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Relaxation of Hamilton-Jacobi Equations

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Abstract

We study the relaxation of Hamilton-Jacobi equations. The relaxation in our terminology is the following phenomenon: the pointwise supremum over a certain collection of subsolutions, in the almost everywhere sense, of a Hamilton-Jacobi equation yields a viscosity solution of the ``convexified'' Hamilton-Jacobi equation. This phenomenon has recently been observed in [13] in eikonal equations. We show in this paper that this relaxation is a common phenomenon for a wide range of Hamilton-Jacobi equations.

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Authors and Affiliations

  1. Department of Mathematics School of Education, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo, 169-8050, Japan

    Hitoshi Ishii

  2. Dipartimentodi Metodi e Modelli Matematici per le Scienze Applicate, Università degli Studi di Roma ``La Sapienza'', Via Scarpa n, 16, 00161, Roma, Italy

    Paola Loreti

Authors
  1. Hitoshi Ishii
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  2. Paola Loreti
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Correspondence to Hitoshi Ishii.

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Communicated by P.-L. Lions

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Ishii, H., Loreti, P. Relaxation of Hamilton-Jacobi Equations. Arch. Rational Mech. Anal. 169, 265–304 (2003). https://doi.org/10.1007/s00205-003-0268-3

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  • DOI: https://doi.org/10.1007/s00205-003-0268-3

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