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 $$T$$.

MSC:

 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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