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BOUNDS FOR ELLIPTIC OPERATORS IN WEIGHTED SPACES LOREDANA 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 hy potheses 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 Let Ω be a bounded open subset of R n , n ≥ 3, and let L = n  i, j=1 a ij (x) ∂ 2 ∂x i ∂x j + n  i=1 a i (x) ∂ ∂x i + a(x) (1.1) 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 ∈ W 2,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, if p ≥ n, a i , a ∈ L p (Ω)(witha ≤ 0), a classical result of Pucci [4] shows that any solution u of the problem (D) verifies the bound sup Ω u ≤ K f  L p (Ω) , (1.2) where K ∈ R + depends on Ω, 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 2 Bounds for elliptic operators in weighted spaces The case p<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 the a ij ’s are of class VMO, a i = a = 0andp ∈]1,+∞[. This theorem has been extended to the case a i = 0, a = 0in[7]. If Ω is an arbitrary open subset of R n and p ∈]n/2,+∞[, a bound of type (1.2)anda consequent uniqueness result can be found in [1]. In fact, it has been proved there that if the coefficients a ij are bounded and locally VMO, the coefficients a i , a satisfy suitable summability conditions and esssup Ω a<0, then for any solution u of the problem Lu ≥ f , f ∈ L p loc (Ω), u ∈ W 2,p loc (Ω) ∩ C o ( ¯ Ω), u | ∂Ω ≤ 0, limsup |x|→+∞ u(x) ≤ 0ifΩ is unbounded, (D  ) there exist a ball B ⊂⊂ Ω and a constant c ∈ R + such that sup Ω u ≤ c   B −   f −   p dx  1/p , (1.3) where f − is the negative part of f ,  B −   f −   p dx = 1 |B|  B | f − | p dx, (1.4) and c depends on n, p, on the ellipticity constant, and on the regularity of the coefficients of L. The a im 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 function σ ∈ Ꮽ(Ω) ∩ C ∞ (Ω) (see Section 2 for the definition of Ꮽ(Ω)) and s ∈ R, we consider a solution u of the problem Lu ≥ f , f ∈ L p loc (Ω), u ∈ W 2,p loc (Ω), limsup x→x o σ s (x) u(x) ≤ 0 ∀x o ∈ ∂Ω, limsup |x|→+∞ σ s (x) u(x) ≤ 0ifΩ is unbounded. (1.5) If the coefficients a ij are bounded and locally VMO, the functions σa i and σ 2 a are bounded and esssup Ω σ 2 a<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 , (1.6) where c o depends on n, 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. Loredana Ca so 3 2. Notation and function spaces Let Ω be an open subset of R n and let Σ(Ω) be the collection of all Lebesgue measurable subsets of Ω.ForeachE ∈ Σ(Ω), we denote by |E| the Lebesgue measure of E and put E(x,r) = E ∩ B(x,r) ∀x ∈ R n , ∀r ∈ R + , (2.1) where B(x,r) is the open ball in R n of radius r centered at x. Denote by Ꮽ(Ω) the class of measurable functions ρ : Ω → R + such that β −1 ρ(y) ≤ ρ(x) ≤ βρ(y) ∀y ∈ Ω, ∀x ∈ Ω  y,ρ(y)  , (2.2) where β ∈ R + is independent of x and y.Forρ ∈ Ꮽ(Ω), we put S ρ =  z ∈ ∂Ω :lim x→z ρ(x) = 0  . (2.3) It is known that ρ ∈ L ∞ loc ( ¯ Ω), ρ −1 ∈ L ∞ loc  ¯ Ω \ S ρ  , (2.4) and, if S ρ =∅, ρ(x) ≤ dist  x, S ρ  ∀ x ∈ Ω (2.5) (see [2, 6]). Having fixed ρ ∈ Ꮽ(Ω)suchthatS ρ = ∂Ω, it is possible to find a function σ ∈ Ꮽ(Ω) ∩ C ∞ (Ω) ∩ C 0,1 ( ¯ Ω) which is equivalent to ρ and such that σ ∈ L ∞ loc ( ¯ Ω), σ −1 ∈ L ∞ loc (Ω), (2.6) σ(x) ≤ dist(x,∂Ω) ∀x ∈ Ω, (2.7)   ∂ α σ(x)   ≤ c α σ 1−|α| (x) ∀x ∈ Ω, ∀α ∈ N n o , (2.8) γ −1 σ(y) ≤ σ(x) ≤ γσ(y) ∀y ∈ Ω, ∀x ∈ Ω  y,σ(y)  , (2.9) where c α ,γ ∈ R + are independent of x and y (see [6]). For more properties of functions of Ꮽ(Ω)wereferto[2, 6]. If Ω has the property   Ω(x, r)   ≥ Ar n ∀x ∈ Ω, ∀r ∈]0,1], (2.10) where A is a positive constant independent of x and r, it i s possible to consider the space BMO(Ω,t), t ∈ R + , of functions g ∈ L 1 loc ( ¯ Ω)suchthat [g] BMO(Ω,t) = sup x∈Ω r∈]0,t]  Ω(x,r) −     g −  Ω(x,r) − g     dy <+∞, (2.11) where  Ω(x,r) − gdy = 1/|Ω(x,r)|  Ω(x,r) gdy.Ifg ∈ BMO(Ω) = BMO(Ω,t A ), where t A = sup ⎧ ⎪ ⎨ ⎪ ⎩ t ∈ R + :sup x∈Ω r∈]0,t] r n   Ω(x, r)   ≤ 1 A ⎫ ⎪ ⎬ ⎪ ⎭ , (2.12) 4 Bounds for elliptic operators in weighted spaces we will say that g ∈ VMO(Ω)if[g] BMO(Ω,t) → 0fort → 0 + . A function η[g]:R + → R + is called a modulus of continuity of g in VMO(Ω)if BMO(Ω,t) ≤ η[g](t) ∀t ∈ R + , lim t→0 + η[g](t) = 0. (2.13) We say that g ∈ VMO loc (Ω)if(ζg) o ∈ VMO(R n )foranyζ ∈ C ∞ o (Ω), where (ζg) o denotes the zero extension of ζg outside of Ω. A more detailed account of properties of the above defined spaces BMO(Ω)andVMO(Ω) can be found in [5]. 3. An a priori bound Fix p ∈]n/2,+∞[. Let B be an open ball of R n , n ≥ 3, of radius δ. We consider in B the differential operator L B = n  i, j=1 α ij (x) ∂ 2 ∂x i ∂x j + n  i=1 α i (x) ∂ ∂x i + α(x), (3.1) with the following condition on the coefficients: α ij = α ji ∈ L ∞ (B) ∩ VMO(B), i, j = 1, ,n, ∃μ ∈ R + : n  i, j=1 α ij ζ i ζ j ≥ μ|ζ| 2 a.e. in B, ∀ζ ∈ R n , α i ∈ L ∞ (B), i = 1, ,n, α ∈ L ∞ (B), α ≤ 0a.e.inB. (h B ) Let μ 0 ,μ 1 ,μ 2 ∈ R + such that n  i, j=1   α ij   L ∞ (B) ≤ μ 0 , δ n  1=1   α i   L ∞ (B) ≤ μ 1 , δ 2 α L ∞ (B) ≤ μ 2 . (3.2) Note that under the assumption (h B ), the operator L B from W 2,p (B)intoL p (B)is bounded and the estimate   L B u   L p (B) ≤ c 1 u W 2,p (B) ∀u ∈ W 2,p (B) (3.3) holds, where c 1 ∈ R + depends on n, p, μ 0 , μ 1 , μ 2 . Lemma 3.1. Suppose that condition (h B )isverified,andletu beasolutionoftheproblem u ∈ W 2,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 ≤ cδ 2−n/p   φ −   L p (B) , (3.5) Loredana Ca so 5 where c depends on n, p, μ, μ 0 , μ 1 , μ 2 ,[p(α ij )] BMO(R n ,·) ,andwherep(α ij ) is an extension of α ij to R n in L ∞ (R n ) ∩ VMO(R n ). Proof. Put B = B(y,δ), where y is the centre of B,andB ∗ = B(y,1). Consider the function T : B → B ∗ defined by the position T(x) = y + x − y δ = z, (3.6) and for each function g defined on B,putg ∗ = g ◦ T −1 . We observe that L ∗ B u ∗ = δ 2  L B u  ∗ , (3.7) where