Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 51, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
LOCAL ESTIMATES FOR GRADIENTS OF SOLUTIONS TO
ELLIPTIC EQUATIONS WITH VARIABLE EXPONENTS
FENGPING YAO
Abstract. In this article we present local L∞ estimates for the gradient of solutions to elliptic equations with variable exponents. Under proper conditions
on the coefficients, we prove that
|∇u| ∈ L∞
loc
for all weak solutions of
div(g(|∇u|2 , x)∇u) = 0
in Ω.
1. Introduction
Uhlenbeck [26] obtained the interior Hölder regularity estimates for weak solutions of
div(ρ(|∇u|2 )∇u) = 0 in Ω,
(1.1)
where Ω is an open bounded domain in Rn and ρ ∈ C 1 ([0, ∞)) is a non-negative
function satisfying the ellipticity conditions
p
p
K −1 (ξ + c) 2 −1 ≤ ρ(ξ) + 2ρ0 (ξ)ξ ≤ K(ξ + c) 2 −1 ,
0
0
p/2−1−α
|ρ (ξ1 )ξ1 − ρ (ξ2 )ξ2 | ≤ K(ξ1 + ξ2 + c)
(ξ1 − ξ2 )
(1.2)
α
(1.3)
for c ≥ 0, α > 0 and p ≥ 2. Especially when ρ(t) = t(p−2)/2 , (1.1) is reduced to
the well-known p-Laplace equation. In this paper we discuss the nonlinear elliptic
equation of the form
div(g(|∇u|2 , x)∇u) = 0 in Ω,
(1.4)
where g(ξ, x) ∈ C 1 ([0, ∞) × Ω) satisfies the ellipticity conditions
C1 (ξ + c)
p(x)
2 −1
≤ g(ξ, x) + 2ξgξ (ξ, x) ≤ C2 (ξ + c)
|∇x g(ξ, x)| ≤ C3 (ξ + c)
p(x)−1
2
p(x)
2 −1
,
|∇p|| ln(ξ + c)|
(1.5)
(1.6)
for c ≥ 0 and C1 , C2 , C3 > 0. Here p ∈ W 1,s (Ω) for some s > n satisfies
1 < p1 = inf p(x) ≤ p(x) ≤ sup p(x) = p2 < ∞.
Ω
Ω
2000 Mathematics Subject Classification. 35J60, 35J70.
Key words and phrases. Regularity; divergence; nonlinear; elliptic equation; gradient;
variable exponent; p(x)-Laplacian.
c
2013
Texas State University - San Marcos.
Submitted September 3, 2012. Published February 18, 2013.
1
(1.7)
2
F. YAO
Especially when g(t) = |t|
tion
p(x)−2
2
EJDE-2013/51
, (1.4) is reduced to the p(x)-Laplace elliptic equa-
div(|∇u|p(x)−2 ∇u) = 0 in Ω,
whose special case is the well-known elliptic p-Laplace equation
div(|∇u|p−2 ∇u) = 0
(1.8)
in Ω,
which can be derived from the variational problem
Z
Φ(u) = min Φ(v) =: min
|∇v|p dx.
v|∂Ω =ϕ
v|∂Ω =ϕ
Ω
p(x)
We denote by L
(Ω) the variable exponent Lebesgue-Sobolev space
Z
Lp(x) (Ω) = {f : Ω → R : f is measurable and
|f |p(x) dx < ∞}
(1.9)
Ω
equipped with the Luxemburg type norm
Z
f
kf kLp(x) (Ω) = inf λ > 0 :
| |p(x) dx ≤ 1 .
λ
Ω
Furthermore, we define
W 1,p(x) (Ω) = {u ∈ Lp(x) (Ω) : |∇u| ∈ Lp(x) (Ω)}
(1.10)
(1.11)
equipped with the norm
kukW 1,p(x) (Ω) = kukLp(x) (Ω) + k∇ukLp(x) (Ω) .
