Page 124 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Chapter 4

Regularity

4.1 Introduction

We are still considering the problem

½ Z ¾
1,p
(P) inf I (u)= f (x, u (x) , ∇u (x)) dx : u ∈ u 0 + W (Ω) = m
0
Ω
where
n
- Ω ⊂ R is a bounded open set;
n
- f : Ω × R × R −→ R, f = f (x, u, ξ);
- u ∈ u 0 + W 1,p (Ω) means that u, u 0 ∈ W 1,p (Ω) and u − u 0 ∈ W 1,p (Ω).
0 0
We have shown in Chapter 3 that, under appropriate hypotheses on f, u 0
1,p
and Ω, (P) has a minimizer u ∈ u 0 + W (Ω).
0
The question that we will discuss now is to determine whether, in fact, the
Ω . More precisely if the data
minimizer u is not more regular, for example C 1 ¡ ¢
f, u 0 and Ω are sufficiently regular, say C ,does u ∈ C ? Thisisone of the
∞
∞
23 problems of Hilbert that were mentioned in Chapter 0.
The case n =1 will be discussed in Section 4.2. We will obtain some general
results. We will then turn our attention to the higher dimensional case. This is
a considerably harder problem and we will treat only the case of the Dirichlet
integral in Section 4.3. We will in Section 4.4 give, without proofs, some general
theorems.
We should also point out that all the regularity results that we will obtain
here are about solutions of the Euler-Lagrange equation and therefore not only
minimizers of (P).

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