7.9 The Friedrichs Inequality and Poincaré Inequality
441
The inequality (7.9.27) is rewritten as
−
¨
G
v
2 ggdxdy
¨
G
∇u · ∇udxdy −
Γ
v
2 g
∂g
∂n
ds
(7.9.28)
or
−
¨
G
v
2 ggdxdy
¨
G
∇u · ∇udxdy +
Γ
v
2 g
∂g
∂n
ds
(7.9.29)
Putting g = sin
π x
a
sin
π y
b
, then g = −π
2
1
a 2 +
1
b 2
g, and g is not zero on the
domain G. Substituting g into the left-hand side of the inequality (7.9.29), there is
−
¨
G
v
2 ggdxdy = π
2
1
a 2 +
1
b 2
¨
G
u
2 dxdy
(7.9.30)
The second term on the right-hand side the inequality (7.9.29) can be written as
Γ
v
2 g
∂g
∂n
ds
=
Γ
u
2 1
g
∂g
∂n
ds
Γ
u
2
1
g
∂g
∂n
ds
(7.9.31)
Along the boundary Γ ,
1
g
∂g
∂n
is the bounded function, let 0
1
g
∂g
∂n
d on
the boundary Γ , then
Γ
v
2 g
∂g
∂n
ds
d
Γ
u
2 ds
(7.9.32)
Putting
c 1 = π
−2
1
a 2 +
1
b 2
−1
, c 2 = dπ
−2
1
a 2 +
1
b 2
−1
(7.9.33)
Substituting the expression (7.9.30), inequality (7.9.32) and expressions (7.9.33)
into inequality (7.9.29), there is
¨
G
u
2 dxdy c 1
¨
G
∇u · ∇udxdy + c 2
Γ
u
2 ds = c 1
¨
G
(u
2
x + u
2
y )dxdy + c 2
Γ
u
2 ds
(7.9.34)
Quod erat demonstrandum.
Sometimes the inequality (7.9.8) is written in the following form
¨
G
u
2
(x, y)dxdy c[
¨
G
(u
2
x + u
2
y )dxdy +
Γ
u
2
(s)ds]
(7.9.35)