(1.12)
1,p(x)
By W0
(Ω) we denote the closure of C0∞ (Ω) in W 1,p(x) (Ω). Actually, the
1,p(x)
Lp(x) (Ω), W 1,p(x) (Ω) and W0
(Ω) spaces are Banach spaces. There have been
many investigations (see for example [9, 10, 11, 12, 16, 17, 19]) on properties of
such variable exponent Sobolev spaces.
As usual, the solutions of (1.4) are taken in a weak sense. We now state the
definition of weak solutions.
1,p(x)
Definition 1.1. A function u ∈ Wloc (Ω) is a local weak solution of (1.4) in Ω
1,p(x)
if for any ϕ ∈ W0
(Ω), we have
Z
g(|∇u|2 , x)∇u · ∇ϕdx = 0.
Ω
When p(x) is a constant, many authors [3, 4, 8, 13, 14, 21, 20, 23] have studied
the regularity estimates for weak solutions of quasilinear elliptic equations of pLaplacian type and the general case. When p(x) is not a constant, such elliptic
problems (1.4) appear in mathematical models of various physical phenomena, such
as the electro-rheological fluids (see, e.g., [1, 24, 25]). There have been many
investigations [7, 15, 22] on Hölder estimates for the p(x)-Laplacian elliptic equation
(1.4) and the more general case. Moreover, Acerbi and Mingione [2] proved that
|f |p(x) ∈ Lqloc (Ω)
=⇒
|∇u|p(x) ∈ Lqloc (Ω)
for q > 1
of weak solutions of (1.4) under some assumptions. The purpose of this paper is
to extend the results in [6], where Challal and Lyaghfouri obtained the local L∞
estimates of |∇u| for the weak solutions of (1.8).
We assume that p(x) ∈ W 1,s (Ω) for some s > n. Therefore, it follows from
Sobolev embedding theorem that p(x) is Hölder continuous with the exponent α =
1 − ns . Now let us state the main result of this work.
EJDE-2013/51
LOCAL ESTIMATES FOR GRADIENTS
3
1,p(x)
Theorem 1.2. Let u ∈ Wloc (Ω) be a local weak solution of (1.4) in Ω under the
assumptions (1.2)-(1.7). Then
|∇u| ∈ L∞
loc (Ω).
Moreover, for each σ > 0 and δ ∈ (0, sq(1+σ)
− 1) with a constant q ∈ (n, s) there
s−q
exists a positive constant R0 , depending only on n, p1 , p2 , s, σ and k|∇u(·)|p(·) kL1 (Ω) ,
such that, wherever R ≤ R0 and the ball B8R ⊂ Ω,
Z
1+σ i
hZ
p(x)
p(x)
α
(1+σ)δ
−B8R |∇u|p(x) + 1dx
,
sup |∇u|
≤ C −B2R |∇u|
dx + R M
BR/2
where M =
R
B8R
|∇u|p(x) dx + 1 and C depends on n, p1 , p2 , s, δ, kpkW 1,s (Ω) .
2. Proof of main result
In this section we prove Theorem 1.2 by the approximation method. Our approach is much influenced by [2, 5, 6, 27]. We first consider the following approximation problem
div(g( + |∇u |2 , x)∇u ) = 0,
x ∈ BR0 , ∈ (0, 1],
(2.1)
where B8R ⊂ BR0 ⊂ Ω. It is standard that (2.1), with the boundary condition
u = u on ∂BR0 , has a unique solution u for fixed > 0. Similarly to [6], we know
2,2
1,µ
u ∈ Wloc
(Ω). From [15] we can get u ∈ Cloc
(Ω) for some µ ∈ (0, 1) and then have
1,ν
u ∈ C (B R0 ) for some ν ∈ (0, 1) and ku kC 1,ν (B R0 ) ≤ C, where C is a constant
independent of . It follows from Ascoli-Arzelà theorem that there exists a sequence
of {k } converging to 0 and satisfying uk → u uniformly in C 1 (B R0 ). Thus, we
can get the result of Theorem 1.2 by passing to the limit as k → 0 in (2.9) with
uk replacing u. So it is sufficient to prove (2.9). For simplicity, we shall drop the
index on u in the exposition. Actually, from (2.1) we have
g( + |∇u|2 , x)δij + 2gξ ( + |∇u|2 , x)ui uj uij + gxi ( + |∇u|2 , x)ui
(2.2)
=: aij uij + bi ui = 0.
