194 F. Bethuel - P. Caldiroli - M. Guida
Clearly, questions i and ii are elementary for the minimal surface equation 1u = 0.
For the H -equation 13, they are rather involved, because the nonlinearity is “critical”.
5.1. Regularity theory
Here we consider weak solutions of the equation 19
1 u
= 2H uu
x
∧ u
y
on O where O is any domain in R
2
. Owing to the nonlinearity 2H uu
x
∧ u
y
as well as to the variational formulation discussed in the previous section, it is natural to consider
solutions of 19 which are in the space H
1
O, R
3
. The first regularity result for 19 was given by H. Wente [47], for H constant.
T
HEOREM
4. If H is constant, then any u ∈ H
1
O, R
3
solution of 19 is smooth, i.e., u
∈ C
∞
O .
Nowadays, this result is a special case of a more general theorem see Theorem 5 below that will be discussed in the sequel. In any case, we point out that the proof of
Theorem 4 relies on the special structure of the nonlinearity:
u
x
∧ u
y
=
u
2 x
u
3 y
− u
3 x
u
2 y
u
3 x
u
1 y
− u
1 x
u
3 y
u
1 x
u
2 y
− u
2 x
u
1 y
=
u
2
, u
3
u
3
, u
1
u
1
, u
2
. Here we have made use of the notation
{ f, g} = f
x
g
y
− f
y
g
x
which represents the Jacobian of the map x , y 7→ f x, y, gx, y. Thus, consid-
ering the equation 19 with H constant, we are led to study the more general linear equation
1φ = { f, g} in O
where f, g satisfy R
O
|∇ f |
2
+∞ and R
O
|∇g|
2
+∞. Obviously { f, g} ∈ L
1
O but, in dimension two, 1φ
∈ L
1
O implies φ
∈ W
1, p loc
O only for p 2, while the
embedding W
1, p
֒ → L
∞
holds true only as p 2. However, { f, g} has a special
structure of divergence form, and precisely { f, g} =
∂ ∂
x f g
y
− ∂
∂ y
f g
x
, and this can be employed to prove what stated in the following lemmata, which have
been used in various forms since the pioneering work by Wente [47]. L
EMMA
4. Let φ ∈ W
1,1 loc
R
2
be the solution of −1φ = { f, g} on R
2
φ z
→ 0 as
|z| → +∞ .
Parametric surfaces 195
Then kϕk
L
∞
+ k∇ϕk
L
2
≤ Ck∇ f k
L
2
k∇gk
L
2
. Proof. Let
−
1 2π
ln |z| be the fundamental solution of −1. Since the problem is invari-
ant under translations, it suffices to estimate φ0. We have φ
= − 1
2π Z
R
2
ln |z| { f, g} dz.
In polar coordinates, one has { f, g} =
1 r
∂ ∂θ
f g
r
− ∂
∂ r
f g
θ
. Hence, integrating by parts, we obtain
φ =
1 2π
Z
R
2
1 r
f g
θ
d z =
1 2π
Z
+∞
dr r
Z
|z|=r
f g
θ
dθ .
Setting f =
1 2πr
R
|z|=r
f dθ , then, using Cauchy-Schwartz and Poincar´e inequality, Z
|z|=r
f g
θ
dθ =
Z
|z|=r
f − f
g
θ
dθ ≤
Z
|z|=r
f − f
2
dθ
1 2
Z
|z|=r
|g
θ
|
2
dθ
1 2
≤ C Z
|z|=r
| f
θ
|
2
dθ
1 2
Z
|z|=r
|g
θ
|
2
dθ
1 2
≤ Cr
2
Z
|z|=r
|∇ f |
2
dθ
1 2
Z
|z|=r
|∇g|
2
dθ
1 2
. Going back to φ0, using again Cauchy-Schwartz inequality, we have
|φ0| ≤ C Z
+∞
r Z
|z|=r
|∇ f |
2
dθ
1 2
r Z
|z|=r
|∇g|
2
dθ
1 2
dr ≤ C
Z
+∞
Z
|z|=r
|∇ f |
2
dθ r dr
1 2
Z
+∞
Z
|z|=r
|∇g|
2
dθ r dr
1 2
= Ck∇ f k
L
2
k∇gk
L
2
. Hence
kφk
L
∞
≤ Ck∇ f k
L
2
k∇gk
L
2
.
196 F. Bethuel - P. Caldiroli - M. Guida
Finally, multiplying the equation −1φ = { f, g} by φ and integrating over R
2
, we obtain
Z
R
2
|∇φ|
2
≤ k{ f, g}k
L
1
kφk
L
∞
≤ 2kφk
L
∞
k∇ f k
L
2
k∇gk
L
2
≤ Ck∇ f k
2 L
2
k∇gk
2 L
2
.
