ON PERIODIC ELLIPTIC EQUATIONS WITH GRADIENT DEPENDENCE MASSIMILIANO BERTI(NA) , MICHELE MATZEU AND ENRICO VALDINOCI(RM ) Abstract. We construct entire solutions of ∆u = f (x, u, ∇u) which are superpositions of odd, periodic functions and linear ones, with prescribed integer or rational slope.
1. Introduction The purpose of this note is to find solutions of a nonvariational elliptic equation with a periodic nonlinearity which depends on the gradient. We consider a function f : Rn × R × Rn ∋ (x, t, p) → R, which is Zn+1 -periodic in (x, t), and, given any ω ∈ Zn , we look for solutions of the equation (1.1) ∆u(x) = f x, u(x), ∇u(x)
for any x ∈ Rn , which are of the form u(x) = ω · x + U (x), for a Zn -periodic function U (see Theorem 1.1 below for a precise statement). To formalize this result, we take the following assumptions on f . We suppose that f ∈ C(Rn × R × Rn ) ,
(1.2)
that f is periodic in (x, t) under integer translations, that is (1.3)
f (x + k, t + j, p) = f (x, t, p) n
for any (x, t, p) ∈ R × R × Rn and any (k, j) ∈ Zn × Z, and that f is odd in (x, t), that is (1.4)
f (−x, −t, p) = −f (x, t, p)
for any (x, t, p) ∈ Rn × R × Rn . The problem dealt with in this note has been inspired by the work of J. Moser on the elliptic integrands (see [M]) and on its extensions (see, e.g., [Ba] and [Be]). It can also be seen as an infinite-dimensional analogue of Aubry-Mather theory in dynamical systems. Related questions also arise in minimal surface and phase transition models (see, e.g., [CdlLl], [RS] and [V]). The main result we prove is the following one: Theorem 1.1. Let f satisfy (1.2), (1.3) and (1.4). Suppose also that there exist Λ⋆ , λ⋆ ≥ 0 such that
(1.5)
|f (x, t, p)| ≤ Λ⋆ + λ⋆ |p|
This work has been partially supported by MIUR project “Variational methods and nonlinear differential equations”. 1
2
M. BERTI, M. MATZEU AND E. VALDINOCI
for any x ∈ Rn , t ∈ R and p ∈ Rn . Then, there exists c ∈ (0, 1), depending only on n, in such a way that if λ⋆ ≤ c the following holds. 1,2 Given any ω ∈ Zn , there exists u ∈ Wloc (Rn ) which is a weak solution of (1.1) n for any x ∈ R and which enjoys the following properties. If U (x) := u(x) − ω · x, then U (x + ℓ) = U (x) = −U (−x)
(1.6) n
n
for any x ∈ R and ℓ ∈ Z , and kU kW 2,2 (Tn ) ≤ C(Λ⋆ + λ⋆ |ω|) ,
(1.7)
where C depends only on n. An example of nonlinearity satisfying the assumptions of Theorem 1.1 is given by f (x, t, p) = cos(2πx1 ) . . . cos(2πxn ) sin(2πt)(1 + |p|3/4 + ǫ|p|) ,
with |ǫ| small. The proof of Theorem 1.1 will be given in Section 2 and it is based on a Galerkin approximation and Brouwer’s Fixed Point Theorem. For this, we will make a cutoff on the Fourier coefficients to construct a vector field on a finite dimensional space which “points outward” on suitably large spheres, thanks to our assumptions. Brouwer’s Fixed Point Theorem assures the existence of a zero of such a vector field, and a limit procedure will then give the solution of our problem. The result of Theorem 1.1 may be sharpened in several ways. First of all, if the nonlinearity is Lipschitz in (t, p) with a conveniently small Lipschitz constant, then the solution can be found in a rather constructive way by iteratively solving variational problems, in the light of the following result: Theorem 1.2. Let f satisfy (1.2), (1.3), and (1.4). Suppose also that f has zero average in t, that is Z t+j (1.8) f (x, τ, p) dτ = 0 , t
n
n
for any (x, t, p) ∈ R × R × R and any j ∈ Z. Suppose also that there exist Λ⋆ , λ⋆ ≥ 0 such that (1.9)
|f (x, t, p) − f (x, s, q)| ≤ λ⋆ (|t − s| + |p − q|)
for any t, s ∈ R and p, q ∈ Rn . Then, there exists c ∈ (0, 1), depending only on n, in such a way that if λ⋆ ≤ c the theses of Theorem 1.1 hold true and (1.7) takes the strongest form (1.10)
kU kW 2,2 (Tn ) ≤ Cλ⋆ ,
where C depends only on n. 1,2 What is more, the solution u of Theorem 1.1 is the limit in Wloc (Rn ) of a 1,2 sequence of functions uk ∈ Wloc (Rn ) satisfying u0 (x) = ω · x and ∆uk (x) = f (x, uk (x), ∇uk−1 (x)) weakly, for any k ≥ 1.
