On nonuniqueness of viscosity solutions
So-called viscosity solutions to the Dirichlet problem \(- a(x)u''+c(x)u'=f(x)\), \(u(-I)=\gamma_ -\), \(u(I)=\gamma_ +\) are studied when \(a(x)\geq 0\), \(c(x)\geq 0\). The authors establish a number of theorems, the simplest of which is the following: Let \(c(x)>0\) and \(x=0\) be the only point at which \(a(x)=0\); then (a) there exists a viscosity solution to the problem; (b) solutions to the problem are unique for every choice of data if and only if \(\int^ 1_{-1}{dx\over a(x)}=\infty\). More general boundary conditions are also considered. There is a comment on \textit{Ishii}'s example [see Commun. Pure Appl. Math. 42, No. 1, 15-45 (1989; Zbl 0645.35025)] of a special PDE.
scientific article; zbMATH DE number 94952