Lemma 2.1. If g(ξ, x) ∈ C 1 ([0, ∞) × Ω) satisfies the conditions (1.2) and (1.6),
then
C4 (ξ + c)
p(x)
2 −1
≤ g(ξ, x) ≤ C5 (ξ + c)
|gξ (ξ, x)ξ| ≤ C6 (ξ + c)
p(x)
2 −1
,
p(x)
2 −1
for the constants 0 < C4 < C1 , C5 > C2 > 0, C6 > 0, and
p(x)
p(x)
2 −1
2 −1
2
2
2
C4 c + + |∇u|
|ξ| ≤ aij ξi ξj ≤ C5 c + + |∇u|
|ξ|2 .
(2.3)
(2.4)
(2.5)
Proof. We prove only (1.3). First, we find that
Z ξ
Z ξ
1 −1/2 ξ 1/2 g(ξ, x) =
(t1/2 g(t, x))t dt =
t
g(t, x) + 2tgt (t, x) dt.
0
0 2
Moreover, from (1.2) we deduce that
Z
Z
p(x)
p(x)
C1 ξ −1/2
C2 ξ −1/2
−1
1/2
2
t
(t + c)
dt ≤ ξ g(ξ, x) ≤
t
(t + c) 2 −1 dt =: I2 .
I1 =:
2 0
2 0
To estimate of I1 and I2 , we consider two cases.
4
F. YAO
EJDE-2013/51
Case 1: c ≤ ξ. We have
Z
p(x)−3
C1 ξ
(t + c) 2 dt
2 0
p(x)−1
p(x)−1
C1
1
≥
(
)[(ξ + c) 2 − c 2 ]
2 p(x)−1
I1 ≥
2
p(x)−1
p(x)−1
C1
≥
[(ξ + c) 2 − c 2 ].
p(x) − 1
Since 1 < p1 ≤ p(x) ≤ p2 and c ≤ c+ξ
2 , we obtain
p(x)−1
C1 1 p(x)−1 I1 ≥
1−( ) 2
(ξ + c) 2
p2 − 1
2
≥ C10 (ξ + c)
p(x)−1
2
≥ C10 (ξ + c)
p(x)
2 −1
ξ 1/2 .
Moreover, we deduce that
p(x)
C2
I2 ≤
(ξ + c) 2 −1
2
Z
ξ
t−1/2 dt = C2 (ξ + c)
p(x)
2 −1
ξ 1/2
for p(x) ≥ 2
0
and
C2
I2 ≤
2
Z
ξ
t
p(x)−1
−1
2
dt
0
p(x)−1
C2
ξ 2
p(x) − 1
p(x)−1
C2
≤
(ξ + c) 2
p1 − 1
p(x)
C2
=
(ξ + c) 2 −1 (ξ + c)1/2
p1 − 1
=
which implies
for 1 < p(x) < 2,
√
p(x)
p(x)
C2
2C2
−1
1/2
2
I2 ≤
(ξ + c)
(2ξ)
=
(ξ + c) 2 −1 ξ 1/2
p1 − 1
p1 − 1
in view of the fact that ξ + c ≤ 2ξ.
Case 2: c ≥ ξ. Then we have
Z ξ
p(x)−1
C1 −1/2
ξ
(ξ + c)−1/2
(t + c) 2 dt.
