Existence of very weak solutions to elliptic systems of p-Laplacian type

0.2 MB·

by  Miroslav Bulíček

EXISTENCE OF VERY WEAK SOLUTIONS TO ELLIPTIC SYSTEMS OF
p-LAPLACIAN TYPE
MIROSLAVBUL´ICˇEK ANDSEBASTIAN SCHWARZACHER
Abstract. Westudyvectorvaluedsolutionstonon-linearellipticpartialdifferentialequa-
tions with p-growth. Existence of a solution is shown in case the right hand side is the
divergenceofafunctionwhichisonlyq integrable,whereq isstrictlybelowbutclosetothe
duality exponent p′. It implies that possibly degenerate operators of p-Laplacian type are
well posed in a larger class then the natural space of existence. The key novelty here is a
6
1 refined a priori estimate, that recovers a duality relation between the right hand side and
0 thesolution in terms of weighted Lebesgue spaces.
2

n
a
J 1. Introduction

0
3 LetΩ ⊂Rd beaLipschitz domainandS :Ω×Rd×N → Rd×N beaCarath´eodory mapping.
We investigate the existence of a very weak solution u: Ω → RN with N ∈ N of the system

]
P divS(·,∇u) = div|f|p−2f in Ω,
A (1.1)
u= 0 on ∂Ω.
.
h
Here we assume growth, coercivity and monotonicity assumptions on S related to the expo-
t
a
nent p ∈ (1,∞). Explicitly, that there exist constants C ,C > 0 and C ≥ 0, such that for
m 1 2 3
all z ,z ∈ Rd×N and almost all x ∈ Ω there holds
1 2
[
1 (1.2) S(x,z1)·z1 ≥ C1|z1|p−C3, coercivity
v p−1
2 (1.3) |S(x,z1)| ≤ C2|z1|p−1+C3p , boundedness
2
(1.4) (S(x,z )−S(x,z ))·(z −z ) ≥ 0, monotonicity.
1 1 2 1 2
0 The model case is the p-Laplacian system
0
. div|∇u|p−2∇u= div|f|p−2f in Ω,
2
(1.5)
0 u = 0 on ∂Ω.
6
1 It is well known, that if assumptions (1.2)–(1.4) are satisfied and f ∈ Lp(Ω;Rd×N), then a
:
v solution of (1.1) exists with ∇u∈ Lp(Ω). It can for instance be shown by monotone operator
i
X theory. The starting point of our investigations is the following question.
r Q: Does for f ∈ Lq(Ω;Rd×N) and q 6= p a distributional solution u of (1.1) exist, such
a that ∇u∈ Lq(Ω;Rd×N)?
Under general assumptions (1.2)–(1.4) the answer to this question is not affirmative for
all q ∈ (1,∞). This is well known due to the various counterexamples even in the simplest
case p = 2 and f ≡ 0. At the end of exponents smaller then 2 we mention the non-smooth
solutions constructed by Serrin [19]. At the end of exponents larger than 2 the non-smooth
2010 Mathematics Subject Classification. 35D99,35J57,35J60,35A01.
Keywordsandphrases. nonlinearellipticsystems,weightedestimates,existence,veryweaksolution,mono-
tone operator, div–curl–biting lemma, weighted space, Muckenhouptweights.
M. Bul´ıˇcek’s work is supported by the project LL1202 financed by the Ministry of Education, Youth and
Sports, Czech Republic and by the University Centre for Mathematical Modelling, Applied Analysis and
Computational Mathematics (Math MAC) S. Schwarzacher thanks the program PRVOUK P47, financed by
Charles University in Prague. M. Bul´ıˇcek is a member of the Neˇcas Center for Mathematical Modeling.
1
2 M.BUL´ICˇEKANDS.SCHWARZACHER
solutions constructed by Neˇcas [18]. The example of Serrin [19] implies also that there can
be no hope for uniqueness in the large class of W1,q(Ω;RN) with q ∈ (1,p), unless stricter
0
assumptions are available.
However, it turns out that closely around p the existence of solutions with natural inte-
grability is true, even under the minimal assumptions (1.2)–(1.4).
More precisely, we are able to prove the following theorem.
Theorem 1.1. Let Ω ⊂ Rd be a bounded Lipschitz domain and S satisfy (1.2)–(1.4). Then
there is an ε depending on C ,C ,d,N, p and Ω, such that for all q ∈ [p−ε,p] the following
1 2
holds. If f ∈ Lq(Ω;Rd×N), then there exists u ∈ W1,q(Ω;RN) which is a distributional
0
solution to (1.1).
Moreover, there is a constant c only depending on C ,C ,p,q,d,N,Ω and a constant c
1 2 1
depending additionally on C , such that
3
(1.6) k∇uk ≤ ckfk +c .
