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A Regularity of the Weak Solution Gradient of the Dirichlet Problem for Divergent Form Elliptic Equations in Morrey Spaces
Article in Australian Journal of Mathematical Analysis and Applications · October 2021
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AJMAA
A REGULARITY OF THE WEAK SOLUTION GRADIENT OF THE DIRICHLET PROBLEM FOR DIVERGENT FORM ELLIPTIC EQUATIONS IN MORREY
SPACES
NICKY K. TUMALUN AND PHILOTHEUS E. A. TUERAH
Received 27 February, 2021; accepted 9 September, 2021; published 8 October, 2021.
MATHEMATICSDEPARTMENT, UNIVERSITASNEGERIMANADO, TONDANOSELATAN, MINAHASA
REGENCY, SULAWESIUTARAPROVINCE, INDONESIA. nickytumalun@unima.ac.id
pheatuerah@unima.ac.id
ABSTRACT. In this paper we prove that the gradient of the weak solution of the Dirichlet prob- lem for divergent form elliptic equations, with the known term belongs to the Morrey spaces, is the element of the weak Morrey spaces.
Key words and phrases: Dirichlet problem; Morrey spaces.
2010 Mathematics Subject Classification. Primary 35J25, 35B65. Secondary 46E30, 42B35.
ISSN (electronic): 1449-5910
c 2021 Austral Internet Publishing. All rights reserved.
2 N. K. TUMALUN ANDP. E. A. TUERAH
1. INTRODUCTION
Recently, the regularity properties of the Dirichlet problem (1.1)
( Lu=f, u∈H01(Ω),
wheref belongs to some various Morrey spaces, have been studied by some authors (for ex- ample, see [2, 3, 5]). HereΩis bounded domain inRn,H01(Ω)is the Sobolev spaces, andLis divergent form elliptic operator defined inH01(Ω).
By using the assumption that f belongs to the Morrey spacesL1,λ(Ω) for0 < λ < n−2, Di Fazio [3] showed that the weak solution of (1.1) is the element of the weak Morrey spaces wLpλ,λ(Ω), where p1
λ = 1− n−λ2 . For more sharp result, in the sense of inclusion between Morrey spaces, Di Fazio [4] also proved that the weak solution of (1.1) is the element of the weak Morrey spaces wLqλ,λ(Ω), where q1
λ = 12 − n−λ1 , by taking f from the Morrey spaces L2,λ(Ω)for0< λ < n−2.
For the casen−2< λ < nandf is in the Morrey spacesL1,λ(Ω), Cirmi et. al [2] showed that the weak solution (1.1) is bounded essentially inΩand its gradient belongs to the Morrey spacesLp,λ(Ω), for somen−2< p≤λ.
Di Fazio has not investigated the regularity of the weak solution gradient of (1.1). In this paper, we continue his works, which are different from that one by Cirmi et. al in case of parameterλ. We prove that the the weak solution gradient of (1.1) belongs to the weak Morrey spaces wLpλ,λ(Ω), where p1
λ = 1− n−λ1 , by assuming f is in the Morrey spaces L1,λ(Ω) for 0< λ < n−2.
2. DIRICHLETPROBLEM AND MORREY SPACES
LetΩbe an open, bounded, and connected subset ofRnwithn≥ 3. These assumptions are always assumed forΩ. Fora ∈Ωandr >0, we define
B(a, r) ={y ∈Rn :|y−a|< r}, and
Ω(a, r) = Ω∩B(a, r) = {y∈Ω :|y−a|< r}.
For 1 ≤ p < ∞ and 0 ≤ λ ≤ n, the Morrey space L1,λ(Ω) is the set of all functions f ∈L1(Ω)which satisfies
kfkL1,λ = sup
a∈Ω,r>0
1 rλ
Z
Ω(a,r)
|f(y)|dy
.
Meanwhile, the weak Morrey spacewLp,λ(Ω) is the set of all measurable functionsf defined onΩwhich satisfies
sup
a∈Ω,t>0
supt>0t|{x∈Ω(a, r) :f(x)> t}|1p rλp
!
<∞.
