Page 222 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Chapter 4: Regularity 209

while
3
x x 2 x 1 x 2 ¡ 2 2 ¢
1
0
00
= V (|x|)+ 2x +3x V (|x|)
u x 1 x 1 1 2
2 3
|x| |x|
3
x x 1 x 1 x 2 ¡ 2 2 ¢
2
0
= V (|x|)+ 2x +3x V (|x|)
00
2 3
u x 2 x 2 2 1
|x| |x|
4
2 2
2 2
x x x + x x + x 4 2
1 2
1
1 2
00
= V (|x|)+ V (|x|)+ V (|x|) .
0
2 3
u x 1 x 2
|x| |x|
We therefore get that
0 ∞
u x 1 x 1 ,u x 2 x 2 ∈ C (Ω) ,u x 1 x 2 / ∈ L (Ω) .
Exercise 4.3.4. Let V (r)= log |log r|. A direct computation shows that
x 1 x 2
0
= 0 = V (|x|)
u x 1 V (|x|) and u x 2
|x| |x|
and therefore
x 2 1 x 2 2
0
= V (|x|)+ V (|x|)
00
2 3
u x 1x 1
|x| |x|
x 2 2 x 2 1
00
0
= V (|x|)+ V (|x|)
2 3
u x 2 x 2
|x| |x|
x 1 x 2 x 1 x 2
0
= 00 V (|x|) .
2 3
u x 1 x 2 V (|x|) −
|x| |x|
This leads to
0
V (|x|) −1 1
∆u = V (|x|)+ = 2 2 ∈ L (Ω)
00
|x| |x| |log |x||
1
/ ∈ L (Ω). Summarizing the results we indeed have that
while u x 1 x 1 ,u x 1 x 2 ,u x 2x 2
1
u/∈ W 2,1 (Ω) while ∆u ∈ L (Ω). We also observe (compare with Example 1.33
(ii)) that, trivially, u/∈ L ∞ (Ω) while u ∈ W 1,2 (Ω),since
ZZ Z 1/2 dr 2π
2
|∇u| dx =2π = .
2
Ω 0 r |log r| log 2
A much more involved example due to Ornstein [79] produces a u such that
1 1
u x 1 x 1 ,u x 2 x 2 ∈ L (Ω) ,u x 1 x 2 / ∈ L (Ω) .