Lq(Ω;Rd×N) Lq(Ω;Rd×N) 1
As was mentioned before, this result is optimal with respect to the generality of the as-
sumptions (1.2)–(1.4). Let us briefly collect what was already known before.
1,p
In case q > p, the existence of a solution in W (Ω) is obvious. The integrability improve-
0
ment follows by an argument known as Gehring’s Lemma. With the minimal assumptions
(1.2)–(1.4)it was proved in[16, Theorem7.8]. Seealso themoreclassical result[11, Theorem
4.1] and for an overview and more details [17]. For the p-Laplacian, it is known that (1.6)
holds also in the case of large exponents q ∈ [p,∞) [13, 5] and beyond [6, 9].
In the special case p = 2, it was possible to show that there exists an ε depending on
C ,C and Ω alone, such that existence, uniqueness and regularity are available in case
1 2
f ∈ Lq(Ω) and q ∈ [2−ε,2+ε] [3]. However, it needed the stricter assumption, that S is
Lipschitz continuous with respect to z. Existence, Uniqueness and Regularity for the full
range q ∈ (1,∞) has recently been shown to hold, in case p = 2 with S having additional
Uhlenbeck type structure [4].
In the case of p 6= 2 and q < p very little is known about the existence of a distributional
solution. The situation is quite delicate, since even for bounded domains the existence of
any object of solution is not obvious. The only existence result available related to powers
q < p for the p-Laplacian are restricted to the scalar case N = 1 and to a better structure
of the right hand side. More precisely, when the right hand side is a function (or a Radon
measure) [1]. But even for solutions to (1.5) the existence of solutions was not known in
the case of p 6= 2 and q < p. This seems astonishing, since the a priori estimates (1.6) are
available for solutions to (1.5) in case Rd = Ω ever since the seminal work of Iwaniec [14] in
1992. There it is showed, that there is an ε > 0 that provided a distributional solution to the
p-Laplace exists, it holds already (1.6) for q ∈ [p−ε,p]. Greco, Iwaniec and Sbordone could
stretch the existence frame slightly by showing existence provided the right hand side is in
the grande Lebesgue space, f ∈ Lp)(Ω;Rd×N) [12]. It is a space, that is slightly larger then
Lp(Ω;Rd×N) quantified in terms of logarithmic powers. Later the a priori estimates could be
extended to the parabolic case in [15].
After having collected all above efforts we conclude that the existence was open ever since
1992 for q < p and could not be closed ever since. Therefore our main result is the existence
of a distributional solution in case q ∈ [p−ε,p] and not the estimate (1.6), although it is new
in this general form.
The key observation is, that the Lq a priori estimate alone is not suitable to establish
solutions to non-linear operators. This is due to the fact that the a priori estimates inherit
only weak compactness orless andtheonlypossibleway tomach weak compactness andnon-
liniarities is via convexity. In the setting here it is reflected via the use of the monotonicity,
for instance via the Minty method. However, the method seemed lost the moment the
weak limit is not a suitable test function anymore. Only recently a new point of view was
VERY WEAK SOLUTIONS TO SYSTEMS WITH p-GROWTH 3
established, which allowed to regain a duality relation between f and ∇u [4]. The duality
is gained by replacing Lq estimates with weighted Lp estimates, where the weight is chosen
ω
in terms of the right hand side f. It has to be chosen in such a way that it belongs to the
Muckenhoupt class A . This is necessary since it is known, that many linear and sub-linear
q
operators including the Laplace and the Maximal operator are bounded restrictively in these
Muckenhoupt classes.
The key novelty, which is of its sovereign interest is the a priori estimate in the existence
and regularity result below.
Theorem 1.2. Let Ω ⊂ Rd be a bounded Lipschitz domain and S satisfy (1.2)–(1.3). Then
′
Cp
there isanεdepending on 2 , p,d,N and Ω, suchthat forall q ∈ [p−ε,p] the following holds.
C1
If f ∈ Lq(Ω;Rd×N), then there exists u ∈ W1,q(Ω;RN) which is a distributional solution to
0
(1.1).
Moreover, there is a constant c depending on C ,C ,p,q,d,N and Ω, such that1
1 2
(1.7) |∇u|p(M(f +1))q−pdx ≤c |f|qdx+c(C +1).
3
Z Z
Ω Ω
Let us point out that the above estimate measures ∇u more accurate in relation to the
right hand side. For once we find by (5.10) that (1.7) implies (1.6). But it also implies
for instance that non p-integrable singularities of ∇u, can only appear in areas, where f is
large, quantified by the naturally related weight. We therefore believe, that estimates of the
above type will be of increasing importance in the framework of the existence theory in many
applications. Moreover, we wish to indicate its potential for numerical analysis, especial its
use for adaptive schemes.
