LOREDANA CASOReceived 24 November 2004; Accepted 28 September 2005 Some estimates for solutions of the Dirichlet problem for second-order elliptic equations are obtained in this paper..
Trang 1LOREDANA CASO
Received 24 November 2004; Accepted 28 September 2005
Some estimates for solutions of the Dirichlet problem for second-order elliptic equations are obtained in this paper Here the leading coefficients are locally VMO functions, while the hypotheses on the other coefficients and the boundary conditions involve a suitable weight function
Copyright © 2006 Loredana Caso This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
1 Introduction
LetΩ be a bounded open subset ofRn,n ≥3, and let
L =
n
i, j =1
a i j(x) ∂
2
∂x i ∂x j +
n
i =1
a i(x) ∂
be a uniformly elliptic operator with measurable coefficients in Ω Several bounds for the solutions of the problem
Lu ≥ f , f ∈ L p(Ω),
u ∈ W2,p(Ω)∩ C o( ¯Ω),
u | ∂Ω≤0,
(D)
(p ∈] n/2, + ∞[) have been given, and the application of such estimates allows to obtain
certain uniqueness results for (D)
For instance, ifp ≥ n, a i,a ∈ L p(Ω) (with a≤0), a classical result of Pucci [4] shows that any solutionu of the problem ( D) verifies the bound
sup
whereK ∈ R+depends onΩ, n, p, a i L p(Ω)and on the ellipticity constant
Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2006, Article ID 76215, Pages 1 14
DOI 10.1155/JIA/2006/76215
Trang 2The casep < n, where additional hypotheses on the leading coefficients are necessary, has been studied by several authors Recently, a uniqueness result has been obtained in [3] under the assumption that thea i j’s are of class VMO,a i = a =0 andp ∈]1, +∞[ This
theorem has been extended to the casea i =0,a =0 in [7]
IfΩ is an arbitrary open subset ofRnandp ∈] n/2, + ∞[, a bound of type (1.2) and a consequent uniqueness result can be found in [1] In fact, it has been proved there that
if the coefficients a i j are bounded and locally VMO, the coefficients a i,a satisfy suitable
summability conditions and ess supΩa < 0, then for any solution u of the problem
Lu ≥ f , f ∈ Llocp (Ω),
u ∈ Wloc2,p(Ω)∩ C o( ¯Ω),
u | ∂Ω≤0, lim sup
| x |→+∞ u(x) ≤0 ifΩ is unbounded,
(D)
there exist a ballB ⊂⊂ Ω and a constant c ∈ R+such that
sup
Ω u ≤ c
B
−f −p
dx
1/ p
where f −is the negative part of f ,
B
−f −p
dx = 1
| B |
andc depends on n, p, on the ellipticity constant, and on the regularity of the coefficients
ofL.
The aim of this paper is to study a problem similar to that considered in [1], but with boundary conditions depending on an appropriate weight function More precisely, fix a weight functionσ ∈Ꮽ(Ω)∩ C ∞(Ω) (seeSection 2for the definition ofᏭ(Ω)) and s ∈ R,
we consider a solutionu of the problem
Lu ≥ f , f ∈ Llocp (Ω),
u ∈ Wloc2,p(Ω), lim sup
x → x o
σ s(x)u(x) ≤0 ∀ x o ∈ ∂Ω, lim sup
| x |→+∞
σ s(x)u(x) ≤0 ifΩ is unbounded.
(1.5)
If the coefficients ai j are bounded and locally VMO, the functions σa i and σ2a are
bounded and ess supΩσ2a < 0, we will prove that there exist a ball B ⊂⊂Ω and a constant
c o ∈ R+such that
sup
Ω σ s u ≤ c o
B
−σ s+2 f −p
dx
1/ p
wherec o depends onn, p, s, σ, on the ellipticity constant, and on the regularity of the
coefficients of L As a consequence, some uniqueness results are also obtained
Trang 32 Notation and function spaces
LetΩ be an open subset ofRnand letΣ(Ω) be the collection of all Lebesgue measurable subsets ofΩ For each E ∈Σ(Ω), we denote by| E |the Lebesgue measure ofE and put
E(x, r) = E ∩ B(x, r) ∀ x ∈ R n,∀ r ∈ R+, (2.1) whereB(x, r) is the open ball inRnof radiusr centered at x.
