Regularity of Weak Solutions to Some Anisotropic
Elliptic Equations
GAO Yanmin
College of Mathematics and Information Science, Hebei University, Baoding, 071002, China.email:1245712005@qq.com
GAO Hongya
College of Mathematics and Information Science, Hebei University, Baoding, 071002, China.email:ghy@hbu.cn
Abstract
We consider boundary value problem of the form This paper we deals with the system of N partial differential equations
n
X
i=1
Di(aαi(x, Du(x))) = n
X
i=1
DiFiα(x), α= 1,· · ·N.
We show that regularity of boundary datumu∗ forcesuto have regular-ity as well, provided we assume suitable ellipticregular-ity and growth conditions on aαi.
Mathematics Subject Classification: 49N60, 35J60.
Keywords: Regularity, weak solution, anisotropic, elliptic systems.
1
Introduction
In a recent paper [1], Leonetti and Petricca considered the system ofN partial differential equations
n
X
i=1
Di(aαi(x, Du(x))) = 0, x∈Ω, α= 1,2,· · ·, N. (1.1)
Under the boundary condition
the authors showed that higher integrability of the boundary datum u∗ forces solutions u to have higher integrability as well, provided aα
i satisfy suitable
ellipticity and growth conditions. Among all the results, they obtained a the-orem see [1, Thethe-orem 1.3].
In this paper, we consider a more general problem. We refer the reader to [1] for the notations and symbols used in this paper. Let us consider the following system of N partial differential equations
n
X
i=1
Di(aαi(x, Du(x))) = n
X
i=1
DiFiα(x), α= 1,· · ·, N, (1.3)
and suppose that the Carath´eodory functions aα
i(x, z) : Ω×R n
→ R satisfy the monotonicity condition
ν
n
X
i=1
|zi −zei|pi ≤ n
X
i=1
(aαi(x, z)−aαi(x,ze))(ziα−zeiα), (1.4)
for some positive constant ν,and
|aαi(x, z)| ≤c(1 + n
X
j=1
|zj|pj) 1−pi1
, i= 1,2,· · ·, n, α= 1,2,· · ·, N, (1.5)
We work in Marcinkiewicz spaces: if q >1, then the spaceMm(Ω) consists of measurable functions g on Ω such that
sup
t>0
t|{x∈Ω :|g(x)|> t}|m1 <∞.
This condition is equivalently stated as
|||g(x)|||m = sup E⊂Ω,|E|>0
1
|E|m1′
Z
E|g(x)|dx <∞,
where m′ is the conjugate exponent of q, 1 m +
1
m′ = 1. We recall that L q(Ω)
is a proper subspace of Lqweak(Ω), and if g ∈ L q
weak(Ω) for some q > 1, then
g ∈Lq−ε(Ω) for every 0< ε≤q−1.
The following Sobolev type inequality is also proved: there exists a positive constant c, depending only on Ω, such that
kvkLr(Ω) ≤c
n
Y
i=1
k∂ivk 1
n
Lpi(Ω), ∀r ∈[1, p∗], (1.6)
for any v∈C1
0(Ω) where pi >1 for i= 1,2,· · ·, n. In the following the letter c
of (1.1), withm > p′∞, we consider the casep∞= ¯p∗, wherep∞ = max{pn,p¯∗},
pn = max{pi}, and we also consider the elliptic systems problem withf ∈Mm
too, not only this but also our method is different from the previous one.For some recent developments on anisotropic functionals and anisotropic elliptic equations and systems, see[2-3].
2
Preliminary Notes
We mow introduce some symbols used in this paper. Let Tk(u) is the usual
truncation of u at level k >0, that is,
Tk(u) = max{−k,min{k, u}}.
Moreover, let
Gk(u) =u−Tk(u).
Definition 2.1 A function u ∈ u∗+W01,(pi)(Ω;RN) is called a solution to the boundary value problem
n
P
i=1Di(a α
i(x, Du(x))) =fα(x), x∈Ω,
u(x) =u∗(x), x∈∂Ω,
(2.7)
if
Z
Ω n
X
i=1 N
X
α=1
aαi(x, Du(x))·Diϕα(x)dx =
Z
Ω N
X
α=1
fα(x)·ϕα(x)dx, (2.8)
holds true for any ϕ ∈W1,(pi)
0 (Ω;RN), α= 1,2,· · ·, N.
3
Main Results
These are the main results of the paper.