After little restrictions the method can easily be applied on unbounded domains.
Corollary 1.3. Let Ω ⊂ Rd be a Lipschitz domain and S satisfy (1.2)–(1.3), with C = 0.
3
′
Cp
Then there is an ε depending on 2 , p,d,N and Ω, such that for all q ∈ [p−ε,p] the following
C1
holds. If f ∈ Lq(Ω;Rd×N), then there exists u ∈ D1,q(Ω;RN)2 which is a distributional
0
solution to (1.1).
Moreover, there is a constant c only depending on C ,C ,p,q,d,N and Ω, such that
1 2
(1.8) |∇u|q +|∇u|p(Mf)q−pdx ≤ c |f|qdx.
Z Z
Ω Ω
The structure of the paper is as follows. We introduce the necessary notation in the
preliminary below. In Section 3 we introduce a truncation method of Sobolev functions,
relative to an open set, which is the analytical highlight of this article. In Section 4 we
deduce the a-priory estimates and in Section 5 we prove the existence.
2. Preliminary
Throughout the paper all cubes will have sides parallel to the axes. By c,C we denote a
generic constant, i.e. its value may change at every appearance. Its dependencies are either
stated in the results or are indicated in C(...). Let us recall the definition of the Hardy
Littlewood maximal function. For any f ∈ L1 (Rn) we define
loc
1
Mf(x):= sup h|f|i with h|f|i := − |f(y)|dy := |f(y)|dy.
Q Q
Z |Q|Z
{Qacube:x∈Q} Q
Q
It is standard, that the operator is sub linear and continuous from Ls(Ω) → Ls(Ω), for
s ∈ (1,∞]. Further, we say that ω : Rd → R is aweight function if it is ameasurable function
1Here M is theHardy Littlewood maximal operator that is definedin Section 2.
2D1,q(Ω;RN):={u∈L1 (ω;RN):∃uj ∈C∞(Ω;RN) s.t. ∇uj →∇uin Lq(Ω;Rd)}.
0 loc 0
4 M.BUL´ICˇEKANDS.SCHWARZACHER
that is almost everywhere finite and positive. For such a weight and arbitrary measurable
Ω ⊂ Rd we denote the space Lp(Ω;RN) with p ∈ [1,∞) as
ω
1
p
Lpω(Ω,Rd) := (cid:26)f : Ω→ RN, measurable : kfkLpω := (cid:18)Z |f(x)|pω(x)dx(cid:19) < ∞(cid:27).
Ω
We introduce the weighted Sobolev spaces, as
W1,q(Ω;RN) := {u ∈ W1,1(Ω;RN): ∇u∈ Lq(Ω;RN)}
ω ω
W1,q(Ω;RN) := W1,1(Ω;RN)∩W1,q(Ω;RN).
0,ω 0 ω
Next, for p ∈[1,∞), we say that a weight ω belongs to the Muckenhoupt class A if and only
p
if there exists a positive constant A such that for every cube Q ⊂ Rn the following holds
1
(2.1) − ωdx − ω−(p′−1)dx p′−1 ≤ A if p ∈ (1,∞),
(cid:18)Z (cid:19)(cid:18)Z (cid:19)
Q Q
(2.2) Mω(x) ≤ Aω(x) if p = 1.
In what follows, we denote by A (ω) the smallest constant A for which the inequality (2.1),
p
resp. (2.2), holds.
The next result makes a very useful link between the maximal operator and A -weights.
p
Lemma 2.1 (See pages 229–230 in [20] and page 5 in [21]). Let f ∈ L1 (Rn) be such
loc
that Mf < ∞ almost everywhere in Rn. Then for all α ∈ (0,1) we have (Mf)α ∈ A .
1
Furthermore, for all p ∈ (1,∞) and all α ∈ (0,1) there holds (Mf)−α(p−1) ∈ A .
p
3. Relative Truncation
In this section we introduce a relative truncation. It is strongly influenced by the Lips-
chitz truncation method, developed in [10] see also [2, 7] where the concept was refined in
the direction which can be adapted here. In contrast to the Lipschitz truncation, the rela-
tive truncation smoothens the function relative to a different independent function, or more
explicit, it truncates the gradient on any given open set.
The first step is a suitable covering. We take the covering introduced in [8, Proposition
3.17] and proof some more properties needed for our situation.
