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


DOI: 10.1007/s00526-013-0627-3

Cite this article as:
Balogh, Z.M., Calogero, A. & Pini, R. Calc. Var. (2014) 49: 1379. doi:10.1007/s00526-013-0627-3


The purpose of this paper is to bring a new light on the state-dependent Hamilton–Jacobi equation and its connection with the Hopf–Lax formula in the framework of a Carnot group \((\mathbf G ,\circ ).\) The equation we shall consider is of the form
$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} u_{t}+ \Psi (X_{1}u, \ldots , X_{m}u)=0\qquad &{}(x,t)\in \mathbf G \times (0,\infty ) \\ {u}(x,0)=g(x)&{}x\in \mathbf G , \end{array} \right. \end{aligned}$$
where \(X_{1},\ldots , X_{m}\) are a basis of the first layer of the Lie algebra of the group \(\mathbf G ,\) and \(\Psi : \mathbb{R }^{m} \rightarrow \mathbb{R }\) 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
$$\begin{aligned} u(x,t) = \inf _{y\in \mathbf G }\left\{ t \Psi ^\mathbf{G }\left( \delta _{\frac{1}{t}}(y^{-1}\circ x)\right) + g(y) \right\} , \end{aligned}$$
where \(\Psi ^\mathbf{G }:\mathbf G \rightarrow \mathbb{R }\) is the \(\mathbf G \)-Legendre–Fenchel transform of \(\Psi ,\) defined by a control theoretical approach. We recover, as special cases, some known results like the classical Hopf–Lax formula in the Euclidean spaces \(\mathbb{R }^n,\) showing that \(\Psi ^{\mathbb{R }^n}\) is the Legendre–Fenchel transform \(\Psi ^*\) of \(\Psi ;\) moreover, we recover the result by Manfredi and Stroffolini in the Heisenberg group for particular Hamiltonian function \(\Psi .\) In this paper we follow an optimal control problem approach and we obtain several properties for the value functions \(u\) and \(\Psi ^\mathbf G .\)

Mathematics Subject Classification (2000)

Primary 35R03  Secondary 49L25  26B25 22E30 53C17 

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of BernBernSwitzerland
  2. 2.Dipartimento di Matematica e ApplicazioniUniversità degli Studi di Milano-BicoccaMilanItaly

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