On piecewise-linear almost-solutions of elliptic equations

In this paper we define deviation 𝜀𝑄(𝑓𝑁) of piecewise-linear almost-solution of the minimal surface equation 𝑄[𝑓(𝑥)] = 0 and we get a general formula to calculate it. Let 1, 2,..., 𝑘(𝑖) be tetrahedrons which have vertex 𝑃𝑖. We denote Γ𝑖 𝑗1, Γ𝑖 𝑗2,..., Γ𝑖 sides leaving from vertex of the tetrahedron 𝑗, = 1, 2,..., 𝑘(𝑖), and let be Γ𝑖 𝑗𝑛+1 the side of tetrahedron opposite to vertex 𝑃𝑖. We set by 𝑗1, 𝑗2,..., 𝑗𝑛, 𝑗𝑛+1 the external normal vectors of the sides Γ𝑖 𝑗1, Γ𝑖 𝑗2,..., Γ𝑖 relatively of 𝑗. As is linear function in then ∇𝑓𝑁 = ≡ const. Then the following equality holds 𝜀𝑄(𝑓𝑁) = 1 Σ︁ inner 𝑃𝑖 ⃒⃒⃒⃒⃒⃒ Σ︁𝑘(𝑖) 𝑗=1 𝑗, 𝑗𝑛+1⟩ √︁ 1 + |𝜉𝑖 |2 |Γ𝑖 𝑗𝑛+1| ⃒⃒⃒⃒⃒⃒, where Γ𝑖 𝑗𝑛+1 is exterior side relatively 𝑃𝑖. On the basis of this concept it obtained approximation equation 𝜀𝑄(𝑓𝑁) = 0 or = (𝜇𝑖𝑗,5 + 𝜇𝑖𝑗,6)𝑢𝑖+1𝑗 + (𝜇𝑖𝑗,2 + + 𝜇𝑖𝑗,3)𝑢𝑖−1𝑗 + (𝜇𝑖𝑗,1 + 𝜇𝑖𝑗,2)𝑢𝑖𝑗+1 + (𝜇𝑖𝑗,4 + 𝜇𝑖𝑗,5)𝑢𝑖𝑗−1, where 𝜇𝑖𝑗,𝑘 = 1 √︁ 1 + |𝜉𝑖𝑗 |2, = 𝜇𝑖𝑗,1 + 2𝜇𝑖𝑗,2 + 𝜇𝑖𝑗,3 + 𝜇𝑖𝑗,4 + 2𝜇𝑖𝑗,5 + 𝜇𝑖𝑗,6, 𝑖, = 1,...,𝑁 − 1 and proved that the deviation 𝜀𝑄(𝑢𝑁) converges to the integral of the modulus of the mean curvature of the graph of 𝐶2-smooth function 𝑢, that is lim 𝑁→∞ 𝜀𝑄(𝑢𝑁) = ∫︁∫︁ Ω |𝑄[𝑢(𝑥, 𝑦)]| 𝑑𝑥𝑑𝑦. Thus, the obtained system of nonlinear equations aproximate the minimal surface equation.