Proposition 3.1. Let O be an open subset of Rd, with O =6 Rd. Then there exists a countable
family Q of closed, dyadic cubes such that
i
(a) Q = O and all cubes Q have disjoint interiors.
i i i
(b) Sdiam(Qi) < dist(Qi,Oc) ≤4diam(Qi)
(c) If Q ∩Q 6= ∅, then diam(Q ) ≤ 2diam(Q ) ≤ 4diam(Q ).
i j i j i
(d) For given Q , there exists at most 4d−2d cubes Q touching Q (boundaries intersect
i j i
but not the interiors), we define A as the index set of all neighboring cubes of Q .
i i
(e) The family of cubes {3Q } has finite intersection. The family can be split in 4d−2d
2 i i∈N
pairwise disjoint subfamilies.
(f) There is a partition of unity, ψ ∈ C0,1(9Q ), such that χ ≤ ψ ≤ χ and
i 8 i 21Qi i 98Qi
diam(Q )|∇ψ |≤ c(d) uniformly.
i i
Proof. Since (a)–(d) are stated in [8, Proposition 3.17] we just proof the last two. First we
show that the cubes 3Q have finite intersection. Firstly, by (b), we find that 3Q ⊂ O,
2 i 2 i
which implies bu (a), that 3Q = Q . Secondly, we find by (c) that 3Q only intersects
i 2 i i i 2 j
its neighbors which impliesSby (d) thSat each x ∈ Qi is at most covered by 4d −2d cubes of
the family 3Q . Moreover, by (c) we find that the 9Q does not intersect with 1Q , for all
2 j 8 i 2 j
j 6= i. Therefore it is standard to construct a partition of unity as requested. (cid:3)
VERY WEAK SOLUTIONS TO SYSTEMS WITH p-GROWTH 5
Next we introduce the relative truncation of u ∈ W1,p(Ω;RN), with respect to O an open
0
proper set. Since we do not assume, that O ⊂ Ω we extend u by 0 outside Ω and define
u(x) x ∈ Rd\O,
u (x) := 
O  ψiu¯i x ∈ O.
Xi

Where the u¯ are defined via the covering constructed in Proposition 3.1:
i
9
− udx if Q ⊂ Ω,
i
 Z 8
u¯ :=
i  98Qi
0 otherwise.


At first we have to proof the following lemma, which is essential to show that the relative
truncation is stable in Sobolev function spaces.
Lemma 3.2. Let Ω be a Lipschitz domain and let u ∈W1,p(Ω;RN), for p ∈ [1,∞). Then for
0
any Q ,Q which are members of the covering introduced in Proposition 3.1 with non-empty
i j
intersection, we have
|u¯ −u¯ |
j i
(3.1) ≤ c(d,Ω) − |∇u|dx+c(d,Ω) − |∇u|dx
diam(Qi) Z Z
32Qi 23Qj
u¯−u¯ p
i p
(3.2) − ≤ c(d,p,Ω) − |∇u| dx
Z (cid:12)diam(Qi)(cid:12) Z
98Qi(cid:12)(cid:12) (cid:12)(cid:12) 23Qi
Proof. The statement of (3.2), in case u¯ 6= 0 is just the Poincar´e inequality. In case u¯ = 0
i i
we enlarge the cube to 3Q . Since we have a Lipschitz boundary and 9Q ∩O 6= ∅ we find
2 i 8 i
that |3Q | ≤ c(Ω)|3Q ∩Ωc|. This implies, that we can apply Poincar´e’s inequality on u for
2 i 2 i
the cube 3Q , which finishes the proof of (3.2).
2 i
To proof (3.1) observe at first, that since Q and Q are neighbours, we find by (c) that
i j
C = 9Q ∩ 9Q have a comparible maesure, to |3Q | and |3Q |. Since in case u¯ = u¯ = 0,
i,j 8 i 8 j 2 i 2 j i j
there is nothing to show, let us assume that u¯ 6= 0. By Poincar´e we find
i
|u¯ −u¯ | ≤ u¯ − − udx + − u−hui dx
j i (cid:12) j Z (cid:12) (cid:12)Z 98Qi (cid:12)
(cid:12)(cid:12) Ci,j (cid:12)(cid:12) (cid:12)(cid:12)Ci,j (cid:12)(cid:12)
≤ u¯ − − udx +c(d) − |u−hui |dx ≤ u¯ − − udx +c(d)diam(Q ) − |∇u|dx
(cid:12) j Z (cid:12) Z 89Qi (cid:12) j Z (cid:12) i Z
(cid:12)(cid:12) Ci,j (cid:12)(cid:12) 89Qi (cid:12)(cid:12) Ci,j (cid:12)(cid:12) 98Qi
In case u¯ 6= 0 as well, then we find by symmetry
j
u¯ − − udx ≤ c(d)diam(Q ) − |∇u|dx.
j j
(cid:12) Z (cid:12) Z
(cid:12)(cid:12) Ci,j (cid:12)(cid:12) 89Qj
In case u¯ = 0 weenlarge theset C to 3Q for which weknow that |3Q |≤ c(Ω)|3Q ∩Ωc|.
j i,j 2 j 2 j 2 j
Therefore, finally Poincar´e’s inequality implies
− |u|dx ≤ c(d) − |u|dx ≤ c(d,Ω)diam(Q ) − |∇u|dx.
j
Z Z Z
Ci,j 32Qj 32Qj
(cid:3)
This implies, that the relative truncation is stable in Sobolev spaces.
