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3.2. Elliptic Boundary Value Problems 103

we continue to estimate as follows:

|a(u, v)|≤ C 1 {|∇u||∇v| + |u||v|} dx + C 2 |∇u||v| dx .
Ω Ω
! !
=:A 1 =:A 2
The integrand of the first addend is estimated by the Cauchy–Schwarz
2
2
inequality for R , and then the Cauchy–Schwarz inequality for L (Ω) is
applied:

1/2 1/2
2 2 2 2
A 1 ≤ C 1 |∇u| + |u| |∇v| + |v| dx
Ω
1/2 1/2
2
2
2
2
|u| + |∇u| dx |v| + |∇v| dx = C 1 u 1 v 1 .
≤ C 1
Ω Ω
2
Dealing with A 2 , we can employ the Cauchy–Schwarz inequality for L (Ω)
directly:
1/2 1/2
2
2
A 2 ≤ C 2 |∇u| dx |v| dx
Ω Ω
≤ C 2 u 1 v 0 ≤ C 2 u 1 v 1 for all u, v ∈ V.
Thus, the assertion follows.
Remark 3.11 In the proof of the propositions (3.24) and (3.25) it has not
been used that the functions u, v satisfy homogeneous Dirichlet boundary
conditions. Therefore, under the assumptions (3.15) these properties hold
1
for every subspace V ⊂ H (Ω).

Conditions for the V -Ellipticity of a
(A) a is symmetric; that is c = 0 (a.e.): Condition (3.17) then has the
simple form r(x) ≥ r 0 for almost all x ∈ Ω.
(A1) c =0, r 0 > 0:
Because of (3.16) we directly get

2 2 2
a(u, u) ≥ {k 0 |∇u| + r 0 |u| } dx ≥ C 3 u for all u ∈ V,
1
Ω
1
where C 3 := min{k 0 ,r 0 }. This also holds for every subspace V ⊂ H (Ω).
(A2) c =0, r 0 ≥ 0:
According to the Poincar´e inequality (Theorem 2.18), there exists some
1
constant C P > 0, independent of u, such that for u ∈ H (Ω)
0
1/2
2
|∇u| dx .
u 0 ≤ C P
Ω