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.