6 M.BUL´ICˇEKANDS.SCHWARZACHER
Lemma 3.3 (Stability). Let u ∈ W1,p(Ω;RN), then u ∈ W1,p(Ω;RN). Moreover, the
0 O 0
following estimate holds
p p
|∇(u−u )| dx ≤c(d,p,Ω) |∇u| dx.
O
Z Z
Ω O∩Ω
Proof. It is enough to show the estimate since the zero-trace follows from the very definition
of the relative truncation immediately. We define by A the index set of all j, such that
i
9Q ∩ 9Q 6= ∅. Observe here, that #A ≤ 4n−2n by (d). Next we use (a), the fact that we
8 j 8 i j
have a partition of unity, (e), (c) and (f).
p
p p
|∇(u−u )| dx= |∇(u−u )| dx = ∇u− ∇ψ u¯ dx
O O j j
Z ZO Xi ZQi(cid:12)(cid:12) jX∈Ai (cid:12)(cid:12)
(cid:12) (cid:12)
p
p
≤ ∇(u−ψ u¯ ) dx≤ c(d,p) |∇(u−ψ u¯ )| dx
j j j j
Xi ZQi(cid:12)(cid:12)jX∈Ai (cid:12)(cid:12) Xi jX∈AiZQi
(cid:12) (cid:12) u−u p
p j
≤ c(d,p) |∇u| dx+c(d,p) dx
Xi ZQi Xi jX∈AiZQi(cid:12)(cid:12)diam(Qi)(cid:12)(cid:12)
(cid:12) (cid:12)
Now (a),(3.1) and (3.2) imply together with Jensen’s inequality, (e) and (b) that
p p p p
|∇(u−u )| dx ≤ c(d,p) |∇u| dx+c(d,p,Ω) |∇u| dx ≤ c(d,p) |∇u| dx.
O
Z ZO Xi Z32Qi ZO
(cid:3)
4. Uniform a priori estimates
In this section, we will not mention the dependence of the constants on d,N anymore.
Proposition 4.1. Let Ω ⊂ Rd be a bounded Lipschitz domain, S satisfy (1.2)–(1.3) and
′
h ∈ L1(Ω) not identically equal to zero. Then there exists ε > 0 depending only on C2p , p and
0 C1
Ω such that for any ε ∈ (0,ε ) the following holds. If f ∈ Lp(Ω;Rd×N)∩Lp (Ω;Rd×N)
0 (Mh)−ε
and u∈ W1,p(Ω;RN) satisfies (1.1) in a weak sense, then ∇u∈ Lp (Ω;Rd×N) as well.
0 (Mh)−ε
Moreover,
|∇u|p |f|p+C
3
(4.1) dx ≤ C(C ,C ,p,Ω) dx.
Z (Mh)ε 1 2 Z (Mh)ε
Ω Ω
Proof. First, we extend u and f with zero outside Ω. The function h, we will approximate
by g = hχ + δ. This implies, that Mg > δ and therefore f,∇u ∈ Lp (Ω;Rd×N) a
Ω (Mg)−ε
priori. Once the estimate is established for g with constants independent of δ the general
result follows by letting δ → 0 and monotone convergence. For any λ > 0 we define O(λ) :=
{x ∈ Rd : Mg(x) > λ}. Since the maximal function is sub-linear it is an open set. In case
O(λ) = Rn, we define u = 0. Else, we are able to construct the relative truncation u .
O(λ) O(λ)
Testing (1.1) with u which is possible due to Lemma 3.3 we get the identity
O(λ)
S ·∇udx = |f|p−2f ·∇udx+ |f|p−2f ·∇u dx
O(λ)
Z Z Z
{Mg≤λ} {Mg≤λ} {Mg>λ}
(4.2)
− S ·∇u dx.