Forq = 1,2, letW1,q(Ω)is denoted the Sobolev spaces. The closure ofC0∞(Ω)inW1,q(Ω) is denoted byW01,q(Ω). We consider the following second order divergent elliptic operator
(2.1) Lu=−
n
X
i,j=1
∂
∂xj
ai,j∂u
∂xi
, whereu∈W01,2(Ω),
ai,j ∈L∞(Ω), i, j = 1, . . . , n,
AJMAA, Vol. 18 (2021), No. 2, Art. 14, 7 pp. AJMAA
and there existsν >0such that ν|ξ|2 ≤
n
X
i=1
ai,j(x)ξiξj ≤ν−1|ξ|2,
for every ξ = (ξ1, . . . , ξn) ∈ Rn and for almost every x ∈ Ω. We also assume a regularity condition of the coefficientsai,j of the operatorL, that is,
|ai,j(x)−ai,j(y)| ≤ω(|x−y|), ∀x, y ∈Ω, whereω : (0,∞)−→(0,∞)is non-decreasing, satisfies
ω(2t)≤Cω(t) for a constantC > 0and for allt >0, and
Z ∞
0
ω(t)
t dt <∞.
Letf ∈L1,λ(Ω). We are interested in investigating the following Dirichlet problem (2.2)
( Lu=f, u∈W01,2(Ω), whereLis defined by (2.1).
The functionu∈W01,2(Ω)is called the weak solution of equation (2.2) if (2.3)
Z
Ω n
X
i,j=1
ai,j(x)∂u(x)
∂xi
∂φ(x)
∂xj
dx= Z
Ω
f(x)φ(x)dx,
for allφ∈C0∞(Ω).
3. TOOLS
Theorem 3.1 (Grüter and Widman, [6]). There exists a unique functionG: Ω×Ω−→R∪{∞}, G≥0, such that for eachy ∈Ωand anyr >0
(3.1) G(·, y)∈W1,2(Ω\B(y, r))∩W01,1(Ω), and for allφ ∈C0∞(Ω),
(3.2)
Z
Ω n
X
i,j=1
ai,j(x)∂G(x, y)
∂xi
∂φ(x)
∂xj
dx=φ(y).
Furthermore, there exists a positive constantC0 =C(n, ν)andC1 =C1(n, ν, ω,Ω)such that
(3.3) G(x, y)≤C0 1
|x−y|n−2, and
(3.4) |∇G(x, y)| ≤C1 1
|x−y|n−1 for allx, y ∈Ω, withx6=y.
4 N. K. TUMALUN ANDP. E. A. TUERAH
The function G in Theorem 3.1 is called the Green function for L and Ω. Fix y ∈ Ω.
According to (3.1),G(·, y)has a weak derivative inΩ, which is denoted by ∂G(x,y)∂x
i . Therefore (3.5)
Z
Ω
∂G(x, y)
∂xi φ(x)dx=− Z
Ω
G(x, y)∂φ(x)
∂xi dx, for allφ∈C0∞(Ω).
LetM be the Hardy-Littlewood maximal operator, defined by M(f)(x) := sup
r>0
1
|B(x, r)|
Z
B(x,r)
|f(y)|dy,
for everyf ∈L1loc(Rn).
Theorem 3.2. [1] Let0< λ < n,a∈Ω, andr >0. Iff ∈L1,λ(Ω), then there exists a positive constantC, which is independent fromaandr, such that
sup
t>0
t|{x∈Ω(a, r) :M(f)(x)> t}| ≤CrλkfkL1,λ. (3.6)
The first proof of Theorem 3.2 was given by [1]. For more simple and elegant proof of this theorem, we refer to [7].
4. AN INTEGRAL OPERATOR AND MAINRESULT
From now on, we always assume that0< λ < n−2. LetGbe the Green function forLand Ω. Forf ∈L1,λ(Ω), we define
(4.1) u(x) =
Z
Ω
G(x, y)f(y)dy
for everyx∈Ω. Next we define
(4.2) ui(x) =
Z
Ω
∂G(x, y)
∂xi f(y)
dy, for everyi= 1, . . . , nandx∈Ω.
We note that the functionu which is defined by (4.1) is the unique weak solution of (2.2).