Denote byᏭ(Ω) the class of measurable functions ρ : Ω → R+such that
β −1ρ(y) ≤ ρ(x) ≤ βρ(y) ∀ y ∈Ω,∀ x ∈Ωy, ρ(y)
whereβ ∈ R+is independent ofx and y For ρ ∈Ꮽ(Ω), we put
S ρ = z ∈ ∂Ω : limx → z ρ(x) =0 . (2.3)
It is known that
ρ ∈ L ∞loc( ¯Ω), ρ −1∈ L ∞loc¯
Ω\ S ρ
and, ifS ρ = ∅,
ρ(x) ≤dist
x, S ρ
(see [2,6]) Having fixedρ ∈ Ꮽ(Ω) such that S ρ = ∂Ω, it is possible to find a function
σ ∈Ꮽ(Ω)∩ C ∞(Ω)∩ C0,1( ¯Ω) which is equivalent to ρ and such that
σ ∈ L ∞loc( ¯Ω), σ −1∈ L ∞loc(Ω), (2.6)
∂ α σ(x) ≤ c α σ1−| α |(x) ∀ x ∈Ω,∀ α ∈ N n
γ −1σ(y) ≤ σ(x) ≤ γσ(y) ∀ y ∈Ω, ∀ x ∈Ωy, σ(y)
wherec α,γ ∈ R+are independent ofx and y (see [6]) For more properties of functions
ofᏭ(Ω) we refer to [2,6]
IfΩ has the property
Ω(x,r) ≥ Ar n ∀ x ∈Ω, ∀ r ∈]0, 1], (2.10) whereA is a positive constant independent of x and r, it is possible to consider the space
BMO(Ω,t), t∈ R+, of functionsg ∈ L1loc( ¯Ω) such that
[g]BMO(Ω,t)= sup
x ∈Ω
r ∈]0,t]
Ω(x,r)
g −
Ω(x,r)
− g
d y < + ∞, (2.11)
where
Ω(x,r)
− gd y =1/ |Ω(x,r)| Ω(x,r) gd y If g ∈BMO(Ω)=BMO(Ω,t A), where
t A =sup
⎧
⎪
⎪t ∈ R+: sup
x ∈Ω
r ∈]0,t]
r n
Ω(x,r) ≤ A1
⎫
⎪
Trang 4we will say thatg ∈VMO(Ω) if [g]BMO(Ω,t) →0 fort →0+ A functionη[g] :R +→ R+is
called a modulus of continuity of g in VMO(Ω) if
BMO(Ω,t) ≤ η[g](t) ∀ t ∈ R+, lim
We say thatg ∈VMOloc(Ω) if (ζg)o ∈VMO(Rn) for anyζ ∈ C o ∞(Ω), where (ζg)odenotes the zero extension ofζg outside ofΩ A more detailed account of properties of the above defined spaces BMO(Ω) and VMO(Ω) can be found in [5]
3 An a priori bound
Fix p ∈] n/2, + ∞[ Let B be an open ball ofRn,n ≥3, of radiusδ We consider in B the
differential operator
L B =
n
i, j =1
α i j(x) ∂
2
∂x i ∂x j+
n
i =1
α i(x) ∂
with the following condition on the coefficients:
α i j = α ji ∈ L ∞(B) ∩VMO(B), i, j =1, , n,
∃ μ ∈ R+:
n
i, j =1
α i j ζ i ζ j ≥ μ | ζ |2 a.e inB, ∀ ζ ∈ R n,
α i ∈ L ∞(B), i =1, , n, α ∈ L ∞(B), α ≤0 a.e inB.
