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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 righthand 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
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Daniel
20101220