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## 5.3.2 Viscosity solutions of PDEs

We are now ready to introduce the concept of a viscosity solution for PDEs. Consider a PDE of the form (5.30)

where is a continuous function. A viscosity subsolution of the PDE (5.34) is a continuous function such that (5.31)

As we know, this is equivalent to saying that at every we must have for every test function such that has a local minimum at . Similarly, is a viscosity supersolution of (5.34) if (5.32)

or, equivalently, at every we have for every function such that has a local maximum at . Finally, is a viscosity solution if it is both a viscosity subsolution and a viscosity supersolution.

The above definitions of a viscosity subsolution and supersolution impose conditions on only at points where , respectively , is nonempty. We know that the set of these points is dense in the domain of . At all points where is differentiable, the PDE must hold in the classical sense. If is Lipschitz, then by Rademacher's theorem it is differentiable almost everywhere. Note the lack of symmetry in the definition of a viscosity solution: the sign convention used when writing the PDE is important. In the above example, if we rewrite the PDE as , then it is easy to see that is no longer a viscosity solution.

The terminology viscosity solutions" is motivated by the fact that a viscosity solution of the PDE (5.34) can be obtained from smooth solutions to the family of PDEs (5.33)

(parameterized by ) in the limit as . The operator on the right-hand side of (5.37) denotes the Laplacian ( ); in fluid mechanics it appears in the PDE describing the motion of a viscous fluid. To understand the basic idea behind this convergence result, let for some . Consider a test function such that and has a local maximum at . Assume that (if not, approximate it by a function). If is close to for small , then has a local maximum at some near , implying that and . Since solves (5.37), this gives Taking the limit as , by continuity of we have , which means that is a supersolution of (5.34). The argument showing that is a subsolution is similar.     Next: 5.3.3 HJB equation and Up: 5.3 Viscosity solutions of Previous: 5.3.1 One-sided differentials   Contents   Index
Daniel 2010-12-20