Electronic Journal of Differential Equations,
Conf. 10 (2002), pp. 153-161.

Title: A selfadjoint hyperbolic boundary-value problem

Author: Nezam Iraniparast (Western Kentucky Univ., USA) 
 
Abstract: 
 We consider the eigenvalue wave equation
 $$u_{tt} - u_{ss} = \lambda pu,$$
 subject to $ u(s,0) = 0$,
 where $u\in\mathbb{R}$, is a function of $(s, t) \in \mathbb{R}^2$,
 with $t\ge 0$.
 In the characteristic triangle $T =\{(s,t):0\leq t\leq 1, t\leq s\leq 2-t\}$
 we impose a boundary condition along characteristics so that
 $$
 \alpha u(t,t)-\beta \frac{\partial u}{\partial n_1}(t,t) = \alpha u(1+t,1-t)
 +\beta\frac{\partial u}{\partial n_2}(1+t,1-t),\quad 0\leq t\leq1.
 $$
 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
 $p\equiv 1$ 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 $p\geq 0$ is an arbitrary
 continuous function in $T$.


Published February 28, 2003.
Math Subject Classifications: 35L05, 35L20, 35P99.
Key Words: Characteristics; eigenvalues;
 eigenfunctions; Green's function; Fredholm alternative.