Page 135 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

122 Regularity

where γ denotes a constant independent of h. Returning to (4.14), we choose

2u (x) − u (x + h) − u (x − h)
v (x)= (D −h (D h u)) (x)=
2
|h|
n
n
and observe that, since u ∈ W 1,2 (R ), v ∈ W 1,2 (R ). We therefore find
Z Z
h∇u (x); ∇ (D −h (D h u)) (x)i dx = g (x)(D −h (D h u)) (x) dx . (4.15)
R n R n
Let us express differently the left hand side of the above identity and write
Z
h∇u; ∇ (D −h (D h u))i dx
R n
Z
1
= h∇u (x);2∇u (x) −∇u (x + h) −∇u (x − h)i dx
2
|h| R n
Z
2 h 2 i
= |∇u (x)| − h∇u (x); ∇u (x + h)i dx
2
|h| R n
Z
1 2
= |∇u (x + h) −∇u (x)| dx
2
|h| R n
whereweused, forpassing from the first to the second identity and from the
second to the third one, respectively
Z Z
[h∇u (x); ∇u (x + h)i] dx = [h∇u (x); ∇u (x − h)i] dx
R n R n
Z Z
2 2
|∇u (x)| dx = |∇u (x + h)| dx .
R n R n
Returning to (4.15) we just found that
Z Z
1 2
h∇u (x); ∇ (D −h (D h u)) (x)i dx = |∇u (x + h) −∇u (x)| dx
2
R n |h| R n
Z
2
= |(D h ∇u)(x)| dx
R n
Z
= g (x)(D −h (D h u)) (x) dx .
R n
Applying Cauchy-Schwarz inequality and the properties of the operator D h we
get
2
kD h ∇uk L 2 ≤ kgk L 2 kD −h (D h u)k L 2 ≤ kgk L 2 kD h ∇uk L 2
and hence
kD h ∇uk L 2 ≤ kgk L 2 .