ELLIPTIC EQUATIONS WITH GRADIENT DEPENDENCE
3
An example of nonlinearity satisfying the assumptions of Theorem 1.2 is given by 2
f (x, t, p) = ǫ cos(2πx1 ) . . . cos(2πxn ) sin(2πt)e−|p| , as long as |ǫ| is suitably small. The proof of Theorem 1.2 will be given in Section 3 and will exploit the iteration method of [DFGM] and [GM] applied recursively to a variational problem. That is, we iteratively fix the gradient term and then construct a solution of the associated variational problem by direct minimization. We will then obtain a sequence of uk ’s satisfying ∆uk = f (x, uk , ∇uk−1 ). Uniform energy estimates will then yield the convergence of the above sequence. Another extension of Theorem 1.1 consists in obtaining pointwise estimates on the oscillation of the solution, as the next result states: Corollary 1.3. Let a0 := min{2/n, 1}. Let f satisfy (1.2), (1.3) and (1.4). Assume that there exists Λ0 ≥ 0 and
(1.11)
in such a way that (1.12)
a ∈ (0, a0 )
|f (x, t, p)| ≤ Λ0 (1 + |p|a ) ,
for any x ∈ Rn , t ∈ R and p ∈ Rn . Then, the theses of Theorem 1.1 hold true. What is more, the solution u of Theorem 1.1 lies in C 1,α , with α ∈ (0, 1). More precisely,
(1.13)
kU kC 1,α (Tn ) ≤ Ca,Λ0 (1 + |ω|)
with α := 1 − (na/2). The constant Ca,Λ0 only depends on n, a and Λ0 . An example of nonlinearity satisfying the assumptions of Corollary 1.3 is given by f (x, t, p) = cos(2πx1 ) . . . cos(2πxn ) sin(2πt)(1 + |p|a ) , with a as in (1.11). We observe that (1.13) implies that the function U (x) = u(x) − ω · x lies in L∞ (Rn ) and so the graph of the solution is trapped in between two hyperplanes. Thus, following the terminology of [CL], we may say that this solution is “plane-like”. In the dynamical system setting, the fact that U is bounded may be interpreted as the existence of a “rotation number” (thus one may think ω to be a “frequency”). In this respect, note that, when n = 1, one might naively conceive the gradient dependence as a friction term; however, in this case, structural conditions as the ones we take here are essential for the existence of solutions of given rotation numbers (for instance, a pendulum with friction only admits the rotation number zero). Section 4 is devoted to the proof of Corollary 1.3, which uses the De Giorgi-NashMoser theory to get pointwise bounds and then Calder´ on-Zygmund estimates. The following result on “rational frequencies” ω ∈ Qn also holds: Corollary 1.4. Let f satisfy (1.2), (1.3), (1.4) and (1.12). Fix ω ∈ Qn and N ∈ N in such a way that ωN ∈ Zn . 1,2 1,α Then, there exists u ∈ Wloc (Rn )∩Cloc (Rn ), u(x) = ω ·x+ U (x), which is a weak solution of (1.1). Also, U (x + N ℓ) = U (x) = −U (−x) for any x ∈ Rn and ℓ ∈ Zn .
The proof of Corollary 1.4 will be performed in Section 5 by a scaling argument.
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M. BERTI, M. MATZEU AND E. VALDINOCI
2. Proof of Theorem 1.1 In the sequel, we will denote by Ca,b,etc. positive constants depending only on a, b, etc. (the dependence on the dimension n will be omitted). Such constants may change from line to line. 2.1. Construction of a vector field. Given φ ∈ L2 (Tn ) which is odd (and thus with zero average), we consider the Fourier expansion X φ(x) = φk sin(2πk · x) k∈Nn
and we define the projection on its kth Fourier coefficient as Πk (φ) := φk . Given N ∈ N, we set NN := {k ∈ Nn s.t. 1 ≤ k1 + · · · + kn ≤ N }
and we denote by m(N ) the cardinality of NN . Fixed X = (X1 , . . . , Xm(N ) ) ∈ Rm(N ) , we define for any x ∈ Rn X X ℓXℓ cos(2πℓ · x) . Xℓ sin(2πℓ · x), ω + 2π gX (x) := f x, ω · x + ℓ∈NN
ℓ∈NN
2
n
Note that gX ∈ L (T ) thanks to (1.5) and that X X ℓXℓ cos(2πℓ · x) Xℓ sin(2πℓ · x), ω + 2π gX (−x) = f − x, −ω · x − ℓ∈NN
ℓ∈NN
=
X X ℓXℓ cos(2πℓ · x) Xℓ sin(2πℓ · x), ω + 2π −f x, ω · x + ℓ∈NN
ℓ∈NN
=
−gX (x) ,
due to (1.4). Also, gX (x + ℓ) = gX (x), for any ℓ ∈ Zn , due to (1.3). Hence, gX is periodic and odd, and so it has zero average. Thus, for any k ∈ Nn we may define wk (X) := Πk (gX ) .
