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

H\"older estimates in space-time for viscosity solutions of Hamilton-Jacobi equations

It is well-known that solutions to the basic problem in the calculus of variations may fail to be Lipschitz continuous when the Lagrangian depends on t. Similarly, for viscosity solutions to time-dependent Hamilton-Jacobi equations one cannot expect Lipschitz bounds to hold uniformly with respect to the regularity of coefficients. This phenomenon raises the question whether such solutions satisfy uniform estimates in some weaker norm. We will show that this is the case for a suitable H\"older norm, obtaining uniform estimates in (x,t) for solutions to first and second order Hamilton-Jacobi equations. Our results apply to degenerate parabolic equations and require superlinear growth at infinity, in the gradient variables, of the Hamiltonian. Proofs are based on comparison arguments and representation formulas for viscosity solutions, as well as weak reverse H\"older inequalities

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.Comment: 37 pages. Third version. Presentation of the commutation property changed. Proof of the main theorem made cleare

Bolza Problems with discontinuous Lagrangians and Lipschitz continuity of the value function, Preprint Dip. Matematica Pura ed Applicata, Univ. degli Studi di Padova

Abstract. We study the local Lipschitz–continuity of the value function v associated with a Bolza Problem in presence of a Lagrangian L(x, q), convex and uniformly superlinear in q, but only Borel–measurable in x. Under these assumptions, the associated integral functional is not lower semicontinuous with respect to the suitable topology which assures the existence of minimizers, so all results known in literature fail to apply. Yet, the Lipschitz regularity of v does not depend on the existence of minimizers. In fact, it is enough to control the derivatives of quasi–minimal curves, but the problem is non–trivial due to the general growth conditions assumed here on L(x, ·). We propose a new approach, based on suitable reparameterization arguments, to obtain suitable a priori estimates on the Lipschitz constants of quasi–minimizers. As a consequence of our analysis, we derive the Lipschitz–continuity of v and a compactness result for value functions associated with sequences of locally equi–bounded discontinuous Lagrangians. 1