(h B)
Letμ0,μ1,μ2∈ R+such that
n
i, j =1
α i j
L ∞(B) ≤ μ0, δ
n
1=1
α i
L ∞(B) ≤ μ1, δ2 α L ∞(B) ≤ μ2. (3.2)
Note that under the assumption (h B), the operator L B from W2,p(B) into L p(B) is
bounded and the estimate
L B u
L p(B) ≤ c1 u W2,p(B) ∀ u ∈ W2,p(B) (3.3) holds, wherec1∈ R+depends onn, p, μ0,μ1,μ2
Lemma 3.1 Suppose that condition ( h B ) is verified, and let u be a solution of the problem
u ∈ W2,p(B),
L B u ≥ φ, φ ∈ L p(B),
u | ∂B ≤0.
(3.4)
Then there exists c ∈ R+such that
sup
B
u ≤ cδ2− n/ pφ −
Trang 5where c depends on n, p, μ, μ0,μ1,μ2, [p(α i j)]BMO(R n,·), and where p(α i j ) is an extension
of α i j toRn in L ∞(Rn)∩VMO(Rn ).
Proof Put B = B(y, δ), where y is the centre of B, and B ∗ = B(y, 1).
Consider the functionT : B → B ∗defined by the position
T(x) = y + x − y
and for each functiong defined on B, put g ∗ = g ◦ T −1
We observe that
L ∗ B u ∗ = δ2
L B u∗
where
L ∗ B =
n
i, j =1
α ∗ i j(z) ∂
2
∂z i ∂z j +δ
n
i =1
α ∗ i (z) ∂
∂z i+δ2α ∗(z). (3.8) Denote byp(α i j) an extension ofα i jtoRnsuch that
p
α i j
∈ L ∞
Rn
∩VMO
Rn
(3.9) (for the existence of such function see [5, Theorem 5.1]) Since
p
α i j
∗
∈ L ∞
Rn
∩VMO
Rn
, p
α i j
∗
| B ∗ = α ∗ i j, (3.10)
it follows that
α ∗ i j ∈ L ∞(B ∗)∩VMO(B ∗). (3.11) Moreover, the condition (h B) yields that
α ∗ i j = α ∗ ji, i, j =1, , n,
n
i, j =1
α ∗ i j ζ i ζ j ≥ μ | ζ |2 a.e inB ∗,∀ ζ ∈ R n,
α ∗ i ∈ L ∞(B ∗), i =1, , n, α ∗ ∈ L ∞(B ∗), α ∗ ≤0 a.e inB ∗
(3.12)
We observe that the condition (3.12) implies that forr, s ∈]1, +∞[ the modulus of
con-tinuity ofδα ∗ i in L r(B ∗) and that ofδ2α ∗in L s(B ∗) depend only on δα ∗ i L ∞(B ∗) and
δ2α ∗ L ∞(B ∗), respectively
Thus, applying (3.10), (3.12), and [7, Theorem 2.1], it follows that the problem
L ∗ B v = ψ ∈ L p(B ∗),
v ∈W2,p(B ∗)∩ W o1,p(B ∗) (3.13)
Trang 6has a unique solutionv satisfying the estimate
v W2,p(B ∗)≤ K ψ L p(B ∗), (3.14)
whereK depends on n, p, μ, μ0,μ1,μ2, [p(α i j)∗]BMO(R n,·)
The estimate (3.5) follows now from (3.14) using the same arguments of the proof of Lemma 3.2 [1] in order to obtain there (e B) from [1, (3.23)]
4 Hypotheses and preliminary results
LetΩ be an open subset ofRn,n ≥3 Fixρ ∈Ꮽ(Ω)∩ L ∞(Ω) such that Sρ = ∂Ω
Consider a functiong ∈ C ∞ o( ¯R +) satisfying the condition
0≤ g ≤1, g(t) =1 ift ≥1, g(t) =0 ift ≤1
For anyk ∈ N, we put
η k(x) =1
k ζ k(x) +
1− ζ k(x)
whereζ k(x) = g(kσ(x)), x ∈ Ω Clearly, η k ∈ C ∞(Ω) for any k∈ Nand
η k(x) =
⎧
⎪
⎪
1
k ifx ∈Ω¯k,
where
Ωk =
x ∈ Ω : σ(x) >1
k
In the following we will use the notation
f x =
n
i =1
f2
x i
1/2
, f xx =
n
i, j =1
f2
x i x j
1/2
It is easy to show that for eachk ∈ N,
σ(x) ≤ η k(x) ≤2σ(x), x ∈Ω\Ω¯k, (4.6)
c k σ(x) ≤ η k(x) ≤ σ(x), x ∈Ωk, (4.7)
η k(x)
x ≤ c1
σ(x)
η k(x)
xx ≤ c2
σ(x) 2
x+σ(x)
σ(x)
xx
Trang 7wherec k ∈ R+depends onk and σ, and c1,c2∈ R+depend only onn Moreover, for any
s ∈ R, we have
η s k(x)
x
η s
k(x) ≤ c3
η k(x)
x
η s k(x)
xx
η s k(x) ≤ c3
η k(x) 2
x+η k(x)
η k(x)
xx
wherec3∈ R+depends ons and n.