We consider the vector field v (N ) : Rm(N ) → Rm(N ) whose kth component is (N )
vk (X) := 4π 2 |k|2 Xk + wk (X) .
2.2. Topological argument. We now obtain some bounds with the scope of showing that the vector field v (N ) points outwards on large spheres (see Lemma 2.2 below). Lemma 2.1. For any X ∈ Rm(N ) , we have that X X |k|2 Xk2 . |wk (X)|2 ≤ C Λ2⋆ + λ2⋆ |ω|2 + λ2⋆ k∈Nn
k∈NN
Proof. Fix X ∈ Rm(N ) . For any x ∈ Rn , let X kXk cos(2πk · x) . ηX (x) := k∈NN
Then,
kηX k2L2 (Tn ) ≤ C
X
k∈NN
|k|2 Xk2
ELLIPTIC EQUATIONS WITH GRADIENT DEPENDENCE
5
Also, by (1.5), |gX (x)| ≤ C(Λ⋆ + λ⋆ |ω| + λ⋆ |ηX (x)|) .
Therefore, X
(wk (X))2
X
=
k∈Nn
(Πk (gX ))2
k∈Nn
≤ CkgX k2L2 (Tn )
≤ C(Λ2⋆ + λ2⋆ |ω|2 + λ2⋆ kηX k2L2 (Tn ) ) X ≤ C Λ2⋆ + λ2⋆ |ω|2 + λ2⋆ |k|2 Xk2 , k∈NN
as desired.
Lemma 2.2. There exists r > 0 in such a way that v (N ) (X) · X ≥ 0 as long as |X| ≥ r. Proof. Given X ∈ Rm(N ) , we define
|X|⋆ := Note that |X|⋆ ≥ Thus, exploiting Lemma 2.1, X
k∈NN
as long as
s X
k∈NN
s X
k∈NN
|k|2 Xk2 .
Xk2 = |X| .
|wk (X)|2 ≤ Cλ2⋆ |X|2⋆
|X| ≥
s
Λ2⋆ + |ω|2 =: r . λ2⋆
As a consequence, v (N ) (X) · X
=
(N )
X
vk (X)Xk
k∈NN
=
X
k∈NN
≥ 4π 2 ≥ 4π
2
4π 2 |k|2 Xk2 +
X
k∈NN
|X|2⋆ 2
|k|2 Xk2 −
X
wk (X)Xk
k∈NN
s X
k∈NN
|wk (X)|2
s X
Xk2
k∈NN
− Cλ⋆ |X|⋆ |X|
≥ (4π − Cλ⋆ ) |X|2⋆ ,
which yields the claim provided that λ⋆ is small enough.
The above estimate and Brouwer’s Fixed Point Theorem yield the existence of a zero of the vector field v (N ) : Corollary 2.3. There exists X (N ) ∈ Rm(N ) in such a way that v (N ) (X (N ) ) = 0. Proof. The claim plainly follows from Lemma 2.2 here above and the Lemma on page 493 of [E].
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M. BERTI, M. MATZEU AND E. VALDINOCI
We now localize the zeroes of the vector field: Lemma 2.4. If X = (X1 , . . . , Xm(N ) ) ∈ Rm(N ) is so that v (N ) (X) = 0, then s X |k|2 Xk2 ≤ C(Λ⋆ + λ⋆ |ω|) . k∈NN
Proof. By Lemma 2.1, X X wk (X)Xk |k|2 Xk2 = − 4π 2 k∈NN
k∈NN
s X
≤
k∈NN
|wk
(X)|2
s X
k∈NN
≤ C Λ⋆ + λ⋆ |ω| + λ⋆
Thence,
s X
k∈NN
Xk2
s X
k∈NN
|k|2 Xk2
|k|2 Xk2 ≤ C Λ⋆ + λ⋆ |ω| + λ⋆
s X
k∈NN
s X
k∈NN
which implies the claim as long as λ⋆ is small enough.
