Approximation of a Multivalued Solution of the Hamilton–Jacobi Equation

E. A. Kolpakova<sup>ab

a</sup> Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
^b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg

Abstract: The paper deals with the construction of a multivalued solution of the Cauchy problem for the Hamilton–Jacobi equation with discontinuous Hamiltonian with respect to the phase variable. The constructed multivalued solution is approximated by a sequence of continuous solutions of auxiliary Cauchy problems of the Hamilton–Jacobi equation with Hamiltonian which is Lipschitz with respect to the phase variable. The results of the study are illustrated by an example.

Keywords: Hamilton–Jacobi equation, multivalued solution, minimax/viscosity solution, viable set.

UDC: 517.951

Received: 25.06.2019
Revised: 11.12.2019

DOI: 10.4213/mzm12492

 English version: Mathematical Notes, 2020, 108:1, 77–86

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