A selfadjoint hyperbolic boundary-value problem

Abstract

We consider the eigenvalue wave equation $$u_{tt} - u_{ss} = lambda pu,$$ subject to $ u(s,0) = 0$, where $uinmathbb{R}$, is a function of $(s, t) in mathbb{R}^2$, with $tge 0$. In the characteristic triangle $T ={(s,t):0leq tleq 1, tleq sleq 2-t}$ we impose a boundary condition along characteristics so that $$ alpha u(t,t)-beta frac{partial u}{partial n_1}(t,t) = alpha u(1+t,1-t) +betafrac{partial u}{partial n_2}(1+t,1-t),quad 0leq tleq1. $$ The parameters $alpha$ and $beta$ are arbitrary except for the condition that they are not both zero. The two vectors $n_1$ and $n_2$ are the exterior unit normals to the characteristic boundaries and $frac{partial u}{partial n_1}$, $frac{partial u}{partial n_2}$ are the normal derivatives in those directions. When $pequiv 1$ we will show that the above characteristic boundary value problem has real, discrete eigenvalues and corresponding eigenfunctions that are complete and orthogonal in $L_2(T)$. We will also investigate the case where $pgeq 0$ is an arbitrary continuous function in $T$.

Keywords

• Characteristics
• eigenvalues
• eigenfunctions
• Green's function
• Fredholm alternative.