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
Famous quotes containing the word definition:
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)