O(λ)
Z
{Mg>λ}
Next, we focus on the estimates of integrals on the set where Mg > λ. Consider arbitrary
G ∈ Lp′(Ω;Rd×N) and arbitrary α∈ (0,1). We get using (f)–the property of the partition of
VERY WEAK SOLUTIONS TO SYSTEMS WITH p-GROWTH 7
unity, (3.1), (e) H¨older’s inequality and (b)
(4.3)
|G·∇u |dx ≤ |G| u¯ ∇ψ dx
O(λ) j j
Z{Mg>λ} Xi ZQi (cid:12)(cid:12)jX∈Ai (cid:12)(cid:12)
(cid:12) (cid:12)
= |G| (u¯ −u¯ )∇ψ dx
j i j
Xi ZQi (cid:12)(cid:12)jX∈Ai (cid:12)(cid:12)
(cid:12) (cid:12)
≤ c(Ω) |Q | − |G|dx − |∇u|dx+ − |∇u|dx
i
(cid:18)Z (cid:19)(cid:18) Z Z (cid:19)
Xi jX∈Ai Qi 32Qi 23Qj
|G| α |∇u| α |∇u| α
= c(Ω)Xi jX∈Ai|Qi|(cid:18)ZQ−i (Mg)αp (Mg)p dx(cid:19)(cid:18) 32Z−Qi (Mg)pα′ (Mg)p′ dx+ 23Z−Qj (Mg)pα′ (Mg)p′ dx(cid:19)
|G|p′ p1′ |∇u|p p1 |∇u|p p1
≤ c(Ω) |Q | − dx − dx + − dx
Xi jX∈Ai i (cid:18)ZQi (Mg)αpp′ (cid:19) (cid:18)(cid:18) 32ZQi (Mg)αpp′ (cid:19) (cid:18) 32ZQi (Mg)αpp′ (cid:19) (cid:19)
× − (Mg)αdx .
(cid:18)Z (cid:19)
5Qi
We estimate the last integral on the right hand side using two properties. First, since by
Lemma 2.1 (Mg)α is an A Muckenhoupt weight, M(Mg)α ≤ C(α,d)(Mg)α. Second, by (a)
1
c c
we know that 9Q ∩O(λ) 6= ∅. Consequently, for some x ∈ 9Q ∩O(λ) , we have
i 0 i
− (Mg)αdx ≤ c − (Mg)αdx≤ cM(Mg)α(x )
0
Z Z
5Qi 9Qi
≤ c(α)(Mg)α(x ) ≤ c(α)λα.
0
We can use this to estimate (4.3) in the following way. For i ∈ N and j ∈ A , we deduce by
i
Young’s inequality
λα|Q | − |G|p′ dx p1′ − |∇u|p dx p1
i (cid:18)Z (Mg)αpp′ (cid:19) (cid:18) Z (Mg)αpp′ (cid:19)
Qi 32Qj
(4.4)
′
λαpp |G|p′ λαpp′ |∇u|p
≤ c(Ω,α)|Q | − dx+ − dx .
i (cid:18)Z (Mg)αpp′ Z (Mg)αpp′ (cid:19)
Qi 23Qj
Therefore (4.3) implies using (e), that
′
λαpp |G|p′ λαpp′ |∇u|p
(4.5) |G·∇u |dx ≤ c(Ω,α) dx+ dx.
Z{Mg>λ} O(λ) Z{Mg>λ}∩Ω (Mg)αpp′ (Mg)αpp′
Consequently, we get from (4.2) that
(4.6)
′
λαpp (|f|p+|S|p′) λαpp′ |∇u|p
S ·∇udx ≤ |f|p−1|∇u|dx+c(Ω,α) + dx.
Z{Mg≤λ} Z{Mg≤λ} Z{Mg>λ} (Mg)αpp′ (Mg)αpp′
8 M.BUL´ICˇEKANDS.SCHWARZACHER
We set (p−1) := min((p−1),(p−1)−1) and take ε ∈ (0,α(p−1)). We multiply the above
inequality by λ−1−ε and integrate over λ ∈(0,∞) to deduce
∞ S ·∇u ∞ |f|p−1|∇u|
dxdλ ≤ dxdλ
Z Z λ1+ε Z Z λ1+ε
0 {Mg≤λ} 0 {Mg≤λ}
(4.7) ′
∞ λαpp −1−ε(|f|p+|S|p′) λαpp′−1−ε|∇u|p
+c(Ω,α) + dxdλ.
