1,697 research outputs found
On the (non) existence of viscosity solutions of multi--time Hamilton--Jacobi equations
We prove that the multi--time Hamilton--Jacobi equation in general cannot be
solved in the viscosity sense, in the non-convex setting, even when the
Hamiltonians are in involution.Comment: 15 page
Existence and regularity of strict critical subsolutions in the stationary ergodic setting
We prove that any continuous and convex stationary ergodic Hamiltonian admits critical subsolutions, which are strict outside the random Aubry set. They make up, in addition, a dense subset of all critical subsolutions with respect to a suitable metric. If the Hamiltonian is additionally assumed of Tonelli type, then there exist strict subsolutions of class C^1,1 in R^N. The proofs are based on the use of Lax–Oleinik semigroups and their regularizing properties in the stationary ergodic environment, as well as on a generalized notion of Aubry set
Existence and regularity of strict critical subsolutions in the stationary ergodic setting
We prove that any continuous and convex stationary ergodic Hamiltonian admits
critical subsolutions, which are strict outside the random Aubry set. They make
up, in addition, a dense subset of all critical subsolutions with respect to a
suitable metric. If the Hamiltonian is additionally assumed of Tonelli type,
then there exist strict subsolutions of class \CC^{1,1} in . The proofs
are based on the use of Lax--Oleinik semigroups and their regularizing
properties in the stationary ergodic environment, as well as on a generalized
notion of Aubry set
A REBO-potential-based model for graphene bending by -convergence
An atomistic to continuum model for a graphene sheet undergoing bending is
presented. Under the assumption that the atomic interactions are governed by a
harmonic approximation of the 2nd-generation Brenner REBO (reactive empirical
bond-order) potential, involving first, second and third nearest neighbors of
any given atom, we determine the variational limit of the energy functionals.
It turns out that the -limit depends on the linearized mean and
Gaussian curvatures. If some specific contributions in the atomic interaction
are neglected, the variational limit is non-local
An atomistic-based F\"oppl-von K\'arm\'an model for graphene
We deduce a non-linear continuum model of graphene for the case of finite
out-of-plane displacements and small in-plane deformations. On assuming that
the lattice interactions are governed by the Brenner's REBO potential of 2nd
generation and that self-stress is present, we introduce discrete strain
measures accounting for up-to-the-third neighbor interactions. The continuum
limit turns out to depend on an average (macroscopic) displacement field and a
relative shift displacement of the two Bravais lattices that give rise to the
hexagonal periodicity. On minimizing the energy with respect to the shift
variable, we formally determine a continuum model of F\"oppl-von K\'arm\'a
type, whose constitutive coefficients are given in terms of the atomistic
interactions.Comment: arXiv admin note: text overlap with arXiv:1701.0746
The Gaussian stiffness of graphene deduced from a continuum model based on Molecular Dynamics potentials
We consider a discrete model of a graphene sheet with atomic interactions governed by a harmonic approximation of the 2nd-generation Brenner potential that depends on bond lengths, bond angles, and two types of dihedral angles. A continuum limit is then deduced that fully describes the bending behavior. In particular, we deduce for the first time an analytical expression of the Gaussian stiffness, a scarcely investigated parameter ruling the rippling of graphene, for which contradictory values have been proposed in the literature. We disclose the atomic-scale sources of both bending and Gaussian stiffnesses and provide for them quantitative evaluations
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