This fact can be seen in [3].
Theorem 4.1. There exists a constantC >0such that
(4.3) tλ−n+1λ−n |{x∈Ω(a, r) :|ui(x)|> t}| ≤Crλkfk1−
λ−n−1 λ−n+1
L1,λ ,
for everya ∈Rn,r >0, andt >0.
Proof. Letx∈Ωandδ >0. By virtue of (3.4), we first estimate Z
Ω
|f(y)|
|x−y|n−1dy= Z
Ω(x,2δ)
|f(y)|
|x−y|n−1dy+ Z
Ω\B(x,2δ)
|f(y)|
|x−y|n−1dy=I1+I2. We boundI1 by using the Hedberg estimation, that is,
I1 ≤C(n)δM(f)(x).
AJMAA, Vol. 18 (2021), No. 2, Art. 14, 7 pp. AJMAA
Now we compute the bound ofI2 as follows, I2 =
Z
2δ≤|x−y|
|f(y)χΩ(y)|
|x−y|n−1 dy=
∞
X
k=1
Z
2kδ≤|x−y|<2k+1δ
|f(y)χΩ(y)|
|x−y|n−1 dy
≤C(n)
∞
X
k=1
1 (2k+1δ)n−1
(2k+1δ)λ (2k+1δ)λ
Z
2kδ≤|x−y|<2k+1δ
|f(y)χΩ(y)|dy
≤C(n, λ)kfkL1,λδλ−n+1
∞
X
k=1
2λ 2n−1
k
=C(n, λ)kfkL1,λδλ−n+1. Therefore
(4.4)
Z
Ω
|f(y)|
|x−y|n−1dy≤C(n, λ)
δM(f)(x) +δλ−n+1kfkL1,λ ,
by using the estimations ofI1andI2. We choose δ=
M(f)(x) kfkL1,λ
λ−n1
to minimize the right hand side of (4.4). Then Z
Ω
|f(y)|
|x−y|n−1dy≤C(n, λ)M(f)(x)λ−n+1λ−n kfk
λ−n−1 λ−n
L1,λ . (4.5)
Note that, according to (3.4) and (4.5), then (4.6) |ui(x)| ≤C
Z
Ω
|f(y)|
|x−y|n−1dy ≤CM(f)(x)λ−n+1λ−n kfk
λ−n−1 λ−n
L1,λ ,
for everyx∈Ω, whereC =C(n, λ, C1). Leta ∈Ωandr >0. For everyt >0, we have
|{x∈Ω(a, r) :|ui(x)|> t}| ≤
x∈Ω(a, r) :M(f)(x)> Ctλ−n+1λ−n kfk
λ−n−1 λ−n+1
L1,λ
. Now we use Theorem 3.2 to obtain
x∈Ω(a, r) :M(f)(x)> Ctλ−n+1λ−n kfk
λ−n−1 λ−n+1
L1,λ
≤C rλkfkL1,λ Ctλ−n+1λ−n kfk
λ−n−1 λ−n+1
L1,λ
=Crλkfk1−
λ−n−1 λ−n+1
L1,λ
tλ−n+1λ−n ,
whereC > 0is independent froma,r, andt. This means
tλ−n+1λ−n |{x∈Ω(a, r) :|ui(x)|> t}| ≤Crλkfk1−
λ−n−1 λ−n+1
L1,λ
for everya∈Ω,r >0, andt >0.
Lemma 4.2. ui ∈L1(Ω)for everyi= 1, . . . , n.
6 N. K. TUMALUN ANDP. E. A. TUERAH
Proof. Leta ∈Ω. SinceΩis bounded, then we can chooser >0such thatΩ⊆B(a, r). This meansΩ = Ω(a, r). By the Cavalieri Principle, we have
Z
Ω
|ui(x)|dx= Z
Ω(a,r)
|ui(x)|dx
= Z ∞
0
|{x∈Ω(a, r) :|ui(x)|> t}}|dt
=
Z |Ω(a,r)|
0
|{x∈Ω(a, r) :|ui(x)|> t}}|dt +
Z ∞
|Ω(a,r)|
|{x∈Ω(a, r) :|ui(x)|> t}}|dt.
