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3.2. Elliptic Boundary Value Problems 101

Finally, there exists a constant r 0 ≥ 0 such that
1
r(x) − ∇· c(x) ≥ r 0 for (almost) every x ∈ Ω . (3.17)
2
Boundary Conditions
As in Section 0.5, suppose Γ 1 , Γ 2 , Γ 3 is a disjoint decomposition of the
boundary ∂Ω (cf. (0.39)):

∂Ω= Γ 1 ∪ Γ 2 ∪ Γ 3 ,
where Γ 3 is a closed subset of the boundary. For given functions g j :Γ j →
R ,j =1, 2, 3, and α :Γ 2 → R we assume on ∂Ω

• Neumann boundary condition (cf. (0.41) or (0.36))
on Γ 1 , (3.18)
K∇u · ν = ∂ ν K u = g 1
• mixed boundary condition (cf. (0.37))

u + αu = g 2 on Γ 2 , (3.19)
K∇u · ν + αu = ∂ ν K
• Dirichlet boundary condition (cf. (0.38))

u = g 3 on Γ 3 . (3.20)
Concerning the boundary data the following is assumed: For the classical
approach we need

g j ∈ C(Γ j ) , j =1, 2, 3 , α ∈ C(Γ 2 ) , (3.21)
whereas for the variational interpretation,
2
∞
g j ∈ L (Γ j ) , j =1, 2, 3 , α ∈ L (Γ 2 ) (3.22)
is sufficient.

3.2.1 Variational Formulation of Special Cases
The basic strategy for the derivation of the variational formulation of
boundary value problems (3.12) has already been demonstrated in Sec-
tion 2.1. Assuming the existence of a classical solution of (3.12) the
following steps are performed in general:
Step 1: Multiplication of the differential equation by test functions that
are chosen compatible with the type of boundary condition and
subsequent integration over the domain Ω.
Step 2: Integration by parts under incorporation of the boundary condi-
tions in order to derive a suitable bilinear form.
Step 3: Verification of the required properties like ellipticity and continuity.