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

Pierpaolo Soravia

doi:10.3934/mcrf.2012.2.399

Article Contents

2012, Volume 2, Issue 4: 399-427. Doi: 10.3934/mcrf.2012.2.399

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

Pierpaolo Soravia^1,

1.
Dipartimento di Matematica, Università di Padova, via Trieste 63, 35121 Padova

Received: July 31, 2011

Revised: July 31, 2012

Published: November 2012

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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.

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Mathematics Subject Classification: Primary: 35J70, 35J20; Secondary: 35D40, 49J10, 49N90.

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