Some maximum properties for a family of singular hyperbolic operators (Q1063852)

This paper concerns the maximum properties of the operator \(L_{p,q,c}\) defined as follows: \[ L_{p,q,c}u\equiv u_{xx}-h^ 2(x)u_{tt}+ph'(x)u_ t+q\frac{h'(x)}{h(x)}u_ x+c(x,t)u, \] where p and q are two real parameters while c(x,t) and h(x) are two known functions. A segment AB of the t-axis and the characteristic curves \(\Gamma_ 1\) and \(\Gamma_ 2\) which pass through A and B, respectively, define a domain E. One supposes that \(t_ B>t_ A\), the slope of \(\Gamma_ 1\) is positive while that of \(\Gamma_ 2\) is negative, and x- coordinate on \(\Gamma_{1,2}\) is always positive. If certain conditions on u in E imply \(\max_{AB} u=\max_{\bar E}u\), then one says that the operator \(L_{p,q,c}\) has a maximum property. The author considers three types of such properties and gives some necessary and sufficient conditions related to these properties. The main results are stated in three theorems.