On the Dirichlet problem for first order quasilinear equations on a manifold
Abstract: We study the Dirichlet problem for a first order quasilinear equation on a smooth manifold with boundary. Existence and uniqueness of a generalized entropy solution are established. The uniqueness is proved under some additional requirement on the field of coefficients. It is shown that generally the uniqueness fails. The non-uniqueness occurs because of presence of the characteristics not outgoing from the boundary (including closed ones). The existence is proved in general case. Moreover, we establish that among generalized entropy solutions laying in the ball ‖u‖∞ ≤ R there exist the unique maximal and minimal solutions. To prove our results, we use the kinetic formulation similar to one by C. Imbert and J. Vovelle.