We consider inΩ the differential operator
L =
n
i, j =1
a i j(x) ∂
2
∂x i ∂x j+
n
i =1
a i(x) ∂
and put
L o =
n
i, j =1
a i j(x) ∂
2
We will make the following assumption on the coefficients of L:
a i j = a ji ∈ L ∞(Ω)∩VMOloc(Ω), i, j=1, , n,
∃ ν,ν0∈ R+:
n
i, j =1
a i j
L ∞(Ω)≤ ν0,
n
i, j =1
a i j ζ i ζ j ≥ ν | ζ |2 a.e inΩ,∀ ζ ∈ R n,
∃ ν1,ν2∈ R+: ess sup
Ω
σ(x)
n
i =1
a i(x)≤ ν1, ess sup
Ω
σ2(x) | a(x) |≤ ν2,
∃ a o ∈ R+: ess sup
Ω
σ2(x)a(x)
= − a o
(h1)
Fixeds ∈ R, let u be a solution of the problem
Lu ≥ f , f ∈ Llocp (Ω), u∈ Wloc2,p(Ω), lim sup
x → x o
σ s(x)u(x) ≤0 ∀ x o ∈ ∂Ω, lim sup
| x |→+∞ σ s(x)u(x) ≤0 ifΩ is unbounded.
(P)
For anyk ∈ N, we put
w k(x) = η s
Trang 8Lemma 4.1 Suppose that condition ( h1) holds Then, for any k ∈ N there exist functions
b i k(i =1, , n), b k,g k and positive constants β1and β2such that
ess sup
Ω
σ(x)
n
i =1
b k
i(x)≤ β1, (4.15)
ess sup
Ω
where β1 depends on s, n, ν0, ν1 and β2 depends on s, n, ν0, ν2 Moreover, the function
w k,k ∈ N , satisfies the following conditions:
w k ∈ Wloc2,p(Ω), limsup
x → x o
w k(x) ≤0 ∀ x o ∈ ∂Ω, lim sup
| x |→+∞ w k(x) ≤0 if Ω is unbounded, (4.18)
L o w k+
n
i =1
b k i
w k
x i+b k w k ≥ g k in Ω. (4.19)
Proof Fix k ∈ N From (4.6)–(4.11) and from (2.6), (2.8), it easily follows that the func-tionw k, defined by (4.14), verifies (4.18)
Moreover, observe that
L o w k − uL o η s k −2
n
i, j =1
a i j
η k s
x j u x i+
n
i =1
a i
η s k u
x i
− u
n
i =1
a i
η s k
x i+aη s k u = η s k Lu, x ∈ Ω.
(4.20)
Since
η s k
x j u x i =η s k u
x i
η s k
x j
η s k −
η s k
x i(η s k)x j
η s k 2
η s k u
from (4.20), (4.19) follows, where we have put
b k i = a i −2
n
j =1
a i j
η k s
x j
η k s , i =1, , n,
b k = a + 2
n
i, j =1
a i j
η s k
x i
η s k
x j
η s k
n
i, j =1
a i j
η s k
x i x j
η s k
,
g k = η k s f +
n
i =1
a i
η s k
x i
η s k w k .
