Page 137 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

124 Regularity

Show that
0 ∞
u x 1 x 1 ,u x 2 x 2 ∈ C (Ω) ,u x 1 x 2 / ∈ L (Ω)
2
0
∈ C ,while u/∈ C ,infact u is not even
which implies that ∆u = u x 1 x 1 + u x 2 x 2
in W 2,∞ .
Exercise 4.3.4 The present exercise (in the spirit of the preceding one) will
1
give an example of a function u/∈ W 2,1 ,with ∆u ∈ L .Let
© 2 ª
Ω = x ∈ R :0 < |x| < 1/2
u (x)= u (x 1 ,x 2 )= log |log |x|| ,if x ∈ Ω .
1
Show that u/∈ W 2,1 (Ω) while ∆u ∈ L (Ω).

4.4 Some general results

The generalization of the preceding section to integrands of the form f =
f (x, u, ∇u) is a difficult task. We will give here, without proof, a general the-
orem and we refer for more results to the literature. The next theorem can be
found in Morrey [75] (Theorem 1.10.4).
n
n
Theorem 4.11 Let Ω ⊂ R be a bounded open set and f ∈ C ∞ (Ω × R × R ),
¡ ¢
and similarly for the
f = f (x, u, ξ).Let f x =(f x 1 , ..., f x n ), f ξ = f ξ 1 , ..., f ξ n
n
n
higher derivatives. Let f satisfy, for every (x, u, ξ) ∈ Ω × R × R and λ ∈ R ,
⎧ p p
α 1 V − α 2 ≤ f (x, u, ξ) ≤ α 3 V
⎪
⎪
⎪
⎪
⎪
⎪
⎪ p−1 p−2
⎨ |f ξ | , |f xξ | , |f u | , |f xu | ≤ α 3 V , |f uξ | , |f uu | ≤ α 3 V
(C)
⎪
⎪ n
⎪ X
⎪ 2 2
⎪ p−2 p−2
⎪ α 4 V |λ| ≤ (x, u, ξ) λ i λ j ≤ α 5 V |λ|
⎪ f ξ i ξ j
⎩
i,j=1
2
2
2
where p ≥ 2, V =1 + u + |ξ| and α i > 0, i =1, ..., 5, are constants.
Then any minimizer of
½ Z ¾
1,p
(P) inf I (u)= f (x, u (x) , ∇u (x)) dx : u ∈ u 0 + W 0 (Ω)
Ω
is in C ∞ (D), for every D ⊂ D ⊂ Ω.
Remark 4.12 (i) The last hypothesis in (C) implies a kind of uniform convexity
of ξ → f (x, u, ξ); it guarantees the uniform ellipticity of the Euler-Lagrange
equation. The example of the preceding section, obviously, satisfies (C).
(ii) For the regularity up to the boundary, we refer to the literature.