1,720,990 research outputs found
Weak KAM theory for nonregular commuting Hamiltonians
No full textIn 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,C-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
On the relaxation of a class of functionals defined on Riemannian distances
No full textIn this paper we study the relaxation of a class of functionals defined on distances induced by isotropic Riemannian metrics on an open subset of R-N. We prove that isotropic Riemannian metrics are dense in Finsler ones and we show that the relaxed functionals admit a specific integral representation
Bolza Problems with discontinuous Lagrangians and Lipschitz continuity of the value function
No full text
SMOOTH APPROXIMATION OF WEAK FINSLER METRICS
No full textSmooth Finsler metrics are a natural generalization of Riemannian ones and have been widely studied in the framework of differential geometry. The definition can be weakened by allowing the metric to be only Borel measurable. This generalization is necessary in view of applications, such as, for instance, optimization problems. In this paper we show that smooth Finsler metrics are dense in Borel ones, generalizing the results obtained in [15]. The case of degenerate Finsler distances is also discussed
On calibrations for Lawson's cones
No full textIn this paper a calibration method is recalled and applied to Lawson's cones to prove their minimality. The original proof of Bombieri, De Giorgi and Giusti is reinterpreted and made simpler
Existence and uniqueness of solutions to parabolic equations with superlinear Hamiltonians
Full text linkWe give a proof of existence and uniqueness of viscosity solutions to parabolic quasi- linear equations for a fairly general class of nonconvex Hamiltonians with superlinear growth in the gradient variable. The approach is mainly based on classical techniques for uniformly parabolic quasilinear equations and on the Lipschitz estimates provided in [S. N. Armstrong and H. V. Tran, Viscosity solutions of general viscous Hamilton–Jacobi equations, Math. Ann. 361 (2015) 647–687], as well as on viscosity solution arguments
A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations
No full textWe consider the Hamilton–Jacobi equation ?_t u + H(x, Du) = 0 in (0, +?) × T^N , where T^N is the flat N -dimensional torus, and the Hamiltonian H(x, p) is assumed continuous in x and strictly convex and coercive in p. We study the large time behavior of solutions, and we identify the limit through a Lax-type formula. Some convergence results are also given for H solely convex. Our qualitative method is based on the analysis of the dynamical properties of the Aubry set, performed in the spirit of [A. Fathi and A. Siconolfi, Calc. Var. Partial Differential Equations, 22 (2005), pp. 185–228]. This can be viewed as a generalization of the techniques used in [A. Fathi, C. R. Acad. Sci. Paris Ser. I Math., 327 (1998), pp. 267–270] and [J. M. Roquejoffre, J. Math. Pures Appl. (9), 80 (2001), pp. 85–104]. Analogous results have been obtained in [G. Barles and P. E. Souganidis, SIAM J. Math. Anal., 31 (2000), pp. 925–939] using PDE methods
Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case
No full textWe perform a qualitative investigation of critical Hamilton-Jacobi equations, with stationary ergodic Hamiltonian, in dimension 1. We show the existence of approximate correctors, give characterizing conditions for the existence of correctors, provide Lax-type representation formulae and establish comparison principles. The results are applied to look into the corresponding effective Hamiltonian and to study a homogenization problem. In the analysis a crucial role is played by tools from stochastic geometry such as, for instance, closed random stationary sets
On the vanishing discount problem from the negative direction
Full text linkIt has been proved in [10] that the unique viscosity solution of
\begin{equation}\label{abs}\tag{*} \lambda u_\lambda+H(x,d_x
u_\lambda)=c(H)\qquad\hbox{in }, \end{equation} uniformly converges, for
, to a specific solution of the critical equation
where is a closed and connected
Riemannian manifold and is the critical value. In this note, we consider
the same problem for . In this case, viscosity
solutions of equation \eqref{abs} are not unique, in general, so we focus on
the asymptotics of the minimal solution of \eqref{abs}. Under the
assumption that constant functions are subsolutions of the critical equation,
we prove that the also converges to as . Furthermore, we exhibit an example of for which equation \eqref{abs}
admits a unique solution for as well.Comment: 14 page
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