Page 124 -

3.2. Elliptic Boundary Value Problems 107

In order to ensure the V -ellipticity of a we need, besides the obvious
conditions
1
ν · c ≥ 0 on Γ 1 and α + ν · c ≥ 0 on Γ 2 , (3.32)
2
the following corollary from Theorem 3.13.
d ˜
Corollary 3.14 Suppose Ω ⊂ R is a bounded Lipschitz domain and Γ ⊂
∂Ω has a positive (d − 1)-dimensional measure. Then there exists some
1
constant C F > 0 such that for all v ∈ H (Ω) with v| ˜ Γ =0,
1/2
2
|∇v| dx = C F |v| 1 .
v 0 ≤ C F
Ω
This corollary yields the same results as in the case of homogeneous
Dirichlet boundary conditions on the whole of ∂Ω.
If |Γ 3 | d−1 =0, then by tightening conditions (3.32) for c and α,the
application of Theorem 3.13 as done in (II) may be successful.

Summary
We will now present a summary of our considerations for the case of
homogeneous Dirichlet boundary conditions.
d
Theorem 3.15 Suppose Ω ⊂ R is a bounded Lipschitz domain. Under the
assumptions (3.15), (3.16), (3.22) with g 3 =0, the boundary value problem
(3.12), (3.18)–(3.20) has one and only one weak solution u ∈ V ,if
1
(1) r − ∇· c ≥ 0 in Ω .
2
(2) ν · c ≥ 0 on Γ 1 .
1
(3) α + ν · c ≥ 0 on Γ 2 .
2
(4) Additionally, one of the following conditions is satisfied:
(a) |Γ 3 | d−1 > 0 .
˜
˜
(b) There exists some Ω ⊂ Ω with |Ω| d > 0 and r 0 > 0 such that
˜
1
r − ∇· c ≥ r 0 on Ω.
2
˜
˜
(c) There exists some Γ 1 ⊂ Γ 1 with |Γ 1 | d−1 > 0 and c 0 > 0 such
˜
that ν · c ≥ c 0 on Γ 1 .
˜
˜
(d) There exists some Γ 2 ⊂ Γ 2 with |Γ 2 | d−1 > 0 and α 0 > 0 such
˜
1
that α + ν · c ≥ α 0 on Γ 2 .
2
Remark 3.16 We point out that by using different techniques in the proof,
it is possible to weaken conditions (4)(b)–(d) in such a way that only the
following has to be assumed:

1
(b)
x ∈ Ω: r − ∇· c> 0
> 0 ,
2 d

(c)
{x ∈ Γ 1 : ν · c> 0}
> 0 ,
d−1

1
(d)
x ∈ Γ 2 : α + ν · c> 0
> 0 .
2 d−1