Showing papers by "Stanley Osher published in 1973"
TL;DR: In this article, the authors examined the properties of hyperbolic systems in regions with corners and obtained conditions equivalent to the validity of certain energy estimates for a general class of systems.
Abstract: . In the previous paper in this series we obtained conditions equivalent to the validity of certain energy estimates for a general class of hyperbolic systems in regions with corners. In this paper we examine closely the phenomena which occur near the corners if these conditions are violated. These phenomena include: the development of strong singularities (lack of existence), travelling waves which pass unnoticed through the corner (lack of uniqueness), existence and uniqueness if and only if additional conditions are imposed at the corner, and weak solutions which are not strong solutions. We also systematically analyze the conditions for certain important problems. We discuss the physical and computational significance of these results.
34 citations
TL;DR: In this paper, it was shown that the Green's function has no nodal lines for the annular region with inner radius Y and outer radius 1, and that the principal eigenfunction has a diametric nodal line if Y is small enough (y. < l/71 5).
18 citations
TL;DR: In this article, the authors gave necessary and sufficient conditions for existence, uniqueness, and the validity of certain energy estimates for the solutions of a general class of ill-posed hyperbolic equations.
Abstract: The purpose of this note is to give a simple example of an ill posed problem for a hyperbolic equation to be solved in a region whose boundary has a corner. In [2] we gave necessary and sufficient conditions for existence, uniqueness, and the validity of certain energy estimates for the solutions of a general class of these problems. Analogous conditions for problems in regions with smooth boundaries were obtained by Kreiss [1]. Our example below is somewhat unusual in that bounded C°° initial data lead to a solution which is exponentially unbounded at the corner for any positive time. Consider the equation
16 citations