Theorem 3.1 Let f ∈ Mm(Ω;RN), u
∗ ∈W1,1(Ω;RN) with Diu∗ ∈Mpim,
i= 1,2,· · ·, n, and under previous assumptions (1.4)−(1.5),
i) If m> np, then there exists a weak solution u for the problem (1.3), andu−u∗
is bounded;
ii) If m= n
p, then there exists a weak solution u for the problem (1.3) and a
constant β >0 such that Z
Ωe
β|u−u∗|<∞; iii) If (p∗)′
< m < np, then there exists a weak solution u for the problem(1.3)
and u−u∗ belongs to Ms with s satisfies
s = mp
∗(p−1)
mp+p∗−mp∗ =
mn(p−1)
Proof of Theorem 3.1. Let us fix the componentβ ∈ {1,2,· · ·, N}and let us consider the test function
ϕ = (0,· · ·,0, ϕβ,0,· · ·,0),
where, for k ∈(0,+∞) we takeϕβ =G
k(uβ −uβ∗) in (2.8). We have
n X i=1 Z Ωa β
i(x, Du)DiGk(uβ−uβ∗) = Z
Ωf β
Gk(uβ−uβ∗),
because n X i=1 Z Ωa β
i(x, Du)DiGk(uβ−uβ∗) =
n
X
i=1
Z
Ak
aβi(x, Du)Di(uβ−uβ∗) = Z
Ak
fβ(uβ−uβ∗),
where Ak ={|uβ−uβ∗| > k}. Hence by (1.4), (1.5) and Young inequality we obtain that ˜ ν n X i=1 Z Ak
|Diuβ −Diuβ∗|pi
≤ n X i=1 Z Ak
(aβi(x, Du)−aβi(x, Du∗))(Diuβ−Diuβ∗)
= Z
Ak
fβ(uβ −uβ∗)−
n
X
i=1
Z
Ak
aβi(x, Duβ∗)(Diuβ −Diuβ∗)
≤
Z
Ak
|fβ(uβ −uβ∗)|+
n
X
i=1
Z
Ak
|aβi(x, Du∗)||Diuβ−Diuβ∗|
≤
Z
Ak
|fβ(uβ −uβ∗)|+c
n X i=1 Z Ak (1 + N X j=1
|Diu∗|pj)1−
1
pi(Diuβ −Diuβ
∗)
≤
Z
Ak
|fβ(uβ −uβ∗)|+c(ε)
n X i=1 Z Ak (1 + n X i=1
|Diu∗|pi) +ε
n
X
i=1
Z
Ak
|Diuβ−Diuβ∗|pi.
Then Z
Ak
|Diuβ−Diuβ∗|pi ≤c(
Z
Ak
|fβ(uβ−u∗β)|+|Ak|+ n
X
i=1
Z
Ak
|Diu∗|pi). (3.10)
Therefore, by (1.6), with r=p∗, H¨older inequality and (3.10), we get
c( Z
Ak
|uβ−uβ∗|p∗
)p1∗
≤ n Y i=1 ( Z Ak
|Diuβ−Diuβ∗|pi)
1
pin
≤ (
Z
Ak
|fβ(uβ −u∗β)|+|Ak|+ n
X
i=1
Z
Ak
|Diu∗|pi)
1
p
≤ (
Z
Ak
|fβ|(p∗)′)
1 (p∗)′
( Z
Ak
|uβ−uβ∗|p∗
)p1∗ +|A k|+
n
X
i=1
Z
Ak
|Diu∗|pi]
1
p.
Since f ∈Mm(Ω) and D
iu∗ ∈Mpim, and m≥(p∗)′, we have Z
Ak
|f|(p∗)′
≤c|Ak|1− (p∗)′
m ,
Z
Ak
|Diu∗|pi ≤c|Ak|1− 1
m.
Then by applying Young inequality, |Ak| 1
m ≤ |Ω|
1
m and (3.11) becomes
c( Z
Ak
|uβ−uβ∗|p∗
)pp∗
≤ (|Ak|1− (p∗)′
m )
1 (p∗)′
( Z
Ak
|uβ−uβ∗|p∗
)p1∗ +|A
k|+|Ak|1− 1
m
≤ |Ak| 1 (p∗)′−
1
m
( Z
Ak
|uβ −uβ∗|p∗)p1∗ + 1
k|Ak|
1 (p∗)′−
1
m · |Ak|
1
m(
Z
Ak
|uβ−uβ∗|p∗)p1∗
+1
k|Ak|
−m1 Z
Ak
|uβ −uβ∗| ≤ |Ak|
1 (p∗)′−m1
( Z
Ak
|uβ −uβ∗|p∗
)p1∗ +c|A k|
1 (p∗)′−m1
( Z
Ak
|uβ−uβ∗|p∗
)p1∗
+c|Ak|− 1
m(
Z
Ak
|uβ −uβ∗|p∗
)p1∗|A k|
1 (p∗)′
≤ c(ε)|Ak| ( 1
(p∗)′− 1
m)(p)
′
+ε( Z
Ak
|uβ−uβ∗|p∗
)pp∗.
