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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.

##### MSC:
 35L35 Initial-boundary value problems for higher-order hyperbolic equations
##### Keywords:
representation formulae