5.5 Introduction to the Famous Scientists
363
5.20 Find the Euler equation and the boundary condition of the functional J =
1
2
˝
V (ε|∇ϕ|
2
− 2ρϕ)d V +
1
2
◦ S ε( f 1 ϕ
2
− 2 f 2 ϕ)d S, where, ε is a constant.
5.21 Let Γ be the fixed boundary of the domain D, find the Euler equation and the natural boundary condition of the functional J [u] =
˜
D (u
2
x + u
2
y + u x ϕ + u y ψ)d x d y, where, ϕ, ψ ∈ C
1
(D).
363
5.20 Find the Euler equation and the boundary condition of the functional J =
1
2
˝
V (ε|∇ϕ|
2
− 2ρϕ)d V +
1
2
◦ S ε( f 1 ϕ
2
− 2 f 2 ϕ)d S, where, ε is a constant.
5.21 Let Γ be the fixed boundary of the domain D, find the Euler equation and the natural boundary condition of the functional J [u] =
˜
D (u
2
x + u
2
y + u x ϕ + u y ψ)d x d y, where, ϕ, ψ ∈ C
1
(D).