of the energies for the solutions.Show more Item Compactness for a class of integral functionals with interacting local and non-local terms(2022-12-20) Braides, Andrea; Dal Maso, Gianni; mathematicsShow more We prove a compactness result with respect to -convergence for a class of integral functionals which are expressed as a sum of a local and a non-local term. The main feature is that, under our hypotheses, the local part of the -limit depends on the interaction between the local and non-local terms of the converging subsequence. The result is applied to concentration and homogenization problems.Show more Item Asymptotic behaviour of the capacity in two-dimensional heterogeneous media(2022-06-13) Braides, Andrea; Brusca, G.C.; mathematicsShow more We describe the asymptotic behaviour of the minimal inhomogeneous two-capacity of small sets in the plane with respect to a fixed open set Ω. This problem is gov erned by two small parameters: ε, the size of the inclusion (which is not restrictive to assume to be a ball), and δ, the period of the inhomogeneity modelled by oscillating coefficients. We show that this capacity behaves as C| log ε| −1. The coefficient C is ex plicitly computed from the minimum of the oscillating coefficient and the determinant of the corresponding homogenized matrix, through a harmonic mean with a proportion depending on the asymptotic behaviour of | log δ|/| log ε|.Show more Item A note on the homogenization of incommensurate thin films(2022-12-21) Anello, Irene; Braides, Andrea; Caragiulo, Fabrizio; mathematicsShow more Dimension-reduction homogenization results for thin films have been obtained under hy potheses of periodicity or almost-periodicity of the energies in the directions of the mid-plane of the film. In this note we consider thin films, obtained as sections of a periodic medium with a mid-plane that may be incommensurate; that is, not containing periods other than oggi si 0. A geometric almost-periodicity argument similar to the cut-and-project argument used for quasicrystals allows to prove a general homogenization result.Show more Item Dissipative solutions to Hamiltonian systems(2022-09-12) Bianchini, Stefano; Leccese, Giacomo Maria; mathematicsShow more Item An example of a weakly mixing BV vector field which is not strongly mixing(2022-08-14) Zizza, Martina; mathematicsShow more We give an example of a weakly mixing vector field b ∈ L∞([0, 1], BV(T2)) which is not strongly mixing, in the setting first introduced in [3]. The example is based on a work of Chacon [4] who constructed a weakly mixing automorphism which is not strongly mixing on ([0, 1], B([0, 1]), | · |), where B([0, 1]) are the Borel subsets of [0, 1] and | · | is the one-dimensional Lebesgue measure.Show more Item Algebra of Operators in AdS-Rindler(2022-08-08) Bahiru, Eyoab D.; physicsShow more We discuss the algebra of operators in AdS-Rinlder wedge, particularly in AdS5/CFT4. We explicitly construct the algebra at N = 8 limit and discuss its Type III1 nature. We will consider 1/N corrections to the theory and describe how several divergences can be renormalized and the algebra becomes Type II8. This will make it possible to associate a density matrix to any state in the Hilbert space and thus a von Neumann entropy.Show more Item Continuity of some non-local functionals with respect to a convergence of the underlying measures(2022-04-04) Braides, Andrea; Dal Maso, Gianni; mathematicsShow more We study some non-local functionals on the Sobolev space W1,p0(Ω) involving a double integral on Ω × Ω with respect to a measure µ. We introduce a suitable notion of convergence of measures on product spaces which implies a stability property in the sense of Γ-convergence of the corresponding functionals.Show more Item Asymptotic behavior of the dirichlet energy on poisson point clouds(2022-03-23) Braides, Andrea; Caroccia, Marco; mathematicsShow more We prove that quadratic pair interactions for functions defined on planar Poisson clouds and taking into account pairs of sites of distance up to a certain (large-enough) threshold can be almost surely approximated by the multiple of the Dirichlet energy by a deterministic constant. This is achieved by scaling the Poisson cloud and the corresponding energies and computing a compact discrete-to-continuum limit. In order to avoid the effect of exceptional regions of the Poisson cloud, with an accumulation of sites or with ‘disconnected’ sites, a suitable ‘coarse-grained’ notion of convergence of functions defined on scaled Poisson clouds must be given.Show more