278 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 LaxâOleinik 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
Convergence of the solutions of the discounted equation: the discrete case
We derive a discrete version of the results of our previous work. If is a
compact metric space, a continuous cost function
and , the unique solution to the discrete
-discounted equation is the only function
such that We prove that there exists a unique constant
such that the family of is
bounded as and that for this , the family uniformly
converges to a function which then verifies The proofs make
use of Discrete Weak KAM theory. We also characterize in terms of Peierls
barrier and projected Mather measures.Comment: 15 page
A 2D metamaterial with auxetic out-of-plane behavior and non-auxetic in-plane behavior
Customarily, in-plane auxeticity and synclastic bending behavior (i.e.
out-of-plane auxeticity) are not independent, being the latter a manifestation
of the former. Basically, this is a feature of three-dimensional bodies. At
variance, two-dimensional bodies have more freedom to deform than
three-dimensional ones. Here, we exploit this peculiarity and propose a
two-dimensional honeycomb structure with out-of-plane auxetic behavior opposite
to the in-plane one. With a suitable choice of the lattice constitutive
parameters, in its continuum description such a structure can achieve the whole
range of values for the bending Poisson coefficient, while retaining a
membranal Poisson coefficient equal to 1. In particular, this structure can
reach the extreme values, and , of the bending Poisson coefficient.
Analytical calculations are supported by numerical simulations, showing the
accuracy of the continuum formulas in predicting the response of the discrete
structure
Convergence of the solutions of the discounted equation: the discrete case
We derive a discrete version of the results of Davini et al. (Convergence of the solutions of the discounted Hamilton-Jacobi equation. Invent Math, 2016). If M is a compact metric space, a continuous cost function and , the unique solution to the discrete -discounted equation is the only function such that We prove that there exists a unique constant such that the family of is bounded as and that for this , the family uniformly converges to a function which then verifies The proofs make use of Discrete Weak KAM theory. We also characterize in terms of Peierls barrier and projected Mather measures
Convergence of the solutions of the discounted equation
We consider a continuous coercive Hamiltonian on the cotangent bundle of
the compact connected manifold which is convex in the momentum. If
is the viscosity solution of the discounted equation
where is the critical
value, we prove that converges uniformly, as , to a
specific solution of the critical equation We characterize in terms of Peierls barrier and projected
Mather measures.Comment: 35 page
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