Page 94 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

The model case: Dirichlet integral 81

The interested reader is referred for further reading to the book of the au-
thor [31] or to Buttazzo [15], Buttazzo-Giaquinta-Hildebrandt [17], Cesari [20],
Ekeland-Témam [41], Giaquinta [47], Giusti [51], Ioffe-Tihomirov [62], Morrey
[75], Struwe [92] and Zeidler [99].

3.2 The model case: Dirichlet integral

The main result is
n
Theorem 3.1 Let Ω ⊂ R be a bounded open set with Lipschitz boundary and
u 0 ∈ W 1,2 (Ω). The problem
½ Z ¾
1 2 1,2
(D) inf I (u)= |∇u (x)| dx : u ∈ u 0 + W 0 (Ω) = m
2 Ω
1,2
has one and only one solution u ∈ u 0 + W (Ω).
0
Furthermore u satisfies the weak form of Laplace equation, namely
Z
h∇u (x); ∇ϕ (x)i dx =0, ∀ϕ ∈ W 1,2 (Ω) (3.1)
0
Ω
n
where h.; .i denotes the scalar product in R .
Conversely if u ∈ u 0 + W 0 1,2 (Ω) satisfies (3.1) then it is a minimizer of (D).

Remark 3.2 (i) We should again emphasize the very weak hypotheses on u 0
1,2
and Ω and recall that u ∈ u 0 + W (Ω) means that u = u 0 on ∂Ω (in the sense
0
of Sobolev spaces).
(ii) If the solution u turns out to be more regular, namely in W 2,2 (Ω) then
(3.1) can be integrated by parts and we get
Z
1,2
∆u (x) ϕ (x) dx =0, ∀ϕ ∈ W (Ω)
0
Ω
which combined with the fundamental lemma of the calculus of variations (The-
orem 1.24) leads to ∆u =0 a.e. in Ω. This extra regularity of u (which will
turn out to be even C ∞ (Ω)) will be proved in Section 4.3.
(iii) As we already said, the above theorem was proved by Hilbert, Lebesgue
andTonelli, but it wasexpressedinadifferent way since Sobolev spaces did not
exist then. Throughout the 19th century there were several attempts to establish
a theorem of the above kind, notably by Dirichlet and Riemann.
Proof. The proof is surprisingly simple.
Part 1 (Existence). We divide, as explained in the Introduction, the proof
into three steps.