(4.22)
On the other hand, using the hypothesis (h1), (4.6)–(4.11), and (2.8) it is easy to show that there existβ1∈ R+depending ons, n, ν0,ν1andβ2∈ R+depending ons, n, ν0,ν2,
Trang 9Now we suppose that the following hypothesis onρ holds:
lim
k →+∞
sup
Ω\Ωk
σ(x)
x+σ(x)
σ(x)
xx
An example of functionρ such that σ satisfies ( h2) is provided in [2]
Lemma 4.2 Suppose that conditions ( h1) and ( h2) hold Then there exists k o ∈ N such that
ess sup
Ω
σ(x)
n
i =1
b k o
i (x)≤ ν1+a o
2, ess sup
Ω
σ2(x)b k o(x)
≤ − a o
2,
g k o(x) ≥ η s k o(x) f (x) − a o
8σ
−2(x)w k
o(x), x ∈ Ω.
(4.23)
Proof From (4.10), (4.11), and hypothesis (h1), we deduce that
σ
n
i, j =1
a i j
η s k
x j
η s k
≤ c4
η k
x,
σ2
n
i, j =1
a i j
η k s
x i
η s k
x j
η s k
2
+σ2
n
i, j =1
a i j
η s k
x i x j
η s k
≤ c5
η k
2
x+η k
η k
xx
,
σ2
n
i =1
a i
η s k
x i
η s k
≤ c6
η k
x,
(4.24)
wherec4,c5∈ R+ depend ons, n, ν0 andc6∈ R+ depends on s, n, ν1 Observing that (η k)x =(η k)xx =0 in ¯Ωk, the statement follows now from (4.8), (4.9), (h1), (h2), and
5 Main results
It is well know that there exists a function ˜α ∈ C ∞(Ω)∩ C0,1( ¯Ω) which is equivalent to dist(·,∂Ω) (see, e.g., [8]) For every positive integerm, we define the function
ψ m:x ∈Ω¯ −→ g
m˜ α(x)
1− g
| x |
2m
whereg ∈ C ∞( ¯R +) verifies (4.1) It is easy to show thatψ m belongs toC o ∞(Ω) for every
m ∈ Nand
0≤ ψ m ≤1, suppψ m ⊆ E2m, ψ m |¯
where
E m =
x ∈Ω :| x | < m, ˜ α(x) > 1
m
Trang 10
Remark 5.1 It follows from hypothesis ( h1) and from [5, Lemma 4.2] that for anym ∈ N
the functions (ψ m a i j)o(obtained as extensions ofψ m a i jtoRnwith zero values out ofΩ) belong to VMO(Rn) and
ψ m a i j
o
BMO(R n,t) ≤ψ m a i j
fort small enough.
In the following we denote byw, b i,b, and g the functions defined by (4.14), (4.22), respectively, corresponding tok = k o, wherek ois the positive integer ofLemma 4.2
We can now prove the main result of the paper
Theorem 5.2 Suppose that conditions ( h1) and ( h2) hold, and let u be a solution of the problem ( P ) Then there exist an open ball B ⊂⊂ Ω and a constant c o ∈ R+such that
sup
Ω σ s(x)u(x) ≤ c o
B
−σ s+2 f −p
dx
1/ p
where c o depends only on n, p, s, γ, ν, ν0, ν1, ν2, a o , η[ψ m a i j] (m ∈ N).
Proof It can be assumed that supΩσ s(x) u(x) > 0 Thus it follows from (4.14) and (4.18) that there exists y ∈Ω such that supΩw(x) = w(y); moreover, there exists R o ∈]0,
dist(y, ∂ Ω)[ such that w(x) > 0 for all x ∈ B(y, R o)
Letλ, α, α o ∈ R+, withα o > 1 (that will be chosen late), such that
λα ≤min{R o,σ(y) }, α = α o σ(y). (5.6)
In the following we denote byB the open ball B(y, αλ).
