Page 130 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

The model case: Dirichlet integral 117

Exercise 4.2.2 Let p> 2q> 2 and

1 p λ q qp q−1 (p − 1)
f (x, u, ξ)= f (u, ξ)= |ξ| + |u| where λ = q
p q (p − q)
p − q p/(p−q)
u (x)= |x|
p
¡ 2 ¢
(note that if, for example, p =6 and q =2,then f ∈ C ∞ R ).
1
2
(i) Show that u ∈ C ([−1, 1]) but u/∈ C ([−1, 1]).
(ii) Find some values of p and q so that
0 p−2 0 q−2 ∞
|u | u , |u| u ∈ C ([−1, 1]) ,
2
although u/∈ C ([−1, 1]).
(iii) Show that u is the unique minimizer of
½ Z ¾
1 p − q
0
(P) inf I (u)= f (u (x) ,u (x)) dx : u (−1) = u (1) = .
u∈W 1,p (−1,1) −1 p
4.3 The model case: Dirichlet integral

n
Let Ω ⊂ R be a bounded open set with Lipschitz boundary and u 0 ∈ W 1,2 (Ω)
and consider the problem
½ Z ¾
1 2 1,2
(P) inf I (u)= |∇u (x)| dx : u ∈ u 0 + W 0 (Ω) .
2 Ω
We have seen in Section 3.2 that there exists a unique minimizer u ∈ u 0 +
1,2
W (Ω) of (P). Furthermore u satisfies the weak form of Laplace equation,
0
namely
Z
1,2
(E w ) h∇u (x); ∇ϕ (x)i dx =0, ∀ϕ ∈ W 0 (Ω)
Ω
n
where h.; .i denotes the scalar product in R .
We will now show that u ∈ C ∞ (Ω) andthatitsatisfies Laplace equation
∆u (x)= 0, ∀x ∈ Ω .

We speak then of interior regularity. If, in addition, Ω is a bounded open set
¡ ¢ ¡ ¢
with C ∞ boundary and u 0 ∈ C ∞ Ω one can show that in fact u ∈ C ∞ Ω ;we
speak then of regularity up to the boundary.
Dirichlet integral has been studied so much that besides the books that we
have quoted in the introduction of the present chapter, one can find numerous