This thesis is divided into three parts. In the first part, we study the boundary value problem with measures for the Hamilton-Jacobi equation (E1) $-\Delta u+g(|\nabla u|)=0$ in a bounded domain $\Omega$ in ${\mathbb R}^N$, satisfying (E2) $u = \mu$ on $\partial \Omega$ and provide a condition on $g$ for which the problem (E1)-(E2) can be solved with any positive bounded measure. When $g(r) \geq r^q$ with $q>1$, we prove that any positive solution of (E1) admits a boundary trace which is an outer regular Borel measure, not necessarily bounded. When $g(r)=r^q$ with $1
