Basic Properties
The three basic properties of viscosity solutions are existence, uniqueness and stability.
- The uniqueness of solutions requires some extra structural assumptions on the equation. Yet it can be shown for a very large class of degenerate elliptic equations. It is a direct consequence of the comparison principle. Some simple examples where comparison principle holds are
- with H uniformly continuous in x.
- (Uniformly elliptic case) so that is Lipschitz with respect to all variableas and for every and, for some .
- The existence of solutions holds in all cases where the comparison principle holds and the boundary conditions can be enforced in some way (through barrier functions in the case of a Dirichlet boundary condition). For first order equations, it can be obtained using the vanishing viscosity method or for most equations using Perron's method.
- The stability of solutions in holds as follows: a locally uniform limit of a sequence of solutions (or subsolutions, or supersolutions) is a solution (or subsolution, or supersolution).
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