On instability of extremals of potential energy functional

Abstract

The paper is devoted to studying the stability and instability of extremals of a potential energy functional. A particular case of this functional is the area type functionals. The potential energy functional is the sum of functionals of area type and of volume density of forces. The potential energy functional is constructed in such way in order to take into consideration the loads on the surface from outside and inside. The stability is defined as the sign-definiteness of the second variation. In this paper we prove the formulae for the first and second variations of the functional. We also prove that the extremal surface can be locally minimal and locally maximal depending on the sign of matrix $G$. Using the $G$-capacity and the second variation of the functional, we obtain the conditions for the instability of the extremals of the potential energy functional. This technique was developed in works by V.M. Miklyukov and V.A. Klyachin. For $G$-parabolic extremal surfaces we prove the degeneracy into the plane. This result is an analogue of the theorems by M. do Carmo and C.K. Peng. By an example of $n$-dmensional surfaces of revolution we demonstrate the formulae for the first and second variations of the functional. We also prove the criteria of stability and instability for $n$-dimensional surfaces of revolution. Similar extremal surfaces arise in applications, in physical problems (e.g. soap films, capillary surfaces, magnetic liquids in a gravitational field with a potential), and the properties of extreme surfaces are used in applied problems (e.g. modeling of awning coverings).