I1 ≥
2
0
Furthermore,
p(x)
p(x)
C1 −1/2
1 p(x)−1
I1 ≥
ξ
(ξ + c) 2 −1 ( ) 2 ξ ≥ C100 (ξ + c) 2 −1 ξ 1/2
2
2
since
1
1
t + c ≥ c ≥ (2c) ≥ (ξ + c).
2
2
Since the result (1.3) is trivial when c = 0. Without loss of generality we may as
well assume that c > 0. Moreover, we first have
Z
p(x)−1
C2 −1/2 ξ −1/2
I2 ≤
c
t
(t + c) 2 dt
2
0
EJDE-2013/51
LOCAL ESTIMATES FOR GRADIENTS
≤
p(x)−1
C2 −1/2
c
(ξ + c) 2
2
Z
5
ξ
t−1/2 dt,
0
which implies
I2 ≤ C2 c−1/2 (ξ + c)
p(x)−1
2
ξ 1/2
p(x)
ξ + c 1/2
= C2 (
) (ξ + c) 2 −1 ξ 1/2
c
√
p(x)
≤ 2C2 (ξ + c) 2 −1 ξ 1/2 .
Thus, from Cases 1 and 2 we have
g(ξ, x) ≥ min{C10 , C100 }(ξ + c)
and
p(x)
2 −1
=: C4 (ξ + c)
p(x)
2 −1
.
√
p(x)
p(x)
2C2 √
, 2C2 }(ξ + c) 2 −1 =: C5 (ξ + c) 2 −1 ,
p1 − 1
which completes the proof.
g(ξ, x) ≤ max{
Now we denote
aij
.
(c + + |∇u|2 )p(x)/2−1
Then, from the lemma above we have
af
ij =
2
C4 |ξ|2 ≤ af
ij ξi ξj ≤ C5 |ξ|
(2.6)
for each ξ ∈ Rn .
(2.7)
Lemma 2.2. Let v = (c + + |∇u|2 )p(x)/2 . Then
1
2
2 1/2 ∇p(x)
af
·
div(
ij · ∇v) ≥ div aij · (c + + |∇u| ) · ln(c + + |∇u| )
p(x)
p(x)
=: div F,
where
h
p(x)(1+σ) 2
|F | ≤ C 1 + (c + + |∇u|2 )
· |∇p|
for any σ > 0.
Proof. We first find that
vx j
p(x)
= (c + + |∇u|2 ) ln(c + + |∇u|2 )1/2 pxj (x) + p(x)ukj uk (c + + |∇u|2 ) 2 −1
and
div(
1
af
ij · ∇v)
p(x)
= div(aij · (c + + |∇u|2 ) · ln(c + + |∇u|2 )1/2 ·
∇p(x)
) + (aij ukj uk )xi .
p(x)
Moreover, differentiating (2.1) with respect to xk , we have
(aij ukj )xi + (bk ui )xi = 0,
(2.8)
where bk is defined in (2.2). Therefore, from (2.8) and Lemma 2.1 we have
1
div
af
ij · ∇v
p(x)
∇p(x) = div aij · (c + + |∇u|2 ) · ln(c + + |∇u|2 )1/2 ·
+ aij ukj uki − (bk ui )xi
p(x)
6
F. YAO
EJDE-2013/51
∇p(x) ≥ div aij · (c + + |∇u|2 ) · ln(c + + |∇u|2 )1/2 ·
− (bk ui )xi
p(x)
=: div F,
where
1/2 ∇p(x)
·
F = aij · c + + |∇u|2 · ln c + + |∇u|2
− bk ui
p(x)
Actually, for any σ > 0, we deduce that
|F | ≤ C |aij |(c + + |∇u|2 )| ln(c + + |∇u|2 )1/2 ||∇p| + |bk ||∇u|
p(x)
≤ C c + + |∇u|2 2 | ln(c + + |∇u|2 )1/2 ||∇p|
p(x)
p(x)
≤ C c + + |∇u|2 2 | ln(c + + |∇u|2 ) 2 ||∇p|
p(x)(1+σ) 2
≤ C 1 + (c + + |∇u|2 )
|∇p|.
