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0.5. Boundary and Initial Value Problems 15

specify a particular situation where a unique solution is expected. Boundary
conditions are specifications on ∂Ω, where ν denotes the outer unit normal
• of the normal component of the flux (inwards):
(0.36)
− (C(u) − K(∇u)) · ν = g 1 on Γ 1
(flux boundary condition),
• of a linear combination of the normal flux and the unknown itself:
− (C(u) − K(∇u)) · ν + αu = g 2 on Γ 2 (0.37)
(mixed boundary condition),
• of the unknown itself:

u = g 3 on Γ 3 (0.38)
(Dirichlet boundary condition).
Here Γ 1 , Γ 2 , Γ 3 form a disjoint decomposition of ∂Ω:
∂Ω= Γ 1 ∪ Γ 2 ∪ Γ 3 , (0.39)
where Γ 3 is supposed to be a closed subset of ∂Ω. The inhomogeneities
g i and the factor α in general depend on x ∈ Ω, and for nonstationary
problems (where S(u) =0 holds) on t ∈ (0,T ). The boundary conditions
are linear if the g i do not depend (nonlinearly) on u (see below). If the g i
are zero, we speak of homogeneous,otherwise of inhomogeneous, boundary
conditions.
Thus the pointwise formulation of a nonstationary equation (where S
does not vanish) requires the validity of the equation in the space-time
cylinder

Q T := Ω × (0,T )
and the boundary conditions on the lateral surface of the space-time
cylinder

S T := ∂Ω × (0,T ) .
Different types of boundary conditions are possible with decompositions
of the type (0.39). Additionally, an initial condition on the bottom of the
space-time cylinder is necessary:
S(u(x, 0)) = S 0 (x) for x ∈ Ω . (0.40)
These are so-called initial-boundary value problems; for stationary prob-
lems we speak of boundary value problems. As shown in (0.34) and (0.35)
flux boundary conditions have a natural relationship with the differential
equation (0.33). For a linear diffusive part K(∇u)= K∇u alternatively
we may require

u := K∇u · ν = g 1 on Γ 1 , (0.41)
∂ ν K