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3.2. Elliptic Boundary Value Problems 109

condition (3.18), and the mixed boundary condition (3.19),
K∇u · ν =0 on Γ 1 ,
K∇u · ν + αu = g 2 on Γ 2 ,
where α := −ν · c.
Now the conditions of Theorem 3.15 can be checked:
1
1
We have r− ∇·c =˜r+ ∇·c; therefore, for the latter term the inequality
2 2
in (1) and (4)(b) must be satisfied. Further, the condition ν · c ≥ 0on
Γ 1 holds due to the characterization of the outflow boundary. Because of
1
1
α + ν · c = − ν · c, the condition (3) is satisfied due to the definition of
2 2
the inflow boundary.
Now we address the case of inhomogeneous Dirichlet boundary
conditions (|Γ 3 | d−1 > 0).
This situation can be reduced to the case of homogeneous Dirich-
let boundary conditions, if we are able to choose some (fixed) element
1
w ∈ H (Ω) in such a way that (in the sense of trace) we have
on Γ 3 . (3.34)
γ 0 (w)= g 3
The existence of such an element w is a necessary assumption for the exis-
1
tence of a solution ˜u ∈ H (Ω). On the other hand, such an element w can
exist only if g 3 belongs to the range of the mapping
1 2
H (Ω) v → γ 0 (v)| Γ 3 ∈ L (Γ 3 ).
2
However, this is not valid for all g 3 ∈ L (Γ 3 ), since the range of the trace
1
2
operator of H (Ω) is a proper subset of L (∂Ω).
Therefore, we assume the existence of such an element w. Since only
the homogeneity of the Dirichlet boundary conditions of the test functions
plays a role in derivation (3.31) of the bilinear form a and the linear form
b, we first obtain with the space V , defined in (3.30), and

1
1
˜
V := v ∈ H (Ω) : γ 0 (v)= g 3 on Γ 3 = v ∈ H (Ω) : v − w ∈ V
the following variational formulation:
˜
Find ˜u ∈ V such that
a(˜u, v)= b(v) for all v ∈ V.
However, this formulation does not fit into the theoretical concept of
˜
Section 3.1 since the space V is not a linear one.
If we put ˜u := u + w, then this is equivalent to the following:
Find u ∈ V such that
˜
a(u, v)= b(v) − a(w, v)=: b(v) for all v ∈ V. (3.35)
Now we have a variational formulation for the case of inhomogeneous
Dirichlet boundary conditions that has the form required in the theory.