|k|2 Xk2 .
|k|2 Xk2 , (N )
2.3. Construction of an approximate solution. If X (N ) = (X1
(N )
, . . . , Xm(N ) )∈
Rm(N ) is as in Corollary 2.3, we define for any x ∈ Rn , X (N ) (2.1) UN (x) := Xk sin(2πk · x) . k∈NN
Lemma 2.5. The function UN satisfies the following properties: UN ∈ C ∞ (Tn ) ,
(2.2)
UN (x) = −UN (−x) = UN (x + ℓ)
(2.3) n
for any x ∈ R and ℓ ∈ Zn ,
∆UN (x) = f (x, ω · x + UN (x), ω + ∇UN (x))
(2.4)
n
for any x ∈ R , and
kUN kW 2,2 (Tn ) ≤ C(Λ⋆ + λ⋆ |ω|) .
(2.5)
Proof. The properties in (2.2) and (2.3) are obvious consequences of (2.1), since NN is a finite set. The fact that v (N ) (X (N ) ) = 0 yields that X (N ) 4π 2 |k|2 Xk sin(2πk · x) ∆UN (x) = − k∈NN
=
X
k∈NN
= =
wk (X (N ) ) sin(2πk · x)
gX (N ) (x) X (N ) X (N ) f x, ω · x + Xℓ sin(2πℓ · x), ω + 2π ℓXℓ cos(2πℓ · x) ℓ∈NN
=
f (x, ω · x + UN (x), ω + ∇UN (x)) ,
ℓ∈NN
ELLIPTIC EQUATIONS WITH GRADIENT DEPENDENCE
7
which checks (2.4). Also, by (2.3), UN is periodic with zero average and so, by Lemma 2.4, (2.6)
kUN kW 1,2 (Tn ) ≤ Ck∇UN kL2 (Tn ) ≤ C
s X
k∈NN
(N ) 2 )
|k|2 (Xk
≤ (Λ⋆ + λ⋆ |ω|) .
Lemmata 2.1 and 2.4 also give that kgX (N ) kL2 (Tn ) ≤ C
sX
k∈Nn
|wk (X (N ) )|2
(2.7)
≤ C Λ⋆ + λ⋆ |ω| + λ⋆ ≤ C(Λ⋆ + λ⋆ |ω|) .
s X
k∈NN
(N ) 2 )
|k|2 (Xk
Elliptic estimates (obtained via Fourier series or by the general result in Theorem 9.11 in [GT]), (2.4), (2.6) and (2.7) yield that kUN kW 2,2 (Tn ) ≤ C(kUN kL2 (Tn ) + kgX (N ) kL2 (Tn ) ) ≤ C(Λ⋆ + λ⋆ |ω|) , that is (2.5).
2.4. End of the proof of Theorem 1.1. By (2.5), there exists a subsequence, say UNj , and a function U ∈ W 1,2 (Tn ), satisfying (1.7), in such a way that ∇UNj converges to ∇U in L2 (Tn ), and UNj , resp. ∇UNj , converges to U , resp. ∇U , almost everywhere. By taking the pointwise limit of UNj in (2.3), one sees that U satisfies (1.6). Given φ ∈ C ∞ (Tn ), we now define hj (x) := f (x, ω · x + UNj , ω + ∇UNj ) φ(x) and h(x) := f (x, ω · x + U, ω + ∇U ) φ(x) . Note that hj converges almost everywhere to h, due to (1.2). Then, by Egoroff Theorem (see, e.g., page 60 of [F]), (2.8)
hj converges to h in measure.
We now observe that the functions hj ’s are uniformly integrable, in the sense that given any ǫ > 0 there exists δ > 0 in such a way that Z |hj | ≤ ǫ (2.9) E
for any measurable set E ⊂ Tn with measure less than δ. Indeed, in order to prove (2.9), we fix ǫ > 0, we take E as above and we make use of (1.5) and (2.5)
8
M. BERTI, M. MATZEU AND E. VALDINOCI
to get that Z
E
|hj | dx
√ δ
≤
sZ
E
|hj |2 dx
≤
√ δ kφkL∞ (Rn )
≤
C
sZ
E
(Λ⋆ + λ⋆ |ω| + λ⋆ |∇UNj |)2 dx
√ δ kφkL∞ (Rn ) (Λ⋆ + λ⋆ |ω| + λ⋆ k∇UNj kL2 (Tn ) ) √ C δ kφkL∞ (Rn ) (Λ⋆ + λ⋆ |ω|) .
