Comparison principle for singular degenerate elliptic equations on unbounded domains
Moto-Hiko Sato · Proceedings of the Japan Academy Series A Mathematical Sciences · 1990
1. Introduction.This paper, a.s a preliminary study for [5], an- nounces a compa.risonprinciple for viscosity solutions of singular degen- erate elliptic equations ( 1 ) u+F(x, u, Vu, Vu)=O in t2 (Iru=grad u, Vu Hessian) where 9 is a domain (not necessarily bounded) in R .A typical exa.mple is (2) u-,u, div ( lr[u)--0 (eR).This equa.tion is derived from the mea.n curva.tureflow equation (2') v--Igvldiv I1 =0 by sein (t, )=e() see also [1, ].The idea. of he proo applies to pa.rabolie equation in [g], so we om.i he deailed proof since i is easily seen from he argumen in [g].The comparison prineiple for viseosity solutions is established by .G. Crandall and P.L. ions [8] for firs order equa.tions,by P.L. ions [1], N. Jensen [10], H. Ishii [7] or second order degenerate elliptic equa.ions(see also [11]), by Y.-G.Chert, Y. Giga.and S. Goo [1] for singular pa.rabolie equations including he mean eurva.ureflow euaions (see a.lso [4]).However so far no resus applied for (2) in an unbounded domain.Z. Comparison principle.et D be a domain in R no necessarily bounded.We consider a degenerate elliptic equation of orm(3) u+F(, u, gu, gu)=O in 9.In this paper we call a continuous function m" [0, )[0, ) a modul if m(0)=0 a.nd it is nondecreasing.We first list a.ssumptions on F= F(x, r, p, X).(F1) F" J(9)=9XRX(Rk{O})xSR is continuous, where S de- notes the spa.ce of real n X n symmetric ma.trices.(F2) F is degenerate elliptic, i.e., F(x, r, p, X+ Y)F(x, r, p, X) in J(9) f YO.(F3) -