Z0 Z{Mg>λ} (Mg)αpp′ (Mg)αpp′
We get on the one hand using the Fubini theorem
1 S ·∇u ∞ S ·∇u ∞ S ·∇u
dx= dλdx = dxdλ;
ε Z (Mg)ε Z Z λ1+ε Z Z λ1+ε
Ω Ω Mg 0 {Mg≤λ}
Thus on the other hand by (4.7) and the Fubini theorem
1 S ·∇u ∞ |f|p−1|∇u|
dx≤ dxdλ
ε Z (Mg)ε Z Z λ1+ε
Ω 0 {Mg≤λ}
′
∞ λαpp −1−ε(|f|p+|S|p′) λαpp′−1−ε|∇u|p
+c(Ω,α) + dλdx
Z0 Z{Mg>λ} (Mg)αpp′ (Mg)αpp′
′
∞ |f|p−1|∇u| Mg λαpp −1−ε(|f|p+|S|p′) λαpp′−1−ε|∇u|p
= dλ+c(Ω,α) + dλdx
ZΩZMg λ1+ε Z0 (Mg)αpp′ (Mg)αpp′
1 |f|p−1|∇u| 1 (|f|p+|S|p′) 1 |∇u|p
= dx+c(Ω,α) + dx
ε ZΩ (Mg)ε ZΩ αpp′ −ε (Mg)ε αpp′ −ε(Mg)ε
1 |f|p−1|∇u| c(Ω,α) |f|p+|S|p′ +|∇u|p
≤ dx+ dx.
ε ZΩ (Mg)ε α(p−1)−ε ZΩ (Mg)ε
By the assumptions (1.2),(1.3) and Young’s inequality we deduce
|∇u|p c(p) |f|p εCp′c(Ω,p) |f|p+|∇u|p+C
(4.8) dx ≤ dx+ 2 3 dx.
ZΩ (Mg)ε C1 ZΩ (Mg)ε C1α((p−1)−ε) ZΩ (Mg)ε
Thus, for ε ∈ (0,ε ), we can absorb the term with |∇u|p on the right hand side of (4.8) into
0
the left hand side and we get (4.1) by letting δ → 0. Here
C (p−1)
1
(4.9) ε = ,
0 p′
C c(Ω,p)
2
since by this choice we can choose for ε ∈ (0,ε ) an α accordingly. Observe, that α → 1
0
(which is the non-stable limit), when ε→ ε . (cid:3)
0
5. Existence of a solution
The proof is making essential use of the following theorem that can be found in [4, Theo-
rem 2.6].
Theorem 5.1 (weighted, biting div–curl lemma). Let Ω ⊂ Rn be an open bounded set.
Assume that for some p ∈ (1,∞) and given ω ∈ A we have a sequence of vector-valued
p
measurable functions (ak,bk)∞ : Ω → Rn×Rn such that
k=1
(5.1) sup |ak|pω+|bk|p′ωdx < ∞.
k∈NZΩ
Furthermore, assume that for every bounded sequence {ck}∞ from W1,∞(Ω) that fulfills
k=1 0
∇ck ⇀∗ 0 weakly∗ in L∞(Ω)
VERY WEAK SOLUTIONS TO SYSTEMS WITH p-GROWTH 9
there holds
(5.2) lim bk ·∇ckdx = 0,
k→∞ZΩ
(5.3) lim ak∂ ck −ak∂ ckdx = 0 for all i,j = 1,...,n.
k→∞ZΩ i xj j xi
Then there exists a subsequence (ak,bk) that we do not relabel and there exists a non-
decreasing sequence of measurable subsets E ⊂ Ω with |Ω\E |→ 0 as j → ∞ such that
j j
(5.4) ak ⇀ a weakly in L1(Ω;Rn),
(5.5) bk ⇀ b weakly in L1(Ω;Rn),
(5.6) ak ·bkω ⇀ a·bω weakly in L1(E ) for all j ∈ N.
j
Proof of Theorem 1.2. Asbefore,weextendeveryfunctionbyzerooutsideΩ,withoutfurther
reference. We approximate a given f ∈ Lq(Ω;Rd×N), by f := min{k,|f|}f/|f|. Then
k
f ∈Lq(Ω;Rd×N)∩L∞(Ω;Rd×N), we find |fk| ր |f| and
k
(5.7) fk → f strongly in Lq(Rn;Rn×N).
Forfk wecanusethestandardmonotoneoperatortheorytofindasolutionuk ∈ W1,p(Ω;RN)
0
fulfilling
(5.8) S(x,∇uk)·∇ϕdx = fk ·∇ϕdx for all ϕ∈ W1,p(Ω;RN).
Z Z 0
Ω Ω
Hence, we fix ε = p−q ∈ (0,ε ). Then we find by (4.1) and the continuity of the maximal
0
function that
|∇u|p |f |p+C
k 3
dx ≤C(C ,C ,p,Ω) dx
Z (M(f +1))ε 1 2 Z (M(f +1))ε
(5.9) Ω Ω
≤c M(f)qdx+cC ≤ c |f|qdx+cC .