≤
By taking δ > 0 conveniently small, the latter quantity is less than ǫ, thus proving (2.9). Then, (2.8), (2.9) and Vitali Convergence Theorem (see, e.g., page 180 of [F]) give that Z Z lim hj dx = h dx . j→+∞
Tn
Tn
Consequently,
0 = =
lim
j→+∞
Z
Tn
Z
Tn
∇UNj · ∇φ + f (x, ω · x + UNj , ω + ∇UNj )φ dx
∇U · ∇φ + f (x, ω · x + U, ω + ∇U )φ dx
for any φ ∈ C ∞ (Tn ). Hence u(x) := U (x) + ω · x satisfies (1.1), concluding the proof of Theorem 1.1. ♦ We remark that, in general, the non variational equation (2.10)
∆V = f (x, ω · x + V, ω + ∇V )
has no solution V ∈ W 1,2 (Tn ) . Indeed, project (2.10) according to the orthogonal decomposition W 1,2 (Tn ) = T ⊕ W01,2 (Tn )
of zero average functions W01,2 (Tn ) plus the constants T, and look for solutions V = α + U of (2.10) with α ∈ T, U ∈ W01,2 (Tn ). Equation (2.10) is equivalent to the system ( ∆U = f (x, ω · x + α + U, ω + ∇U ) − hf i(U ) R (2.11) hf i(U ) := Tn f (x, ω · x + U, ω + ∇U ) dx = 0 .
For each α ∈ Tn , we can find a solution U (α) ∈ W01,2 (Tn ) of the first equation exactly as above, just looking for solutions X (N ) ¯ (N ) = X (N ) . UN = Xk e2πik·x , X k −k k∈Zn 0<|k|≤N
Next, for a non variational nonlinearity f , the real-valued function Ψ : Tn → R defined by Ψ(α) := hf i(U (α) ) could have no zeros (unlike the variational case). This is a statement analogous to the Invariant Curve Theorem in [H] for non Hamiltonian systems.
ELLIPTIC EQUATIONS WITH GRADIENT DEPENDENCE
9
The claim in Theorem 1.1 would hold once we know that there exists α⋆ ∈ T such that Ψ(α⋆ ) = 0. The oddness assumption in (1.4) is thus the easiest topological constraint which makes this possible. 3. Proof of Theorem 1.2 We point out that the “global” condition in (1.9) together with the “zero average condition” in (1.8), implies a uniform bound on the nonlinearity: Lemma 3.1. Conditions (1.2), (1.8) and (1.9) imply that |f (x, t, p)| ≤ λ⋆ ,
n
n
for any x ∈ R , t ∈ R and p ∈ R .
Proof. Fix (x, t, p) and let jt ∈ Z be so that jt ≤ t < jt + 1. By (1.8), Z jt +1 f (x, τ, p) dτ = 0 jt
and so, from (1.2), there exists tx,t,p ∈ [jt , jt + 1] in such a way that f (x, tx,t,p , p) = 0 .
Consequently, by (1.9), as desired.
|f (x, t, p)| = |f (x, t, p) − f (x, tx,t,p , p)| ≤ λ⋆ |t − tx,t,p | ≤ λ⋆ ,
3.1. The variational problem. We now construct the solution of a problem analogous to the one in Theorem 1.1 except for the fact that the gradient term is missing. This problem is variational and it may be solved by direct minimization thanks to the oddness assumption on f . The following result is indeed a variation of Theorem 5.1 of [M]: Lemma 3.2. Let ω ∈ Zn and let g : Rn ×R → R be a measurable function. Assume that g(x, ·) ∈ C(R)
(3.1)
for almost any fixed x ∈ Rn . Suppose that (3.2)
g(x + ℓ, t + j) = g(x, t)
n
for any (x, t) ∈ R × R and any (ℓ, j) ∈ Zn × Z, that (3.3)
n
g(−x, −t) = −g(x, t)
for any (x, t) ∈ R × R, and that Z t+j g(x, τ ) dτ = 0 , (3.4) t
for any (x, t) ∈ Rn × R and any j ∈ Z. Let (3.5)
Γ(x) := sup |g(x, τ )| . τ ∈R
and suppose that (3.6)
Γ ∈ L1 (Tn ) .
10
M. BERTI, M. MATZEU AND E. VALDINOCI 1,2 Then, there exists v ∈ Wloc (Rn ) in such a way that n
v(x + ℓ) = v(x) = −v(−x)
n
for any x ∈ R and ℓ ∈ Z , and which is a weak solution of ∆v(x) = g(x, ω · x + v(x)) .