Hence by applying H¨older inequality with exponentsp∗ and (p∗)′toR
Ω|Gk(uβ−
uβ
∗)|= R
Ak|u
β−uβ
∗| and by simplifying, we obtain
Z
Ω|Gk(u β
−uβ∗)| ≤c|Ak| ( 1
(p∗)′−m1)p−11 +1−p1∗
. (3.12)
We define g(k) =RΩ|Gk(uβ−uβ∗)| and we recall that g
′
(k)=−|Ak|, for almost
every k (see[4], [5]). We obtain, from(3.12), that
g(k)γ1 ≤ −cg′(k),
with γ = ( 1
(p∗)′ − m1)p−11 + 1− p1∗. Therefore
1≤ −cg′(k)g(k)−1γ =− c
1− 1γ(g(k)
1−1
γ)′. (3.13)
If we are in case i) of Theorem 3.1, we note that
1− 1
γ >0.
Therefore, by integrating (3.13) from 0 to k, we get
k ≤ −c[g(k)1−1γ −g(0)1−
1
i.e.
cg(k)1−γ1 ≤ −k+ckuβ−uβ
∗k
1−1γ
L1(Ω).
Sinceg(k) is a non-negative and decreasing function, from the latter inequality we deduce that there exists k0, such that g(k0)=0, and so uβ−uβ∗ ∈ L∞(Ω). In case ii) of Theorem 3.1, since m = n
p,γ = 1, we have
1≤ −cg
′
(x)
g(x). By integrating from 0 to k, we have
k
c ≤log[
kuβ−uβ
∗kL1(Ω)
g(k) ], and since the function t→et increases, we obtain
ekc ≤ ku
β −uβ
∗kL1(Ω)
g(k) ⇒g(k)e
k
c ≤ kuβ −uβ
∗kL1(Ω).
So, recalling that
g(k) = Z
Ω|Gk(u β
−uβ∗)| ≥
Z
A2k
|Gk(uβ−uβ∗)| ≥k|A2k|, (3.14)
Hence, if k≥1, we have
g(k)≥ |A2k| ⇒ |A2k|e
k
c ≤ kuβ−uβ
∗kL1(Ω).
Hence, if k≥2, we get
|Ak|e
k
2c ≤ kuβ−uβ
∗kL1(Ω). (3.15)
We prove now that the previous inequality implies that
+X∞
k=0
ekτ|Ak|<∞,
with 0< τ < 21c. Indeed, by (3.15),
+∞ X
k=0
ekτ|Ak| ≤(1 +e)|Ω|+ +∞ X
k=2
kuβ −uβ
∗kL1(Ω)
ek(21c−τ)
<∞.
Since
+∞ X
k=0
ekτ|Ak|<+∞ ⇒
Z
Ωe β|uβ−uβ
ii) is proved. To conclude, we consider case iii). In this case we have
1− 1
γ <0.
Therefore,
1≤c(g(k)1−γ1)′.
By integration from 0 to k, we obtain
k≤ c[g(k)1−γ1 −g(0)1−
1
γ]≤cg(k)1−
1
γ,
and so
g(k)−1+1γ ≤ c
k ⇒g(k)≤ c k1−γγ
.
Therefore, by (3.14), it holds true that
|A2k| ≤
g(k)
k ≤ c k1−γγk
= c
k1−1γ
.
By recalling the definition of γ, we obtain 1
γ =
mn(p−1)
n−mp =s,
so that uβ −uβ
∗ ∈Ms(Ω). This ends the proof of Theorem 1.1.
References
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[2] F.Leonetti,F.Siepe,Integrability for solutions to some anisotropic elliptic equations,NonlinearAnal.,75(2012),2867-2873.
[3] I.Alessandra,F.Leonetti,Global integrability for weak soiutions to some anisotropic equations,Nonlinear Anallysis,113(2015),430-434.
[4] P.Hartman,G.Stampacchia,On some non-linear elliptic differential func-tional equations, Acta Math.115(1966),271-310.
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