L ∗ B = n  i, j=1 α ∗ ij (z) ∂ 2 ∂z i ∂z j + δ n  i=1 α ∗ i (z) ∂ ∂z i + δ 2 α ∗ (z). (3.8) Denote by p(α ij ) an extension of α ij to R n such that p  α ij  ∈ L ∞  R n  ∩ VMO  R n  (3.9) (for the existence of such function see [5, Theorem 5.1]). Since p  α ij  ∗ ∈ L ∞  R n  ∩ VMO  R n  , p  α ij  ∗ | B ∗ = α ∗ ij , (3.10) it follows that α ∗ ij ∈ L ∞ (B ∗ ) ∩ VMO(B ∗ ). (3.11) Moreover, the condition (h B )yieldsthat α ∗ ij = α ∗ ji , i, j = 1, ,n, n  i, j=1 α ∗ ij ζ i ζ j ≥ μ|ζ| 2 a.e. in B ∗ , ∀ζ ∈ R n , α ∗ i ∈ L ∞ (B ∗ ), i = 1, ,n, α ∗ ∈ L ∞ (B ∗ ), α ∗ ≤ 0a.e.inB ∗ . (3.12) We observe that the condition (3.12) implies that for r, s ∈]1,+∞[themodulusofcon- tinuity of δα ∗ i in L r (B ∗ ) and that of δ 2 α ∗ in L s (B ∗ )dependonlyonδα ∗ i  L ∞ (B ∗ ) and δ 2 α ∗  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 ∈ W 2,p (B ∗ ) ∩ o W 1,p (B ∗ ) (3.13) 6 Bounds for elliptic operators in weighted spaces has a unique solution v satisfying the estimate v W 2,p (B ∗ ) ≤ Kψ L p (B ∗ ) , (3.14) where K depends on n, p, μ, μ 0 , μ 1 , μ 2 ,[p(α ij ) ∗ ] 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 Let Ω be an open subset of R n , n ≥ 3. Fix ρ ∈ Ꮽ(Ω) ∩ L ∞ (Ω)suchthatS ρ = ∂Ω. Consider a function g ∈ C ∞ o ( ¯ R + ) satisfying the condition 0 ≤ g ≤ 1, g(t) = 1ift ≥ 1, g(t) = 0ift ≤ 1 2 . (4.1) For any k ∈ N,weput η k (x) = 1 k ζ k (x)+  1 − ζ k (x)  σ(x), x ∈ Ω, (4.2) where ζ k (x) = g(kσ(x)), x ∈ Ω.Clearly,η k ∈ C ∞ (Ω)foranyk ∈ N and η k (x) = ⎧ ⎪ ⎨ ⎪ ⎩ 1 k if x ∈ ¯ Ω k , σ(x)ifx ∈ Ω \ Ω 2k , (4.3) where Ω k =  x ∈ Ω : σ(x) > 1 k  , k ∈ N. (4.4) In the following we will use the notation f x =  n  i=1 f 2 x i  1/2 , f xx =  n  i, j=1 f 2 x i x j  1/2 . (4.5) It is easy to show that for each k ∈ N, σ(x) ≤ η k (x) ≤ 2σ(x), x ∈ Ω \ ¯ Ω k , (4.6) c  k σ(x) ≤ η k (x) ≤ σ(x), x ∈ Ω k , (4.7)  η k (x)  x ≤ c 1  σ(x)  x , x ∈ Ω, (4.8)  η k (x)  xx ≤ c 2  σ(x)  2 x + σ(x)  σ(x)  xx σ(x) , x ∈ Ω, (4.9) Loredana Ca so 7 where c  k ∈ R + depends on k and σ,andc 1 ,c 2 ∈ R + depend only on n.Moreover,forany s ∈ R,wehave  η s k (x)  x η s k (x) ≤ c 3  η k (x)  x σ(x) , x ∈ Ω, (4.10)  η s k (x)  xx η s k (x) ≤ c 3  η k (x)  2 x + η k (x)  η k (x)  xx σ 2 (x) , x ∈ Ω, (4.11) where c 3 ∈ R + depends on s and n. We consider in Ω the differential operator L = n  i, j=1 a ij (x) ∂ 2 ∂x i ∂x j + n  i=1 a i (x) ∂ ∂x i + a(x), (4.12) and put L o = n  i, j=1 a ij (x) ∂ 2 ∂x i ∂x j . (4.13) We will make the following assumption on the coefficients of L: a ij = a ji ∈ L ∞ (Ω) ∩ VMO loc (Ω), i, j = 1, ,n, ∃ν,ν 0 ∈ R + : n  i, j=1   a ij   L ∞ (Ω) ≤ ν 0 , n  i, j=1 a ij ζ i ζ j ≥ ν|ζ| 2 a.e. in Ω, ∀ζ ∈ R n , ∃ν 1 ,ν 2 ∈ R + :esssup Ω  σ(x) n  i=1   a i (x)    ≤ ν 1 ,esssup Ω  σ 2 (x)|a( x)|  ≤ ν 2 , ∃a o ∈ R + :esssup Ω  σ 2 (x) a(x)  =− a o . (h 1 ) Fixed s ∈ R,letu be a solution of the problem Lu ≥ f , f ∈ L p loc (Ω), u ∈ W 2,p loc (Ω), limsup