Proof. We consider the space of odd and periodic functions n 1,2 X := w ∈ Wloc (Rn ) s.t. w(x + ℓ) = w(x) = −w(−x) o for any x ∈ Rn and ℓ ∈ Zn .
Obviously, functions in X belong to W 1,2 (Tn ) and have zero average. In particular, kwkL2 (Tn ) ≤ Ck∇wkL2 (Tn )
(3.7)
if w ∈ X . We set
G(x, τ ) :=
Z
τ
g(x, s) ds .
0
Note that G is Zn+1 -periodic, thanks to (3.2) and (3.4). Thus, by (3.5), sup |G(x, τ )| = sup |G(x, τ )| ≤ Γ(x) τ ∈R
and so, if w ∈ X , (3.8)
Z
Tn
τ ∈[0,1]
G(x, ω · x + w(x)) dx ≤ kΓkL1 (Tn ) .
We consider the functional |∇w|2 + G(x, ω · x + w(x)) dx . 2 Tn We now follow the standard procedure of the direct methods and take a minimizing sequence vk ∈ X , that is Z
F (w) :=
lim F (vk ) = inf F (w) .
(3.9)
k→+∞
w∈X
Comparing with F (0) and exploiting (3.7), (3.8) and (3.6), one sees that kvk k2W 1,2 (Tn ) ≤ C kΓkL1(Tn ) ,
which, by compactness, gives the existence of a subsequence, say vkj , and a function v ∈ X , in such a way that ∇vkj converges to ∇v weakly in L2 (Tn ) and vkj converges to v almost everywhere. Then, Z Z (3.10) lim inf |∇v|2 dx . |∇vkj |2 dx ≥ j→+∞
Tn
Tn
Also, if hj (x) := g(x, ω ·x+vkj (x)), we have that hj (x) converges to g(x, ω ·x+v(x)) almost everywhere, due to (3.1). Accordingly, by Fatou’s Lemma, Z Z (3.11) lim inf hj (x) dx ≥ g(x, ω · x + v(x)) dx . j→+∞
Tn
Tn
So, by (3.9), (3.10) and (3.11), we infer that v ∈ X satisfies F (v) = inf F ≤ F(v + ǫφ) , X
ELLIPTIC EQUATIONS WITH GRADIENT DEPENDENCE
11
for any ǫ ∈ R and any φ ∈ X and so Z ∇v(x)∇φ(x) + g(x, ω · x + v(x))φ(x) dx = 0 (3.12) Tn
for any odd test function φ ∈ C ∞ (Tn ). Note that the maps x 7→ ∇v(x) and x 7→ g(x, ω · x + v(x)) are, respectively, even and odd by construction (recall (3.3) too), therefore (3.12) holds also for any even test function φ ∈ C ∞ (Tn ). Now, if φ ∈ C ∞ (Tn ) is any test function, we deduce that (3.12) holds for φ since it holds for φ(x) ± φ(−x) φ± (x) := 2 and φ = φ+ + φ− . 3.2. Iteration. We now iterate Lemma 3.2 via a recursion on the gradient term, by following a scheme used in [DFGM] and [GM]. 1,2 Lemma 3.3. There exists a sequence of functions uk ∈ Wloc (Rn ), with u0 = ω · x, in such a way that uk is a weak solution of
∆uk (x) = f (x, uk (x), ∇uk−1 (x))
(3.13)
for any x ∈ Rn and any k ∈ Z, k ≥ 1. Furthermore, if Uk (x) := uk (x) − ω · x, then, Uk (x + ℓ) = Uk (x) = −Uk (−x) ,
(3.14)
n
for any x ∈ R and any ℓ ∈ Zn .
Proof. We set u0 (x) := ω ·x, f0 (x, t) := f (x, t, ∇u0 (x)) = f (x, t, ω) and we suppose, 1,2 iteratively, that, for some k ∈ N, we have found uk ∈ Wloc (Rn ) which solves weakly ∆uk (x) = fk−1 (x, uk (x)) ,
with fk (x, t) := f (x, t, ∇uk (x)) ,
and satisfies
uk (x) − ω · x = Uk (x) = Uk (x + ℓ) = −Uk (−x) for any x ∈ R and any ℓ ∈ Zn . We would like now to apply Lemma 3.2 to the nonlinearity fk . To this extent, we check that fk satisfies the same hypothesis as the function g in the statement of Lemma 3.2. Indeed, fk (x, ·) = f (x, ·, ∇uk (x)) ∈ C(R) thanks to (1.2), and so fk satisfies (3.1). Furthermore, exploiting (1.3), (1.4) and (1.8), we infer that n
that
fk (x + ℓ, t + j) = f (x + ℓ, t + j, ω + ∇Uk (x + ℓ)) = f (x, t, ω + ∇Uk (x + ℓ)) = f (x, t, ω + ∇Uk (x)) = fk (x) , fk (−x, −t) = f (−x, −t, ω + ∇Uk (−x)) =
and that
−f (x, t, ω + ∇Uk (−x)) = −f (x, t, ω + ∇Uk (x)) = −fk (x, t) Z
t
t+j
fk (x, τ ) dτ =
Z
t
t+j
for any x ∈ Rn , t ∈ R, ℓ ∈ Zn and j ∈ Z.
