Thèse pour obtenir le grade de docteur de l'université de Tours
Quasilinear degenerate parabolic equations and Hamilton-Jacobi
equations:
geometrical equations and propagating fronts
Olivier Ley
Abstract:
In the first part, we study quasilinear degenerate parabolic
equations set
in $\R^N\times (0,T)$ like the mean curvature
equation for graphs. We use the
level-set approach to interpret
the time-evolution of the unbounded solutions as
a propagating
front in $\R^{N+1}.$ We prove that uniqueness is equivalent to
the
non-fattening of the front.
Existence of discontinuous viscosity solutions is obtained
from a $L^\infty$ local bound given by the level-set approach.
A spectacular
application is the existence of a unique continuous
viscosity solution for any convex
initial data.
Working directly on the equation, we get existence and uniqueness
results in the
one-dimensional case. By imposing some polynomial-type
growth restriction
on the initial data in $\R^N,$ we prove the
well-posedness of a large class of
equations among functions with
the same growth.
The second part concerns time-dependent Hamilton-Jacobi equations.
First, for
equations set in the whole space $\R^N,$ we establish
lower gradient bounds for the
solutions. We exploit them to obtain
regularity properties of the propagating fronts
associated by the
level-set approach. These bounds ensure the non-fattening but
we
show they are not sufficient to imply sharper regularity even
for semiconcave
functions.
Secondly, we consider these equations in a smooth bounded set with
Neumann
boundary conditions. Using the corresponding control problem
with reflection,
we show that the discontinuous uniqueness result
which holds for such equations
set in $\R^N$ is not true in this case.