SISSA Preprints
http://hdl.handle.net/1963/35281
SISSA authors' not-referred manuscripts of the research outputSat, 23 Sep 2023 19:58:03 GMT2023-09-23T19:58:03ZHomogenisation problems for free discontinuity functionals with bounded cohesive surface terms
http://hdl.handle.net/1963/35462
Homogenisation problems for free discontinuity functionals with bounded cohesive surface terms
Dal Maso, Gianni; Toader, Rodica
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.
Tue, 11 Jul 2023 00:00:00 GMThttp://hdl.handle.net/1963/354622023-07-11T00:00:00ZDiscrete approximation of nonlocal-gradient energies
http://hdl.handle.net/1963/35461
Discrete approximation of nonlocal-gradient energies
Braides, Andrea; Causin, Andrea; Solci, Margherita
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.
SISSA 09/2023/MATE
Sun, 22 Jan 2023 00:00:00 GMThttp://hdl.handle.net/1963/354612023-01-22T00:00:00ZAnother look at elliptic homogenization
http://hdl.handle.net/1963/35460
Another look at elliptic homogenization
Braides, Andrea; Cosma Brusca, Giuseppe; Donati, Davide
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.
Wed, 21 Jun 2023 00:00:00 GMThttp://hdl.handle.net/1963/354602023-06-21T00:00:00ZValidity and failure of the integral representation of Γ-limits of convex non-local functionals
http://hdl.handle.net/1963/35459
Validity and failure of the integral representation of Γ-limits of convex non-local functionals
Braides, Andrea; Dal Maso, Gianni
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.
Preprint number SISSA/06/2023
Tue, 09 May 2023 00:00:00 GMThttp://hdl.handle.net/1963/354592023-05-09T00:00:00Z