Showing papers by "Stanley Osher published in 1974"

An ill-posed problem for a strictly hyperbolic equation in two unknowns near a corner

Abstract: In an earlier note [4] we gave a simple example of an ill-posed problem for a system of hyperbolic equations in a region whose boundary has a corner. The system was diagonal with coupling only at the boundary. Earlier we derived necessary and sufficient conditions for well-posedness [2] for a wide class of constant coefficient hyperbolic systems in such regions. In [3] we examined in some detail the phenomena which occur when these conditions are violated. The fundamental work for hyperbolic problems in regions with smooth boundaries was done by Kreiss [1]. It was pointed out by Sarason and Smoller [5] that the work of Strang [6] for the half-space problem implies that the corner problem is well posed for a strictly hyperbolic system in two unknowns iff the corresponding half-space problems are well posed. They constructed, using geometrical optics, a four dependent variable ill-posed example, where the half-space extensions were well posed. In all the above-mentioned work, the boundary conditions imposed were local, i.e., of the form Bu—f at ^ = 0 , where B is a matrix and u is the unknown vector on the boundary. We have noticed that much of the theory can be extended to nonlocal pseudo-differential boundary conditions. In particular, conditions of the form