x→x o σ s (x) u(x) ≤ 0 ∀x o ∈ ∂Ω, limsup |x|→+∞ σ s (x) u(x) ≤ 0ifΩ is unbounded. (P) For any k ∈ N,weput w k (x) = η s k (x) u(x), x ∈ Ω. (4.14) 8 Bounds for elliptic operators in weighted spaces Lemma 4.1. Suppose that condition (h 1 )holds.Then,foranyk ∈ N there exist functions b k i (i = 1, ,n), b k , g k and positive constants β 1 and β 2 such that esssup Ω  σ(x) n  i=1   b k i (x)    ≤ β 1 , (4.15) esssup Ω  σ 2 (x)   b k (x)    ≤ β 2 , (4.16) g k ∈ L p loc (Ω), (4.17) where β 1 depends on s, n, ν 0 , ν 1 and β 2 depends on s, n, ν 0 , ν 2 .Moreover,thefunction w k , k ∈ N, satisfies the following conditions: w k ∈ W 2,p loc (Ω), limsup x→x o w k (x) ≤ 0 ∀x o ∈ ∂Ω, limsup |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)andfrom(2.6), (2.8), it easily follows that the func- tion w k ,definedby(4.14), verifies (4.18). Moreover, observe that L o w k − uL o η s k − 2 n  i, j=1 a ij  η s k  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  , (4.21) from (4.20), (4.19) follows, where we have put b k i = a i − 2 n  j=1 a ij  η s k  x j η s k , i = 1, ,n, b k = a +2 n  i, j=1 a ij  η s k  x i  η s k  x j  η s k  2 − n  i, j=1 a ij  η s k  x i x j η s k , g k = η s k 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 (h 1 ), (4.6)–(4.11), and (2.8)itiseasytoshow that there exist β 1 ∈ R + depending on s, n, ν 0 , ν 1 and β 2 ∈ R + depending on s, n, ν 0 , ν 2 , such that (4.15), (4.16), (4.17)hold.  Loredana Ca so 9 Now we suppose that the following hypothesis on ρ holds: lim k→+∞  sup Ω\Ω k  σ(x)  x + σ(x)  σ(x)  xx   = 0. (h 2 ) An example of function ρ such that σ satisfies (h 2 )isprovidedin[2]. Lemma 4.2. Suppose that conditions (h 1 )and(h 2 ) hold. Then there exists k o ∈ N such that esssup Ω  σ(x) n  i=1   b k o i (x)    ≤ ν 1 + a o 2 , esssup Ω  σ 2 (x) b k o (x)  ≤− a o 2 , g k o (x) ≥ η s k o (x) f (x) − a o 8 σ −2 (x)   w k o (x)   , x ∈ Ω. (4.23) Proof. From (4.10), (4.11), and hyp othesis (h 1 ), we deduce that σ       n  i, j=1 a ij  η s k  x j η s k       ≤ c 4  η k  x , σ 2       n  i, j=1 a ij  η s k  x i  η s k  x j  η s k  2       + σ 2       n  i, j=1 a ij  η s k  x i x j η s k       ≤ c 5  η k  2 x + η k  η k  xx  , σ 2       n  i=1 a i  η s k  x i η s k       ≤ c 6  η k  x , (4.24) where c 4 ,c 5 ∈ R + depend on s, n, ν 0 and c 6 ∈ R + depends on s, n, ν 1 . Observing that (η k ) x = (η k ) xx = 0in ¯ Ω k , the statement follows now from (4.8), (4.9), (h 1 ), (h 2 ), and (4.24).  