f (x, τ, ∇uk (x)) dτ = 0
12
M. BERTI, M. MATZEU AND E. VALDINOCI
Hence, fk satisfies (3.2), (3.3) and (3.4). Moreover, if Γk (x) := sup |fk (x, τ )| , τ ∈R
1
we infer from Lemma 3.1 that Γk ∈ L (Tn ), therefore, (3.6) is fulfilled by Γk . The above considerations show that fk satisfies the assumptions of Lemma 3.2. 1,2 Then, we take Uk+1 ∈ Wloc (Rn ) to be the function obtained by Lemma 3.2 and which satisfies weakly ∆Uk+1 (x) = fk (x, ω · x + Uk+1 (x)) = f (x, uk+1 (x), ∇uk (x)) , with uk+1 (x) := ω · x + Uk+1 (x), and so that Uk+1 (x) = Uk+1 (x + ℓ) = −Uk+1 (−x) for any x ∈ Rn and any ℓ ∈ Zn , thus completing the iteration. We remark that the method used up to now is enough to obtain a uniform bound of kUk kW 1,2 (Tn ) , which would yield a convergence, up to subsequences, of both uk and ∇uk (almost everywhere and weakly, respectively). This is not enough for our purposes, since we need to pass to the limit (3.13): indeed, we would need both the convergence of ukj and ∇ukj −1 , for a suitable subsequence kj . This problem will now be overcome by showing that the whole sequences uk and ∇uk converge. To this extent, we will use the assumption in (1.9) and the smallness of λ⋆ . 3.3. Energy estimates. The following result is closely related to analogous estimates performed in [DFGM] and [GM] for bounded domains. Lemma 3.4. Let Uk be as in the statement of Lemma 3.3. Then, kUk+1 − Uk kW 1,2 (Tn ) ≤
1 kUk − Uk−1 kW 1,2 (Tn ) , 2
for any k ≥ 1. Proof. By Lemma 3.3, Z Z [f (x, uk , ∇uk−1 ) − f (x, uk+1 , ∇uk )] uj dx . (∇uk+1 − ∇uk ) · ∇uj dx = Tn
Tn
Taking j = k, k + 1, and subtracting, we obtain Z |∇(uk+1 − uk )|2 dx Z ≤ |f (x, uk , ∇uk−1 ) − f (x, uk+1 , ∇uk )| |uk+1 − uk | dx Tn Z ≤ λ⋆ (|uk+1 − uk | + |∇uk − ∇uk−1 |) |uk+1 − uk | dx Tn
≤ Cλ⋆ (kuk+1 − uk k2W 1,2 (Tn ) + kuk − uk−1 k2W 1,2 (Tn ) ) ,
thanks to (1.9). Note that uk+1 − uk = Uk+1 − Uk ∈ W 1,2 (Tn ) has zero average, since so does Uk , due to (3.14). Therefore, we deduce from the estimate above that kUk+1 − Uk k2W 1,2 (Tn ) ≤ Cλ⋆ (kUk+1 − Uk k2W 1,2 (Tn ) + kUk − Uk−1 k2W 1,2 (Tn ) ) , which gives the desired result, provided that λ⋆ is suitably small.