We put
ϕ(x) =1 +λ2− | x − α2y |2, x ∈ B,¯ (5.7) and observe that
1≤ ϕ(x) ≤1 +λ2≤2, x ∈ B,¯ (5.8)
ϕ x i ≤2
α, ϕ x i ϕ x j ≤4 2
α2 , i, j =1, , n, (5.9)
ϕ x i x j =0 ifi = j, ϕ x i x j = −2
Consider now the functionv defined by
Obviously,
v | = w | − w(y) ≤0, v(y) = λ2w(y). (5.12)
Trang 11It is easy to show that
L o(ϕw) − wL o ϕ −2
n
i, j =1
a i j ϕ x j w x i+
n
i =1
b i(ϕw) x i
−
n
i =1
b i ϕ x i w + bϕw = ϕ
L o w +
n
i =1
b i w x i+bw
≥ ϕg inB.
(5.13)
Thus
L o(ϕw) +
n
i =1
d i(ϕw) x i+dϕw ≥ ϕg +
n
i =1
b i ϕ x i w inB, (5.14)
where
d i = b i −2
n
j =1
a i j ϕ x j
d = b + 2
n
i, j =1
a i j ϕ x i ϕ x j
ϕ2 −
n
i, j =1
a i j ϕ x i x j
Therefore we obtain from (5.14) that
L o v +
n
i =1
where
h = ϕg + w
n
i =1
Clearly, (2.9), (5.6), and (5.9) yield that
ϕ x
i ≤2 σ
α2
and hence it follows fromLemma 4.2that
h ≥ ϕη s k o f − a o
8σ
−2ϕw(y) −2γw(y)
ν1+a o 2
1
α2
o
σ −2(y) − dw(y)
≥ ϕη s k o f +
− d −
a
o
4γ
2+ 2γν1
α2
o
+γa o
α2
o
σ −2(y)
w(y).
(5.20)
The constantα ocan be chosen in such a way thatd < − d o σ −2(y) in B, where
d o = a o
4γ
2+ 2γν1
α2
o
+γa o
α2
o
Trang 12In fact, byLemma 4.2, (5.9) and (5.10), we have
d + d o σ −2(y) = b + 2
n
i, j =1
a i j
ϕ x i ϕ x j
ϕ2 −
n
i, j =1
a i j
ϕ x i x j
ϕ +d o σ
−2(y)
≤ − a o
2σ
−2+ 8ν o λ2
α2+ 2ν o 1
α2+d o σ −2(y)
≤
− γ2a o
4 +
10ν o+ 2γ ν1+γa o1
α2
o
σ −2(y),
(5.22)
and hence, fixedα osuch that
1
α2
o
≤ γ2a o
4
10ν o+ 2γ ν1+γa o, (5.23)
it follows that
By (5.11), (5.12), and (5.15)–(5.17), we deduce that the problem
v ∈ W2,p(B),
L o v +
n
i =1
d i v x i+dv ≥ ϕη s
k o f , f ∈ L p(B),
v | ∂B ≤0
(5.25)
satisfies the hypotheses ofLemma 3.1 Therefore, it follows from (5.6), (4.15), and (4.16) that there exists a constantc1∈ R+, depending onn, p, s, γ, ν, ν0,ν1,ν2, [p(a i j | B)]BMO(R n,·), such that
v(x) ≤ c1(λα)2− n/ p
ϕη s
k o f−
L p(B) ∀ x ∈ B. (5.26)
So it follows from (5.8) and from (5.26) withx = y that
λ2w(y) ≤ c1(λα)2− n/ p
ϕη s
k o f−
L p(B) ≤2c1(λα)2− n/ pη s
k o f −
L p(B) (5.27) Thus by (5.6) and (5.27) we have
w(y) ≤ c2(λα) − n/ p α2
o σ2(y)η s
k o f −
L p(B) ≤ c3(λα) − n/ p α2
oσ2η s
k o f −
L p(B), (5.28) wherec2,c3∈ R+depend on the same parameters asc1 Finally from (4.6), (4.7), (4.14), and (5.28) we obtain
sup
Ω σ s u ≤ c4(λα) − n/ p
B
σ2+s f −p
dx
1/ p
≤ c5
B
−σ s+2 f −dx1/ p