Viscosity supersolutions of the evolutionary
-Laplace Equation
Juan J. Manfredi
University of Pittsburgh
manfredi@pitt.edu
url: http://www.pitt.edu/~manfredi
Abstract: Consider the parabolic -Laplacian equation
We study the regularity of the viscosity supersolutions and their
spatial gradients. We give a new proof of the existence of in
Sobolev's sense and of the validity of the equation
for all test functions
. Here is the underlying
domain in
and is a bounded viscosity supersolution in
. The first step of our proof is to establish the above inequality
for the so-called infimal convolution
.
The function
has the advantage of being differentiable with
respect to all its variables
, and , while the
original is merely lower semicontinuous to begin with. The second step is
to pass to the limit as
. It is clear that
but it is delicate to establish a sufficiently
good convergence of the
's.
This is joint work with Peter Lindqvist at Trondheim (Norway).