By (1.6), Lemma 2.1 and
|x|| ln x| ≤ C(1 + |x|1+σ )
for any σ > 0,
we complete the proof.
Next, we shall finish the proof of the main result.
Proof. Using [18, Theorem 8.17], we obtain
p(x)/2
1 Z
p(x)
2 2
2
(c
+
+
|∇u|
)
dx
+
K(R)
,
sup c + + |∇u|
≤C
Rn B2R
BR/2
where
K(R) = R
1− n
q
Z
|F |q dx
1/q
B2R
and q ∈ (n, s) is a positive constant. Moreover, we find that
Z
Z
Z
p(x)(σ+1)q
q
q
2
|F | dx ≤ C
|∇p| dx +
(c + + |∇u|2 )
|∇p|q dx
B2R
B2R
B2R
Furthermore, using Hölder’s inequality and p ∈ W 1,s , we obtain
Z
|F |q dx
B2R
≤C
Z
q/s n Z
|∇p|s dx
B2R
≤C
(c + + |∇u|2 )
p(x)sq(1+σ)
2(s−q)
dx
s−q
s
+R
n(s−q)
s
o
B2R
n Z
(c + + |∇u|2 )
p(x)sq(1+σ)
2(s−q)
s−q
o
n(s−q)
s
dx
+R s
B2R
≤ CR
n(s−q)
s
hZ
i s−q
p(x)sq(1+σ)
s
−B2R (c + + |∇u|2 ) 2(s−q) + 1dx
.
From [2, Theorem 2], for any δ ∈ (0, sq(1+σ)
− 1) there exists a positive constant
s−q
R0 , depending only on n, p1 , p2 , s, δ, σ and k |∇u(·)|p(·) kL1 (Ω) , such that, wherever
R ≤ R0 ,
Z
Z
q(1+σ)
n(s−q)
p(x)
|F |q dx ≤ CR s M q(1+σ)δ −B8R (c + + |∇u|2 ) 2 + 1dx
,
B2R
EJDE-2013/51
LOCAL ESTIMATES FOR GRADIENTS
where
Z
M=
7
|∇u|p(x) dx + 1.
B8R
Thus,
Z
1+σ
p(x)
K(R) ≤ CRα M (1+σ)δ −B8R (c + + |∇u|2 ) 2 + 1dx
,
where the exponent α = 1 − ns . Finally, we conclude that
sup (c + + |∇u|2 )
p(x)
2
BR/2
hZ
p(x)
≤ C −B2R (c + + |∇u|2 ) 2 dx
1+σ i
Z
p(x)
,
+ M (1+σ)δ Rα −B8R (c + + |∇u|2 ) 2 + 1dx
which completes our proof.
(2.9)
Acknowledgements. The author wishes to thank the anonymous reviewers for
their valuable suggestions that improved this article. This work is supported in
part by the NSFC (11001165) and Shanghai Leading Academic Discipline Project
(J50101).
References
[1] E. Acerbi, G. Mingione; Regularity results for a stationary electro-rheologicaluids, Arch.
Ration. Mech. Anal., 164(3)(2002), 213-259.
[2] E. Acerbi, G. Mingione; Gradient estimates for the p(x)-Laplacean system, J. reine angew.
Math., 584(2005), 117-148.
[3] S. Byun, L. Wang; Quasilinear elliptic equations with BMO coefficients in Lipschitz domains,
Trans. Amer. Math. Soc., 359(12)(2007), 5899-5913.
[4] S. Byun, F. Yao, S. Zhou; Gradient estimates in Orlicz space for nonlinear elliptic equations,
J. Funct. Anal., 255(8)(2008), 1851-1873.
[5] S. Challal, A. Lyaghfouri; Second order regularity for the p(x)-Laplace operator, Math.
