A selfadjoint hyperbolic boundary-value problem. (English) Zbl 1029.35156

Summary: 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 \(t\geq 0\). 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 \[ \alpha u(t,t)- \beta \frac{\partial u}{\partial n_1}(t,t)= \alpha u(1+t,1-t)+ \beta \frac{\partial u}{\partial n_2} (1+t,1-t), \quad 0\leq t\leq 1. \] 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 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 also investigate the case where \(p\geq 0\) is an arbitrary continuous function in \(T\).


35L20 Initial-boundary value problems for second-order hyperbolic equations
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35L05 Wave equation
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