Page 160 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Nonparametric minimal surfaces 147

and hence Σ has mean curvature that vanishes everywhere. The converse is also
true; moreover we have uniqueness of such solutions. However, the existence of
a minimizer still needs to be proved because Theorem 5.19 does not deal with
this case. The techniques for solving such a problem are much more analytical
than the previous ones.
Before proceeding with the existence theorems we would like to mention a
famous related problem known as Bernstein problem. The problem is posed in
n
n
n
2
the whole space R (i.e., Ω = R ) and we seek for C (R ) solutions of the
minimal surface equation
⎛ ⎞
n
X ∂ u x i n
(E) ⎝ q ⎠ =0 in R
∂x i 2
i=1 1+ |∇u|
or of its equivalent form Mu =0. In terms of regular surfaces, we are searching
n
for a nonparametric surface (defined over the whole of R )in R n+1 which has
vanishing mean curvature. Obviously the function u (x)= ha; xi+b with a ∈ R n
and b ∈ R, which in geometrical terms represents a hyperplane, is a solution of
the equation. The question is to know if this is the only one.
In the case n =2, Bernstein has shown that, indeed, this is the only C 2
solution (the result is known as Bernstein theorem). Since then several authors
found different proofs of this theorem. The extension to higher dimensions is
however much harder. De Giorgi extended the result to the case n =3,Almgren
to the case n =4 and Simons to n =5, 6, 7. In 1969, Bombieri, De Giorgi and
n
2
Giusti proved that when n ≥ 8, there exists a nonlinear u ∈ C (R ) (and hence
the surface is not a hyperplane) satisfying equation (E). For more details on
Bernstein problem, see Giusti [50], Chapter 17 and Nitsche [78], pages 429-430.
We now return to our problem in a bounded domain. We start by quoting a
result of Jenkins and Serrin; for a proof see Gilbarg-Trudinger [49], page 297.
Theorem 5.26 (Jenkins-Serrin).Let Ω ⊂ R n be a bounded domain with
¡ ¢
C 2,α , 0 <α< 1, boundary and let u 0 ∈ C 2,α Ω . The problem Mu =0 in Ω
with u = u 0 on ∂Ω has a solution for every u 0 if and only if the mean curvature
of ∂Ω is everywhere non negative.
Remark 5.27 (i) We now briefly mention a related result due to Finn and
Osserman. It roughly says that if Ω is a non convex domain, there exists a
continuous u 0 so that the problem Mu =0 in Ω with u = u 0 on ∂Ω has no C 2
solution. Such a u 0 can even have arbitrarily small norm ku 0 k 0.
C
(ii) The above theorem follows several earlier works that started with Bern-
stein (see Nitsche [78], pages 352-358).
We end the present chapter with a simple theorem whose ideas contained in
the proof are used in several different problems of partial differential equations.