Nachr., 284(10)(2011), 1270-1279.
[6] S. Challal, A. Lyaghfouri; Gradient estimates for p(x)-harmonic functions, Manuscripta
Math., 131(3-4)(2010), 403-414.
[7] A. Coscia, G. Mingione; Hölder continuity of the gradient of p(x)-harmonic mappings, C. R.
Acad. Sci. Paris Math., 328(4)(1999), 363-368.
[8] E. DiBenedetto, J. Manfredi; On the higer integrability of the gradient of weak solutions of
certain degenerate elliptic systems, Amer. J. Math., 115(1993), 1107-1134.
[9] L. Diening; Riesz potential and Sobolev embeddings of generalized Lebesgue and Sobolev spaces
Lp(.) and W k,p(.) , Math. Nach., 268(1)(2004), 31-43.
[10] L. Diening, M. Ru̇žička; Calderón-Zygmund operators on generalized Lebesgue spaces Lp(·)
and problems related to fluid dynamics, J. Reine Angew. Math., 563 (2003), 197-220.
[11] L. Diening, M. Ru̇žička; Integral operators on the halfspace in generalized Lebesgue spaces
Lp(.) , part I, J. Math. Anal. . Appl., 298(2)(2004), 559-571.
[12] L. Diening, M. Ru̇žička; Integral operators on the halfspace in generalized Lebesgue spaces
Lp(.) , part II, J. Math. Anal. . Appl., 298(2)(2004), 572-588.
[13] F. Duzaar, G. Mingione; Gradient continuity estimates, Calc. Var. Partial Differential Equations, 39(3-4) (2010), 379-418.
[14] F. Duzaar, G. Mingione; Gradient estimates via non-linear potentials, Amer. J. Math., 133(4)
(2011), 1093-1149.
[15] X. Fan; Global C 1,α regularity for variable exponent elliptic equations in divergence form, J.
Differential Equations, 235 (2007), 397-417.
[16] X. Fan, J. Shen, D. Zhao; Sobolev embedding theorems for spaces W k,p(x) (Ω), J. Math. Anal.
Appl., 262(2001), 749-760.
8
F. YAO
EJDE-2013/51
[17] X. Fan, D. Zhao; On the spaces Lp(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl., 263(2001),
424-446.
[18] D. Gilbarg, N. Trudinger; Elliptic Partial Diferential Equations of Second Order (3rd edition), Springer Verlag, Berlin, 1998.
[19] P. Harjulehto; Variable exponent Sobolev spaces with zero boundary values, Math. Bohem.,
132(2007), 125-136.
[20] T. Iwaniec; Projections onto gradient fields and Lp -estimates for degenerated elliptic operators, Studia Math., 75(1983), 293-312.
[21] J. Kinnunen, S. Zhou; A local estimate for nonlinear equations with discontinuous coefficients, Comm. Partial Differential Equations, 24(1999), 2043-2068.
[22] A. Lyaghfouri; Hölder continuity of p(x)-superharmonic functions, Nonlinear Anal.,
73(8)(2010), 2433-2444.
[23] G. Mingione; Gradient estimates below the duality exponent, Math. Ann., 346(3)(2010), 571627.
◦
[24] K. R. Rajagopal, M. Ružička; Mathematical modeling of electro-rheological materials, Contin.
Mech. Thermodyn., 13(1)(2001), 59-78.
◦
[25] M. Ružička; Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in
Math., vol. 1748, Springer, Berlin, 2000.
[26] K. Uhlenbeck; Regularity for a class of non-linear elliptic systems, Acta Math., 138(1977),
219-240.
[27] L. Wang; Compactness methods for certain degenerate elliptic equations, J. Differential Equations, 107(2)(1994), 341-350.
Fengping Yao
Department of Mathematics, Shanghai University, Shanghai 200444, China
E-mail address: yfp@shu.edu.cn