BOUNDS FOR ELLIPTIC OPERATORS IN WEIGHTED SPACES

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.


Introduction
Let Ω be a bounded open subset of R n , n ≥ 3, and let be a uniformly elliptic operator with measurable coefficients in Ω. Several bounds for the solutions of the problem (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) (with a ≤ 0), a classical result of Pucci [4] shows that any solution u of the problem (D) verifies the bound 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 i j 's are of class VMO, a i = a = 0 and p ∈]1, +∞[. This theorem has been extended to the case a i = 0, a = 0 in [7].
If Ω is an arbitrary open subset of R n and p ∈]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 esssup Ω a < 0, then for any solution u of the problem there exist a ball B ⊂⊂ Ω and a constant c ∈ R + such that and c depends on n, p, on the ellipticity constant, and on the regularity of the coefficients of L. 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 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 If the coefficients a i j 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 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.

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. For each E ∈ Σ(Ω), we denote by |E| the Lebesgue measure of E and put 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 where β ∈ R + is independent of x and y. For ρ ∈ Ꮽ(Ω), we put It is known that (see [2,6]). Having fixed ρ ∈ Ꮽ(Ω) such that S 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 where c α ,γ ∈ R + are independent of x and y (see [6]). For more properties of functions of Ꮽ(Ω) we refer to [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 is possible to consider the space BMO(Ω,t), t ∈ R + , of functions g ∈ L 1 loc (Ω) such that , 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) and VMO(Ω) can be found in [5].

An a priori bound
be an open ball of R n , n ≥ 3, of radius δ. We consider in B the differential operator with the following condition on the coefficients: Note that under the assumption (h B ), the operator L B from W 2,p (B) into L p (B) is bounded and the estimate

. Suppose that condition (h B ) is verified, and let u be a solution of the problem
Then there exists c ∈ R + such that

5)
where c depends on n, p, μ, μ 0 , Proof. Put B = B(y,δ), where y is the centre of B, and B * = B(y,1). Consider the function T : B → B * defined by the position and for each function g defined on B, put g * = g • T −1 .
We observe that (for the existence of such function see [5, Theorem 5.1]). Since Moreover, the condition (h B ) yields that (3.12) We observe that the condition (3.12) implies that for r, s ∈]1, +∞[ the modulus of continuity of δα * i in L r (B * ) and that of δ 2 α * in L s (B * ) depend only on δα * i L ∞ (B * ) and δ 2 α * L ∞ (B * ) , respectively.
Consider a function g ∈ C ∞ o (R + ) satisfying the condition For any k ∈ N, we put where ζ k (x) = g(kσ(x)), x ∈ Ω. Clearly, η k ∈ C ∞ (Ω) for any k ∈ N and where In the following we will use the notation It is easy to show that for each k ∈ N, where c k ∈ R + depends on k and σ, and c 1 ,c 2 ∈ R + depend only on n. Moreover, for any s ∈ R, we have where c 3 ∈ R + depends on s and n.

Main results
It is well know that there exists a functionα ∈ C ∞ (Ω) ∩ C 0,1 (Ω) which is equivalent to dist(·,∂Ω) (see, e.g., [8]). For every positive integer m, we define the function where g ∈ C ∞ (R + ) verifies (4.1). It is easy to show that ψ m belongs to C ∞ o (Ω) for every m ∈ N and where for t small enough.
In the following we denote by w, b i , b, and g the functions defined by (4.14), (4.22), respectively, corresponding to k = k o , where k o is the positive integer of Lemma 4. 2 We can now prove the main result of the paper. Proof. It can be assumed that sup Ω σ s (x) u(x) > 0. Thus it follows from (4.14) and ( and observe that It is easy to show that where Therefore we obtain from (5.14) that   where m 1 is a positive integer such that ψ m1 |B = 1, (5.5) follows from (5.29), (5.30), and from Remark 5.1.  Proof. 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.