The Hopf–Lax formula in Carnot groups: a control theoretic approach

Abstract

The purpose of this paper is to bring a new light on the state-dependent HamiltonJacobi equation and its connection with the Hopf-Lax formula in the framework of a Carnot group (G, •). The equation we shall consider is of the formwhere X 1 , . . . , X m are a basis of the first layer of the Lie algebra of the group G, and Ψ : R m → R is a superlinear, convex function. The main result shows that the unique viscosity solution of the Hamilton-Jacobi equation can be given by the Hopf-Lax formula u(x, t) = infwhere Ψ G : G → R is the G-Legendre-Fenchel transform of Ψ, defined by a control theoretical approach. We recover, as special cases some known results: the classical Hopf-Lax formula in the Euclidean spaces R n showing that Ψ R n is the Legendre-Fenchel trasnsform Ψ * of Ψ; moreover, we recover the result by Manfredi and Stroffolini in the Heisenberg group for particular Hamiltonian function Ψ. In this paper we follow an optimal control problem approach and we obtain several properties for the value functions u and Ψ G : in particular we prove a precise estimate for the horizontal gradient of the solution u, two existence results of the optimal control for the optimal problems and we show that Ψ G is convex.

Stochastic Homogenization for Functionals with Anisotropic Rescaling and Noncoercive Hamilton--Jacobi Equations

“…Following the same arguments of [23, Theorem 2, Section 10.3.3] and [8], we get that u ε fulfills u ε (0, x) = g(x) and it is a viscosity solution of…”

“…We only sketch the proof; for the detailed calculations we refer the reader to [23, Section 10.3.3] and to [8]. First of all we prove that u ε is a solution of (3.1).…”

“…We want to use Theorem 6.1 to derive the convergence of the viscosity solutions of problem (3.1) to the unique solution of the deterministic problem (1.3). Our strategy is to use the Hopf-Lax variational formula from [8].…”

“…This variational approach is motivated by Γ-convergence methods for the random Lagrangian. In order to define the associated variational problem from the Hamilton-Jacobi equation, some form of convexity is needed, but it should be noted that due to the degeneracy the relation is more subtle than the Euclidean Legendre transform, see [8].…”

Lipschitz continuity and convexity preserving for solutions of semilinear evolution equations in the Heisenberg group

“…We refer the reader to Theorem 4.4, which answers a question asked in [26]. A more general question on Lipschitz continuity of viscosity solutions was posed in [2], but it is not clear if our method here immediately applies to that general setting.…”