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3.2. Elliptic Boundary Value Problems 105

As in (I), according to (3.11),

a(u, v):= {K∇u ·∇v + c ·∇uv + ruv} dx + αuv dσ (3.28)
Ω ∂Ω

= b(v):= fv dx + g 2vdσ for all v ∈ V.
Ω ∂Ω
Under the assumptions (3.15), (3.22) the continuity of b and a, respec-
tively, ((3.24) and (3.25)) can easily be shown. The additional new terms
can be estimated, for instance under the assumptions (3.15), (3.22), by
the Cauchy–Schwarz inequality and the Trace Theorem (Theorem 3.4) as
follows:

for all v ∈ V
g 2 vdσ ≤ g 2 0,∂Ω v| ∂Ω 0,∂Ω ≤ C g 2 0,∂Ω v 1

∂Ω
and

2
αuv dσ ≤ α ∞,∂Ω u| ∂Ω 0,∂Ω v| ∂Ω 0,∂Ω ≤ C α ∞,∂Ω u 1 v 1 ,

∂Ω
respectively, for all u, v ∈ V, where C> 0 denotes the constant appearing
in the Trace Theorem.
Conditions for the V -Ellipticity of a
For the proof of the V -ellipticity we proceed similarly to (I)(B), but now
taking into account the mixed boundary conditions. For the convective
term we have

1 2 1 2 1 2
c ·∇uudx = c ·∇u dx = − ∇· cu dx + ν · cu dσ ,
Ω 2 Ω 2 Ω 2 ∂Ω
and thus

1 2 1 2
a(u, u)= K∇u ·∇u + r − ∇· c u dx+ α + ν · c u dσ.
2 2
Ω ∂Ω
1
This shows that α + ν · c ≥ 0on ∂Ω should additionally be assumed. If
2
r 0 > 0 in (3.17), then the V -ellipticity of a follows directly. However, if only
r 0 ≥ 0 is valid, then the so-called Friedrichs’ inequality, a refined version
of the Poincar´e inequality, helps (see [25, Theorem 1.9]).
d
Theorem 3.13 Suppose Ω ⊂ R is a bounded Lipschitz domain and let
˜
the set Γ ⊂ ∂Ω have a positive (d − 1)-dimensional measure. Then there
1
exists some constant C F > 0 such that for all v ∈ H (Ω),
1/2
2
2
v dσ + |∇v| dx . (3.29)
v 1 ≤ C F
˜ Γ Ω
˜
˜
1
If α + ν · c ≥ α 0 > 0for x ∈ Γ ⊂ Γ 2 and Γ has a positive (d − 1)-
2
dimensional measure, then r 0 ≥ 0 is already sufficient for the V -ellipticity.