On the stability of extremal surfaces for a certain area type functional

In this paper we study the some questions about stability of the surfaces, which are the extremals of the area type functional. Stability, means, is called positively or negatively determined of second variation of functional for all any sufficiently small deformations of surface. In the work we obtain of the formulas first and second variation of functional, the capacity feature of instability, the conditions of stability and instability in the terms of the structure of surface image under Gaussian map. We consider some examples by application our results for research of stability of rotation surfaces.