Using the maximum principle, it is possible to derive similarly as obtained by H. Brezis and J.M. Coron [13] the following analogous result:
L
EMMA
5. Assume f, g ∈ H
1
D
2
, R and let φ
∈ W
1,1
D
2
, R be the solution of
−1φ = { f, g} on D
2
φ = 0
on ∂ D
2
. Then
kφk
L
∞
+ k∇φk
L
2
≤ Ck∇ f k
L
2
k∇gk
L
2
. Another proof of the above lemmas can be obtained by using tools of harmonic
analysis. It has been proved Coifman-Lions-Meyer-Semmes [19] that if f, g ∈
H
1
R
2
then { f, g} belongs to the Hardy space H
1
R
2
, a strict subspace of L
1
R
2
, defined as follows:
H
1
R
2
= {u ∈ L
1
R
2
: K
j
u ∈ L
1
for j = 1, 2},
where K
j
= ∂∂x
j
−1
12
. As a consequence, since any Riesz transform R
j
= ∂∂
x
j
−1
−12
maps H
1
R
2
into itself, one has that if −1φ = { f, g} on R
2
then −
∂
2
φ ∂
x
i
∂ x
j
= R
i
R
j
−1φ ∈ H
1
R
2
f or i, j = 1, 2
and hence φ ∈ W
2,1
R
2
⊂ L
∞
R
2
. This argument holds similarly true in the sit- uation of lemma 5 and can be pushed further to obtain the desired estimate, exploit-
ing the fact that the fundamental solution on R
2
to the Laplace equation belongs to BMOR
2
, the dual of H
1
R
2
. We now turn to the case of variable H . Regularity of weak H
1
-solutions has been established under various assumptions on the function H . For instance, H
∈ C
∞
R
3
Parametric surfaces 197
and sup
y ∈R
3
|H y|1 + |y| ≤ α 1 Heinz, [27]
kH k
∞
+∞, H y = H y
1
, y
2
Bethuel-Ghidaglia, [8] kH k
∞
+∞, sup
y ∈R
3
|∇ H y|1 + |y| +∞ Heinz, [28]
kH k
L
∞
+∞, sup
y ∈R
3
∂ H
∂ y
3
y1 + |y
3
| ≤ C Bethuel-Ghidaglia, [9].
However we will describe another regularity theorem, due to F. Bethuel [7]. T
HEOREM
5. If H ∈ C
∞
R
3
satisfies 20
kH k
L
∞
+ k∇ H k
L
∞
+∞ then any solution u
∈ H
1
D
2
, R
3
to 1u = 2H uu
x
∧ u
y
on D
2
is smooth, i.e., u
∈ C
∞
D
2
. The proof of this theorem involves the use of Lorentz spaces, which are borderline
for Sobolev injections, and relies on some preliminary results. Thus we are going to recall some background on the subject, noting that the interest for Lorentz spaces, in
our context, was pointed out by F. H´elein [29], who used them before for harmonic maps.
If is a domain in R
N
and µ denotes the Lebesgue measure, we define L
2, ∞
as the set of all measurable functions f :
→ R such that the weak L
2, ∞
-norm k f k
L
2, ∞
= sup
t 0
{t
1 2
µ {x ∈ : f x t}}
is finite. If L
2,1
denotes the dual space of L
2, ∞
, one has L
2,1
⊂ L
2
⊂
L
2, ∞
, the last inclusion being strict since, for instance, 1r
∈ L
2, ∞
D
2
but 1r ∈
L
2
D
2
. Moreover, if is bounded, then L
2, ∞
⊂ L
p
for every p 2. See
[50] for thorough details. Denoting by B
r
= B
r
z the disc of radius r 0 and center z
∈ R
2
, let now φ
∈ W
1,1
B
r
be the solution of −1φ = { f, g} in B
r
φ = 0
on ∂ B
r
where f, g ∈ H
1
B
r
; recalling lemma 5, one has 21
kφk
L
∞
+ k∇φk
L
2
+ k∇φk
L
2,1
B
r2
≤ Ck∇ f k
L
2
k∇gk
L
2
. The estimate of L
2,1
-norm of the gradient was obtained by L. Tartar [45] using in- terpolation methods, but can also be recovered as a consequence of the embedding
W
1,1
֒ → L
2,1
due to H. Brezis since, as we have already mentioned, the fact that
198 F. Bethuel - P. Caldiroli - M. Guida
{ f, g} belongs to the Hardy space H
1
implies that φ ∈ W
2,1
. Moreover, if g is con- stant on ∂ B
r
, then it can be proved see [7] that 22
k∇φk
L
2
≤ Ck∇ f k
L
2
k∇gk
L
2, ∞
. Finally, we recall the following classical result: if h
∈ L
1
B
r
, then the solution φ
∈ W
1,1
B
r
to −1φ = h in B
r
φ = 0
on ∂ B
r
verifies 23
k∇φk
L
2, ∞
B
r2
≤ Ckhk
L
1
. Proof of Theorem 5. At first we note that the hypothesis 20 grants that
|∇ H u| ≤ C
|∇u| and H u ∈ H
1
. The proof is then divided in some steps. Step 1: Rewriting equation 19.