ELLIPTIC EQUATIONS WITH GRADIENT DEPENDENCE
13
3.4. End of the proof of Theorem 1.2. By Lemma 3.4, uk converges to some u 1,2 in Wloc (Rn ). Also, given φ ∈ C0∞ (Rn ), Z [f (x, uk , ∇uk−1 ) − f (x, u, ∇u)] φ dx Z |uk − u| + |∇uk−1 − ∇u| dx ≤ λ⋆ kφkL∞ (Rn ) spt φ
≤ Cφ (kuk − uk2W 1,2 (Tn ) + kuk−1 − uk2W 1,2 (Tn ) )
thanks to (1.9). Therefore, by Lemma 3.4, for any φ ∈ C0∞ (Rn ), Z 0 = lim ∇uk ∇φ + f (x, uk , ∇uk−1 )φ dx k→+∞ Z = ∇u∇φ + f (x, u, ∇u)φ dx ,
thence u is a weak solution of (1.1) for any x ∈ Rn . Also, the convergence in W 1,2 (Tn ) of Uk implies the existence of a subsequence, say Ukj which converges to U almost everywhere. Since (3.14) holds for Ukj , we conclude that (1.6) holds for U . In particular, U is Zn -periodic and has zero average. Thus, we gather from (1.1) and Lemma 3.1 that Z 1 kU k2W 1,2 (Tn ) ≤ |∇U |2 dx C Tn Z =− f (x, u, ∇u) U dx n (3.15) ZT ≤ λ⋆ |U | Tn
≤ λ⋆ kU kW 1,2 (Tn ) .
Such estimate, Lemma 3.1 and elliptic regularity theory imply that (1.10) holds, thus completing the proof of Theorem 1.2. ♦ 4. Proof of Corollary 1.3 Let c be as in Theorem 1.1 and fix λ∗ = c/2. Note that f satisfies (1.5), thanks to (1.11). More precisely, by Young Inequality, Λ0 (1 + |p|a ) =
Λ0 +
aa Λ0 λa⋆ |p|a · λa⋆ aa
≤
Λ0 +
(1 − a) aa/(1−a) Λ0
1/(1−a)
a/(1−a)
λ⋆
+ λ⋆ |p| ,
thence f satisfies (1.5), with 1/(1−a)
Λ⋆ := Λ0 +
(1 − a) aa/(1−a) Λ0 a/(1−a) λ⋆
with
λ⋆ =
c . 2
We then use Theorem 1.1 to find the desired solution u(x) = ω · x + U (x).
14
M. BERTI, M. MATZEU AND E. VALDINOCI
It only remains to prove (1.13). For this, we set g(x) := f (x, u(x), ∇u(x)) and we note that (4.1)
∆U = g(x)
weakly and that g is Zn -periodic, due to (1.3) and (1.6). Also, if q ∈ (1, 2/a], Z 1/q kgkLq (Tn ) ≤ Ca,q,Λ0 (1 + |ω|a + |∇U |a )q dx Tn
≤ Ca,q,Λ0
(4.2)
a
1 + |ω| +
Z
Tn
aq
|∇U |
1/q ! dx
≤ Ca,q,Λ0 1 + |ω|a + k∇U kaL2 (Tn ) ≤ Ca,q,Λ0 1 + |ω|a + (Λ⋆ + λ⋆ |ω|)a ≤ Ca,q,Λ0 (1 + |ω|a ) ,
thanks to (1.11) and (1.7). We now use (1.7), (4.1), (4.2) and the De Giorgi-Nash-Moser theory (see, e.g., Theorem 4.1 in [HL], applied here with q := 2/a > n/2, where (1.11) is used again) to obtain that sup |U | ≤ Ca,Λ0 kU kL2 (Tn ) + kgkL2/a(Tn ) ≤ Ca,Λ0 (1 + |ω| + |ω|a ) ≤ Ca,Λ0 (1 + |ω|) . Tn
In particular, (4.3)
kU kLp(Tn ) ≤ Ca,p,Λ0 (1 + |ω|) ,
for any p > 1. Thus, by the Calder´ on-Zygmund estimates (see, e.g., Theorem 9.11 in [GT]), kU kW 2,p (Tn ) ≤ Cp (kU kLp(Tn ) + kgkLp(Tn ) ) ≤ Ca,p,Λ0 (1 + |ω|) , for any p ∈ (1, 2/a], thanks to (4.2) and (4.3). Taking p := 2/a > n (recall (1.11) once more) and using the Sobolev Embedding (see, e.g., page 270 of [E]), the above estimate implies (1.13), concluding the proof of Corollary 1.3. ♦ 5. Proof of Corollary 1.4 Let ωN := ωN ∈ Zn and
p . fN (x, t, p) := N 2 f N x, t, N
We apply Corollary 1.3 to get uN (x) = ωN · x + UN (x) which is a weak solution of ∆uN = fN (x, uN , ∇uN ) and which satisfies (1.6). The function u(x) := uN (x/N ) thus fulfills the claim of Corollary 1.4. ♦
ELLIPTIC EQUATIONS WITH GRADIENT DEPENDENCE
15
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[email protected] ` di Roma Tor Vergata, Dipartimento di Matematica, I-00133 Rome, (RM ) Universita Italy, email: matzeu,
[email protected]
