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Extremal properties of solutions of certain hyperbolic equation of third order in trivariate space with applications. (English. Russian original) Zbl 1005.35057
Russ. Math. 43, No. 4, 26-29 (1999); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1999, No. 4, 28-31 (1999).
It is considered an equation of third order in trivariate space \(U_{xyz}=0\). The following notations are used: \[ G^+=\bigl\{(x,y,z): 0<x-y<z<h,\;h>0\bigr\}, \quad G^-=\bigl\{(x,y,z): 0<z<x-y<h\bigr\}, \] \[ H^+=\{(x,y,z): 0<x< z<h, 0<x<y<+ \infty\}, \] \[ H^-=\{(x,y,z): 0<x<z<h, 0<y<x<h\}. \] Using unique solutions of certain boundary value problems in the domains \(G^-\), \(G^+\), \(H^-\), \(H^+\) and \(G^-\cup G^+\), extremal properties as well as representation formulae of solutions are proved.

35L35 Initial-boundary value problems for higher-order hyperbolic equations