Note that Z |Ω(a,r)|
0
|{x∈Ω(a, r) :|ui(x)|> t}}|dt ≤
Z |Ω(a,r)|
0
|Ω(a, r)|dt=|Ω(a, r)|2 <∞.
Using Theorem 4.1 and the fact λ−n+1n−λ + 1 <0, then Z ∞
|Ω(a,r)|
|{x∈Ω(a, r) :|ui(x)|> t}}|dt ≤Crλ Z ∞
|Ω(a,r)|
tλ−n+1n−λ dt
=Crλ|Ω(a, r)|λ−n+1n−λ +1 <∞.
Therefore
Z
Ω
|ui(x)|dx≤ |Ω(a, r)|2+Crλ|Ω(a, r)|λ−n+1n−λ +1 <∞.
This proves the lemma.
Lemma 4.3. Ifuis defined by (4.1), then the weak derivatives ofuis given by
∂u(x)
∂xi = ∂
∂xi Z
Ω
G(x, y)f(y)dy
= Z
Ω
∂G(x, y)
∂xi f(y)dy, for everyi= 1, . . . , n.
Proof. Letφ be an arbitrary element ofC0∞(Ω). We claim that ∂G(x,y)∂x
i f(y)φ(x) ∈L1(Ω×Ω).
This is because Z
Ω
Z
Ω
∂G(x, y)
∂xi f(y)φ(x)
dydx= Z
Ω
|ui(x)||φ(x)|dx
≤max
x∈Ω |φ(x)|
Z
Ω
|ui(x)|dx <∞,
which is concluded from Tonneli’s theorem and Lemma 4.2. Therefore we can use Fubini’s theorem and (3.5) to obtain
Z
Ω
Z
Ω
G(x, y)f(y)dy
∂φ(x)
∂xi dx= Z
Ω
f(y) Z
Ω
G(x, y)∂φ(x)
∂xi dx
dy
=− Z
Ω
f(y) Z
Ω
∂G(x, y)
∂xi
φ(x)dx
dy
=− Z
Ω
Z
Ω
∂G(x, y)
∂xi f(y)dy
φ(x)dx.
The proof is complete.
AJMAA, Vol. 18 (2021), No. 2, Art. 14, 7 pp. AJMAA
The following is our main theorem. This shows that the gradient of the weak solution of (2.2) belongs to the weak Morrey spaces.
Theorem 4.4. Ifuis defined by (4.1), then|∇u| ∈wLpλ,λ(Ω)where p1
λ = 1− n−λ1 . Proof. It is enough to proof that ∂u(x)∂x
i ∈ wLpλ,λ(Ω) for every i = 1, . . . , n. According to Lemma 4.3, (4.2), and Theorem 4.1, we have
tλ−n+1λ−n
x∈Ω(a, r) :
∂u(x)
∂xi
> t
≤tλ−n+1λ−n |{x∈Ω(a, r) :|ui(x)|> t}|
≤Crλkfk1−
λ−n−1 λ−n+1
L1,λ .
Therefore this theorem is proved.
REFERENCES
[1] F. CHIARENZA and M. FRASCA, Morrey spaces and Hardy-Littlewood maximal function, Rend.
Math., 7 (1987), pp. 273-279.
[2] G. R. CIRMI, S. D’ASERO, and S. LEONARDI, Morrey estimates for a class of elliptic equations with drift terms, Adv. Nonlinear Anal., 9 (2020), pp. 1333–1350.
[3] G. DI FAZIO, Regularity for elliptic equations under minimal assumptions, J. Indones. Math. Soc., 26(1) (2020), pp. 101–127.
[4] G. DI FAZIO, On Dirichlet problem in Morrey spaces, J. Differential and Integral Equations, 6 (1993), pp. 383–391.
[5] V. S. GULIYEV, M. N. OMAROVA, M. A. RAGUSA, and A. SCAPELLATO, Regularity of solu- tions of elliptic equations in divergence form in modified local generalized Morrey spaces, Analysis and Mathematical Physics, 11 (2020), pp. 1–20.
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AJMAA, Vol. 18 (2021), No. 2, Art. 14, 7 pp. AJMAA