Let B
2r
z ⊂ D
2
and {H u, u} = {H u, u
1
}, {H u, u
2
}, {H u, u
2
}. The idea is to introduce a Hodge decomposition of 2H u
∇u in B
2r
: 2H u
∇u = ∇ A + ∇
⊥
β w
her e ∇
⊥
= ∂
∂ y
, −
∂ ∂
x .
Since ∂
∂ x
2H uu
y
+ ∂
∂ y
−2H uu
x
= 2{H u, u}, the solution β
∈ W
1,1
B
2r
, R
3
to −1β = {H u, u} in B
2r
β = 0
on ∂ B
2r
belongs, by lemma 5, to H
1
B
2r
, R
3
and satisfies ∂
∂ x
2H uu
y
+ β
x
+ ∂
∂ y
−2H uu
x
+ β
y
= 0. Hence, there exists A
∈ H
1
B
2r
, R
3
such that 24
A
x
= 2H uu
x
− β
y
, A
y
= 2H uu
y
+ β
x
and equation 19, on B
2r
, rewrites: 25
1 u
= A
x
∧ u
y
+ β
y
∧ u
y
. Step 2: “Morrey type” inequality for the L
2, ∞
norm.
Parametric surfaces 199
Since regularity is a local property, as we may reduce the radius r we can assume without loss of generality that
k∇uk
L
2
B
r
ε 1. We are now going to show that
there exists θ ∈ 0, 1 such that
26 k∇uk
L
2, ∞
B
θ r
≤ 1
2 k∇uk
L
2, ∞
B
r
. This is the main step of the proof. Let us consider B
r
⊂ B
r2
and let e
u be the har- monic extension to B
r
of u
|
∂ Br0
. Note that the radius r can be chosen such that
k∇e u
k
L
2
B
r0
≤ Ck∇uk
L
2, ∞
B
r
see [7] for details. In B
r
, using 25, we may write u
= e u
+ ψ
1
+ ψ
2
+ ψ
3
where the functions ψ
1
, ψ
2
, ψ
3
are defined by 1ψ
1
= A
x
∧ u − e u
y
, 1ψ
2
= A
x
∧ e u
y
, 1ψ
3
= β
y
∧ u
y
i n B
r
ψ
1
= ψ
2
= ψ
3
= 0 on ∂ B
r
. Note that, using 24, 21, 20 and the fact that ε 1, computations give
k∇ Ak
L
2
B
r
≤ Ck∇uk
L
2
B
r
. By 22, we have
k∇ψ
1
k
L
2
B
r0
≤ Ck∇ Ak
L
2
B
r
k∇u − e u
k
L
2, ∞
B
r0
≤ Ck∇uk
L
2
B
r
k∇uk
L
2, ∞
B
r
≤ Cεk∇uk
L
2, ∞
B
r
27 and, using 21, we obtain
k∇ψ
2
k
L
2
B
r0
≤ Ck∇ Ak
L
2
B
r0
k∇e u
k
L
2
B
r0
≤ Ck∇uk
L
2
B
r
k∇e u
k
L
2
B
r0
≤ Cεk∇uk
L
2, ∞
B
r
. 28
Using the duality of L
2,1
and L
2, ∞
, 23 and 21 yield k∇ψ
3
k
L
2, ∞
B
r02
≤ Ck∇βk
L
2,1
B
r2
k∇uk
L
2, ∞
B
r2
≤ Cε
2
k∇uk
L
2, ∞
B
r
≤ Cεk∇uk
L
2, ∞
B
r
. 29
By the properties of harmonic functions, one has that 30
∀α ∈ 0, 1 k∇e
u k
L
2
B
α r0
≤ Cαk∇e u
k
L
2
B
r0
≤ Cαk∇uk
L
2, ∞
B
r
. Combining 27–30 and recalling the decomposition of u in B
r
, we finally deduce that
∀α ∈ 0, 1 k∇uk
L
2, ∞
B
α r0
≤ Cε + αk∇uk
L
2, ∞
B
r
and, by a suitable choice of ε and α, 26 follows. Step 3: H¨older continuity.
200 F. Bethuel - P. Caldiroli - M. Guida
From the last result, by iteration, we deduce that there exists µ ∈ 0, 1 such that
k∇uk
L
