arXiv subjects

Andrea Davini

Publications and source records attributed to Andrea Davini.

23 records · Page 2Linked to original sources

arXiv2014

Aubry sets for weakly coupled systems of Hamilton--Jacobi equations

We introduce a notion of Aubry set for weakly coupled systems of Hamilton--Jacobi equations on the torus and characterize it as the region where the obstruction to the existence of globally strict critical subsolutions concentrates. As in the case of a single equation, we prove the existence of critical subsolutions which are strict and smooth outside the Aubry set. This allows us to derive in a simple way a comparison result among critical sub and supersolutions with respect to their boundary data on the Aubry set, showing in particular that the latter is a uniqueness set for the critical system. We also highlight some rigidity phenomena taking place on the Aubry set.

math.AP↗

arXiv2012

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 $\R^N$. 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.

math.AP↗

arXiv2012

Weak KAM theoretic aspects for nonregular commuting Hamiltonians

In this paper we consider the notion of commutation for a pair of continuous and convex Hamiltonians, given in terms of commutation of their Lax- Oleinik semigroups. This is equivalent to the solvability of an associated multi- time Hamilton-Jacobi equation. We examine the weak KAM theoretic aspects of the commutation property and show that the two Hamiltonians have the same weak KAM solutions and the same Aubry set, thus generalizing a result recently obtained by the second author for Tonelli Hamiltonians. We make a further step by proving that the Hamiltonians admit a common critical subsolution, strict outside their Aubry set. This subsolution can be taken of class C^{1,1} in the Tonelli case. To prove our main results in full generality, it is crucial to establish suitable differentiability properties of the critical subsolutions on the Aubry set. These latter results are new in the purely continuous case and of independent interest.

math.AP↗

arXiv2009

A metric analysis of critical Hamilton--Jacobi equations in the stationary ergodic setting

We adapt the metric approach to the study of stationary ergodic Hamilton-Jacobi equations, for which a notion of admissible random (sub)solution is defined. For any level of the Hamiltonian greater than or equal to a distinguished critical value, we define an intrinsic random semidistance and prove that an asymptotic norm does exist. Taking as source region a suitable class of closed random sets, we show that the Lax formula provides admissible subsolutions. This enables us to relate the degeneracies of the critical stable norm to the existence/nonexistence of exact or approximate critical admissible solutions.

math.AP↗

arXiv2009

Weak KAM Theory topics in the stationary ergodic setting

We perform a qualitative analysis of the critical equation associated with a stationary ergodic Hamiltonian through a stochastic version of the metric method, where the notion of closed random stationary set, issued from stochastic geometry, plays a major role. Our purpose is to give an appropriate notion of random Aubry set, to single out characterizing conditions for the existence of exact or approximate correctors, and write down representation formulae for them. For the last task, we make use of a Lax--type formula, adapted to the stochastic environment. This material can be regarded as a first step of a long--term project to develop a random analog of Weak KAM Theory, generalizing what done in the periodic case or, more generally, when the underlying space is a compact manifold.

math.AP↗