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Eigenvalues of the Laplace–Beltrami Operator on the Surface of a Torus 227
Find u ∈H 1 p(Ω 0 )andλ, such that
ρ ∂u ∂v
R + ρ cos θ ∂φ ∂φ + R + ρ cos θ ]
∂u ∂v
dφdθ
ρ ∂θ ∂θ
∫
= λ ρ(R + ρ cos θ)uvdφdθ, (2)
Ω 0
for all v ∈Hp(Ω 1 0 ), with Ω 0 =(0, 2π) × (0, 2π) and with
∫Ω 0
[
Hp(Ω 1 0 )={v | v ∈H 1 (Ω 0 ), v(0,θ)=v(2π, θ), for a.e. θ ∈ (0, 2π),
v(φ, 0) = v(φ, 2π), for a.e. φ ∈ (0, 2π)},
i.e., H 1 p(Ω 0 ) is a space of doubly periodic functions. In the following, keep in
mind that 0