Thèse pour obtenir le grade de docteur de l'université de Tours

Olivier Ley

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.