5. Main results It is well know that there exists a function ˜ α ∈ C ∞ (Ω) ∩ C 0,1 ( ¯ Ω)whichisequivalentto dist( ·,∂Ω) (see, e.g., [8]). For every positive integer m, we define the function ψ m : x ∈ ¯ Ω −→ g  m ˜ α(x)   1 − g  |x| 2m  , (5.1) where g ∈ C ∞ ( ¯ R + )verifies(4.1). It is easy to show that ψ m belongs to C ∞ o (Ω)forevery m ∈ N and 0 ≤ ψ m ≤ 1, suppψ m ⊆ E 2m , ψ m | ¯ E m = 1, (5.2) where E m =  x ∈ Ω : |x| <m, ˜ α(x) > 1 m  . (5.3) 10 Bounds for elliptic oper ators in weighted spaces Remark 5.1. It follows from hypothesis (h 1 )andfrom[5, Lemma 4.2] that for any m ∈ N the functions (ψ m a ij ) o (obtained as extensions of ψ m a ij to R n with zero values out of Ω) belong to VMO( R n )and  ψ m a ij  o  BMO(R n ,t) ≤  ψ m a ij  BMO(Ω,t) , (5.4) for t small enough. In the following we denote by w, b i , b,andg the functions defined by (4.14), (4.22), respectively, c orresponding to k = k o ,wherek o is the positive integer of Lemma 4.2 We can now prove the main result of the paper. Theorem 5.2. Suppose that conditions (h 1 )and(h 2 )hold,andletu 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 , (5.5) where c o depends only on n, p, s, γ, ν, ν 0 , ν 1 , ν 2 , a o , η[ψ m a ij ](m ∈ N). Proof. It can be assumed that sup Ω σ s (x) u(x) > 0. Thus it follows from (4.14)and(4.18) that there exists y ∈ Ω such that sup Ω w(x) = w(y); moreover, there exists R o ∈]0, dist(y,∂Ω)[ such that w(x) > 0forallx ∈ B(y,R o ). Let λ,α,α o ∈ R + ,withα o > 1 (that w ill be chosen late), such that λα ≤ min{R o ,σ(y)}, α = α o σ(y). (5.6) In the following we denote by B the open ball B(y,αλ). We put ϕ(x) = 1+λ 2 − | x − y| 2 α 2 , x ∈ ¯ B, (5.7) and observe that 1 ≤ ϕ(x) ≤ 1+λ 2 ≤ 2, x ∈ ¯ B, (5.8) ϕ x i ≤ 2λ α , ϕ x i ϕ x j ≤ 4λ 2 α 2 , i, j = 1, ,n, (5.9) ϕ x i x j = 0ifi = j, ϕ x i x j =− 2 α 2 if i = j. (5.10) Consider now the function v defined by v(x) = ϕ(x)w(x) − w(y), x ∈ ¯ B. (5.11) Obviously, v | ∂Ω = w | ∂Ω − w(y) ≤ 0, v(y) = λ 2 w(y). (5.12) [...]... Troisi, and A Vitolo, BMO spaces on domains of Rn , Ricerche di Matematica 45 (1996), no 2, 355–378 [6] M Troisi, Su una classe di funzoni peso, Rendiconti Accademia Nazionale delle Scienze detta dei XL Serie V Memorie di Matematica 10 (1986), no 1, 141–152 14 Bounds for elliptic operators in weighted spaces [7] C Vitanza, A new contribution to the W 2,p regularity for a class of elliptic second order... Transirico, On the maximum principle for elliptic operators, Mathematical Inequalities & Applications 7 (2004), no 3, 405–418 [2] L Caso and M Transirico, Some remarks on a class of weight functions, Commentationes Mathematicae Universitatis Carolinae 37 (1996), no 3, 469–477 [3] F Chiarenza, F Frasca, and P Longo, W 2,p -solvability of the Dirichlet problem for nondivergence elliptic equations with VMO... −2 ao 2 s ≥ ϕηko f + − d − γ + 2 2 + 2 σ (y) w(y) 4 αo αo s h ≥ ϕηko f − (5.20) The constant αo can be chosen in such a way that d < −do σ −2 (y) in B, where do = γν1 γao ao 2 γ +2 2 + 2 4 αo αo (5.21) 12 Bounds for elliptic operators in weighted spaces In fact, by Lemma 4.2, (5.9) and (5.10), we have d + do σ −2 (y) = b + 2 n ai j i, j =1 n ϕxi ϕx j ϕxi x j − ai j + do σ −2 (y) 2 ϕ ϕ i, j =1 a o... order equations with discontinuous coefficients, Le Matematiche 48 (1993), no 2, 287–296 [8] W P Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics, vol 120, Springer, New York, 1989 Loredana Caso: Dipartimento di Matematica e Informatica, Facolt` di Scienze MM.FF.NN., a Universit` di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy a E-mail address: lorcaso@unisa.it ... where c2 ,c3 ∈ R+ depend on the same parameters as c1 Finally from (4.6), (4.7), (4.14), and (5.28) we obtain sup σ s u ≤ c4 (λα)−n/ p Ω B σ 2+s f − p 1/ p dx ≤ c5 − σ s+2 f − dx B 1/ p , (5.29) Loredana Caso 13 where c4 ,c5 ∈ R+ depend on the same parameters as c1 and on ao Then, if we choose p ai j |B = ψm1 ai j o , (5.30) where m1 is a positive integer such that ψm1 |B = 1, (5.5) follows from (5.29),.. .Loredana Caso 11 It is easy to show that n n Lo (ϕw) − wLo ϕ − 2 ai j ϕx j wxi + i, j =1 bi (ϕw)xi i =1 n bi wxi + bw ≥ ϕg bi ϕxi w + bϕw = ϕ Lo w + − (5.13) n i =1 in B i=1 Thus n n di (ϕw)xi + dϕw ≥ ϕg + Lo (ϕw) + i=1 bi ϕxi w in B, (5.14) i =1 where n ai j ϕx j , ϕ ai j n ϕxi ϕx j ϕxi x j − ai j ϕ2 ϕ i, j =1 (5.16)... Therefore we obtain from (5.14) that n Lo v + i=1 where n h = ϕg + w bi ϕxi − dw(y) (5.18) i=1 Clearly, (2.9), (5.6), and (5.9) yield that ϕxi ≤ 2γ σ α2 σ 2 (y) o in B, (5.19) and hence it follows from Lemma 4.2 that a 1 a o −2 σ ϕw(y) − 2γw(y) ν1 + o 2 σ −2 (y) − dw(y) 8 2 αo γν1 γao −2 ao 2 s ≥ ϕηko f + − d − γ + 2 2 + 2 σ (y) w(y) 4 αo αo s h ≥ ϕηko f − (5.20) The constant αo can be chosen in such... The result can be obtained applying Theorem 5.2 to the functions u and −u The following uniqueness result is an obvious consequence of Corollary 5.3 Corollary 5.4 If the hypotheses (h1 ) and (h2 ) hold, then the problem Lu = 0, 2,p u ∈ Wloc (Ω), limsup σ s (x)u(x) = 0 ∀xo ∈ ∂Ω, x→xo s (p ) limsup σ (x)u(x) = 0 if Ω is unbounded |x|→+∞ has only the zero solution References [1] L Caso, P Cavaliere, and... αo 4 10νo + 2γν1 + γao (5.23) d < −do σ −2 (y) in B (5.24) it follows that By (5.11), (5.12), and (5.15)–(5.17), we deduce that the problem v ∈ W 2,p (B), n Lo v + i =1 s di vxi + dv ≥ ϕηko f , f ∈ L p (B), (5.25) v|∂B ≤ 0 satisfies the hypotheses of Lemma 3.1 Therefore, it follows from (5.6), (4.15), and (4.16) that there exists a constant c1 ∈ R+ , depending on n, p, s, γ, ν, ν0 , ν1 , ν2 , [p(ai j . BOUNDS FOR ELLIPTIC OPERATORS IN WEIGHTED SPACES LOREDANA CASO Received 24 November 2004; Accepted 28 September 2005 Some estimates for solutions of the Dirichlet problem for second-order elliptic. (2.12) 4 Bounds for elliptic operators in weighted spaces we will say that g ∈ VMO(Ω)if[g] BMO(Ω,t) → 0fort → 0 + . A function η[g]:R + → R + is called a modulus of continuity of g in VMO(Ω)if BMO(Ω,t) ≤. unbounded. (P) For any k ∈ N,weput w k (x) = η s k (x) u(x), x ∈ Ω. (4.14) 8 Bounds for elliptic operators in weighted spaces Lemma 4.1. Suppose that condition (h 1 )holds.Then,foranyk ∈ N there

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