Definition
There are several equivalent ways to phrase the definition of viscosity solutions. See for example the section II.4 of Fleming and Soner's book or the definition using semi-jets in the Users Guide.
An equation in a domain is defined to be degenerate elliptic if for any two symmetric matrices and such that is positive definite, and any values of, and, we have the inequality . For example is degenerate elliptic. Any first order equation is degenerate elliptic.
An upper semicontinuous function in is defined to be a subsolution of a degenerate elliptic equation in the viscosity sense if for any point and any function such that and in a neighborhood of, we have .
An lower semicontinuous function in is defined to be a supersolution of a degenerate elliptic equation in the viscosity sense if for any point and any function such that and in a neighborhood of, we have .
A continuous function u is a viscosity solution of the PDE if it is both a viscosity supersolution and a viscosity subsolution.
Read more about this topic: Viscosity Solution
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