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.