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:
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)