3 3
Z Z
Moreover, by Young’s inequality for the exponents q + p−q = 1, continuity of the maximal
p p
function and the previous we gain
(p−q)q
|∇u|q(M(f +1)) p
|∇u|qdx = dx
Z Z (p−q)q
(5.10) Ω Ω (M(f +1)) p
|∇u|p
≤ c dx+c (M(f +1))qdx ≤ c |f|qdx+c(C +1).
Z (M(f +1))ε Z Z 3
Ω Ω
We define the weight ω : 1 . Using the a priori estimate, the reflexivity of the corre-
(M(f+1))ε
spondingspaces andthegrowthassumption(1.3),wecanpasstoasubsequence(stilldenoted
by uk) such that
(5.11) uk ⇀ u weakly in W1,q(Ω;RN),
0
(5.12) ∇uk ⇀ ∇u weakly in Lp ∩Lq(Ω;Rn×N),
ω
(5.13) S(x,∇uk) ⇀ S weakly in Lp ∩Lq(Ω;Rn×N).
ω
Hence by (5.10),(5.9) and 5.12 and the weak lower semicontinuity we obtain
q p q
(5.14) |∇u| +|∇u| ωdx ≤ c |f| dx+c,
Z Z
Ω Ω
which concludes the a priori estimate.
10 M.BUL´ICˇEKANDS.SCHWARZACHER
We still have to show that u is a distributional solution. Using (5.8), (5.7) and (5.13) it
follows that
(5.15) S ·∇ϕdx = f ·∇ϕdx for all ϕ ∈ C∞(Ω;RN).
0
Z Z
Ω Ω
To complete the proof of Theorem 1.2, it remains to show that
(5.16) S(x) = S(x,∇u(x)) in Ω.
To do so, we use Theorem 5.1. We denote ak := ∇uk and bk := S(x,∇uk). By using (5.14)
and (1.3), we find that (5.1) is satisfied with the weight ω. Since, we assume by (4.9) that
ε < (p−1),wefindbyLemma2.1thatω ∈ A . Alsotheassumption(5.2)holds,whichfollows
p
from 5.7, (5.8) and (5.15). Finally, (5.3) is valid trivially since ak is a gradient. Therefore,
Theorem 5.1can beapplied. Meaning, that wehave anon-decreasingsequenceof measurable
sets E , such that |Ω\E | → 0 and
j j
S(x,∇uk)·∇ukω ⇀ S ·∇uω weakly in L1(E ).
j
This is enough, to apply some variant of the Minty trick. For any G ∈ Lp(Ω;Rn×N) we get
ω
by 5.12 and 5.13
(S(x,∇uk)−S(x,G))·(∇uk −G)ω ⇀ (S −S(x,G))·(∇u−G)ω weakly in L1(E ).
j
Due to the monotonicity condition (1.4) we find that the term on the left hand side is non-
negative and consequently its weak limit is non-negative as well; especially (S−S(x,G))·
Ej
(∇u−G)ωdx ≥ 0 and R
(S −S(x,G))·(∇u−G)ωdx ≥ (S −S(x,G))·(∇u−G)ωdx.
Z Z
Ω Ω\Ej
Letting j → ∞ the Lebesgue dominated convergence theorem implies using the fact that
|Ω\E |→ 0 as j → ∞ we obtain
j
(5.17) (S −S(x,G))·(∇u−G)ωdx ≥ 0 for all G ∈ Lp(Ω;Rn×N).
ω
Z
Ω
Hence, setting G := ∇u−δH where H ∈ L∞(Ω;Rn×N) is an arbitrary function and dividing
(5.17) by δ implies
(S −S(x,∇u−δH))·Hωdx ≥ 0.
Z
Ω
Finally, letting δ → 0 implies bythecontinuity assumptionof S anddominated convergence
+
(S −S(x,∇u))·Hωdx≥ 0 for all H ∈ L∞(Ω;Rn×N).
Z
Ω
Since ω is strictly positive almost everywhere in Ω, the relation (5.16) easily follows by
choosing e.g.,
S −S(x,∇u)
H := − .
1+|S −S(x,∇u)|
Thus u is a distributional solution to (1.1). (cid:3)
Proof of Corollary 1.3. Theproofoftheaprioriestimate(4.1)isexactlythesame. Analogous
to the estimate of (5.9),(5.10) one finds in case C = 0, that
3
(5.18) |∇u |q +|∇u |p(Mf)q−pdx ≤ c |f|q.
k k
Z Z
Ω Ω
The existence proof is also analogous. However, since Theorem 5.1 is only valid on bounded
domains one has to use that S satisfies (5.15) for an arbitrary bounded subset of Ω. This
allowstoshowthatS = S(∇u)inΩ′. Sinceitwasarbitrarilychosentheexistencefollows. (cid:3)