Existence of absolute minimizers for noncoercive Hamiltonians and viscosity solutions of the Aronsson equation

Abstract

In this paper we study absolute minimizers and the Aronsson equation for a noncoercive Hamiltonian. We extend the definition of absolutely minimizing functions (in a viscosity sense) for the minimization of the $L^\infty$ norm of a Hamiltonian, within a class of locally Lipschitz continuous functions with respect to possibly noneuclidian metrics. The metric structure is naturally associated to the Hamiltonian and it is related to the a-priori regularity of the family of subsolutions of the Hamilton-Jacobi equation. A special but relevant case contained in our framework is that of Hamiltonians with a Carnot-Carathéodory metric structure determined by a family of vector fields (CC for short in the following), in particular the eikonal Hamiltonian and the corresponding anisotropic infinity-Laplace equation. In this case, the definition of absolute minimizer can be written in an almost classical way, by the theory of Sobolev spaces in a CC setting. In general open domains and with a prescribed continuous Dirichlet boundary condition, we prove the existence of an absolute minimizer which satisfies the Aronsson equation as a viscosity solution. The proof is based on Perron's method and relies on a-priori continuity estimates for absolute minimizers.