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DERIVATION OF THE ARONSSON EQUATION FOR C1 HAMILTONIANS
 

Summary: DERIVATION OF THE ARONSSON EQUATION FOR C1
HAMILTONIANS
MICHAEL G. CRANDALL, CHANGYOU WANG, YIFENG YU
Abstract. It is proved herein that any absolute minimizer u for a suitable Hamiltonian
H C1
(Rn
× R × U) is a viscosity solution of the Aronsson equation:
Hp(Du, u, x) · (H(Du, u, x))x = 0 in U.
The primary advance is to weaken the assumption that H C2
, used by previous authors,
to the natural condition that H C1
.
1. Introduction
Let U be an open subset of Rn
and H(p, z, x) C(Rn
× R × U). A function u : U R is
said to be an absolute minimizer for H in U if the following two conditions hold:
(i) u is locally Lipschitz continuous in U;
(ii) whenever V is an bounded open subset of U, ŻV U, v C(ŻV ) is locally Lipschitz
continuous in V and u|V = v|V , we have

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara
Wang, Changyou - Department of Mathematics, University of Kentucky

 

Collections: Mathematics