Extremals of the equation for the potential energy functional

To study the surfaces on the stability (or instability) it is necessary to obtain the expression of the first and second functional variation. This article presents the first part of the research of the functional of potential energy. We calculate the first variation of the potential energy functional and prove some consequences of them. They help to build the extreme surface of rotation. Let be an dimensional connected orientable manifold from the class 𝐶2. We consider a hypersurface ℳ = (𝑀, 𝑢), obtained by a 𝐶2 -immersion : → R𝑛+1. Let Ω ⊂ R𝑛+1 be a domain such that ℳ⊂ 𝜕Ω; Φ, Ψ :R𝑛+1 → → R - 𝐶2-smooth function. If the field of unit normals to the surface ℳ, then for any 𝐶2-smooth surfaces ℳ defined functional 𝑊(ℳ) = w ℳ Φ( ) 𝑑ℳ+ w Ω Ψ(𝑥) 𝑑𝑥, which we call the functional of potential energy. It is the main object of study. Theorem of the first variation of the functional. Theorem 3. If 𝑊(𝑡) = 𝑊(ℳ𝑡), then 𝑊′(0) = w ℳ (div(𝐷Φ( ))𝑇 - 𝑛𝐻Φ( ) + Ψ(𝑥))ℎ(𝑥) 𝑑ℳ, where ℎ(𝑥) ∈ 𝐶1 0 (ℳ). Theorem 4 is the the main theorem of this article. It obtained the equations of extremals of the functional of potential energy. Theorem 4. A surfaceℳof class 𝐶2 is extremal of functional of potential energy if and only if Σ︁𝑖=1 𝑘𝑖𝐺(𝐸𝑖,𝐸𝑖) = Ψ(𝑥). Corollary. If an extreme surfaceℳis a plane, then the function Ψ(𝑥) = 0. Theorem 5. If = 𝑥𝑛+1 and Φ( ) = Φ( 𝑛+1), then div(( 𝑛+1Φ′( 𝑛+1) - Φ( 𝑛+1))∇𝑓) = Ψ(𝑥) 𝑛+1.