On the maximum principle for elliptic operators in weighted spaces

Loredana Caso; Roberta D’Ambrosio

doi:10.1186/1687-2770-2014-91

Research
Open Access

On the maximum principle for elliptic operators in weighted spaces

Loredana Caso¹Email author and
Roberta D’Ambrosio¹

Boundary Value Problems20142014:91

https://doi.org/10.1186/1687-2770-2014-91

Received: 16 January 2014
Accepted: 15 April 2014
Published: 6 May 2014

Abstract

We establish a maximum principle for subsolutions of second order elliptic equations. In particular, we consider some linear operators with leading coefficients locally VMO, while the other coefficients and the boundary conditions involve a suitable weight function.

MSC:35J25, 35R05, 35B50.

Keywords

elliptic operators
VMO-coefficients
weighted Sobolev spaces
maximum principle

1 Introduction

It is well known that a priori estimates and uniqueness results, which are necessary in the proof of the well-posedness for boundary value problems for elliptic equations in nondivergence form, are based on Aleksandrov type estimates, i.e., on estimates for the maximum of a solution in terms of the $L^{n}$ -norm of the right-hand side.

If Ω is a bounded domain in

R^{n}

(

n > 2

) and

L = \sum_{i, j = 1}^{n} a_{i j} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} + \sum_{i = 1}^{n} a_{i} \frac{\partial}{\partial x_{i}} + a,

(1.1)

is a uniformly elliptic operator in Ω, the classical result of AD Aleksandrov states that if

u \in C^{0} (\bar{Ω}) \cap W^{2, n} (Ω)

, with

u \leq 0

in ∂Ω, verifies

L u \geq f

, where

a_{i}, a, f \in L^{n} (Ω)

(

a \leq 0

), then

sup_{Ω} u \leq c {∥ f ∥}_{L^{n} (Ω)},

(1.2)

where $c \in R_{+}$ depends only on n, Ω, ${∥ a_{i} ∥}_{L^{n} (Ω)}$ and on the ellipticity constant.

There have been various directions of developments and extensions of Aleksandrov estimate. For example, maximum principles have been established in different types of boundary problems, such as in the stationary oblique derivative problem or in the stationary Venttsel’ problem. Another direction of development of the Aleksandrov ideas is the extension of maximum estimates to equations with lower order coefficients and right-hand sides in other function classes (for example, in spaces with anisotropic norms or weighted spaces). In particular, a large number of works is devoted to the weakening of requirements for the right-hand side of the equation considered (see, for example, [1] and its large bibliography).

In this framework, it is well known that additional hypotheses on the leading coefficients are necessary to obtain the estimates. Several authors have obtained estimates for the maximum of a solution through the $L^{p}$ -norms of the right-hand side ( $p > n / 2$ ) under different conditions on the leading coefficients.

For instance, if Ω is an arbitrary open subset of

R^{n}

and

p \in] n / 2, + \infty [

, a bound of type (1.2) and a consequent uniqueness result can be found in [2]. In fact, it has been proved that, if the coefficients

a_{i j}

are bounded and locally VMO, the coefficients

a_{i}

, a satisfy suitable summability conditions, and

{ess sup}_{Ω} a < 0

, then for any solution u of the problem

{\begin{cases} u \in W_{loc}^{2, p} (Ω) \cap C^{0} (\bar{Ω}), \\ L u \geq f, f \in L_{loc}^{p} (Ω), \\ u_{|_{\partial Ω}} \leq 0, \\ {lim sup}_{| x | \to + \infty} u (x) \leq 0 if Ω is unbounded, \end{cases}

(1.3)

there exist a ball

B \subset \subset Ω

and a constant

c \in R_{+}

such that

sup_{Ω} u \leq c {(⨏_{B} {| f^{-} |}^{p} d x)}^{\frac{1}{p}},

(1.4)

where

f^{-}

is the negative part of f,

⨏_{B} {| f^{-} |}^{p} d x = \frac{1}{| B |} \int_{B} {| f^{-} |}^{p} d x,

and c depends on n, p, on the ellipticity constant and on the regularity of the coefficients of L.

If the boundary of a domain has various singularities, as for example corners or edges, then, in accordance with the linear theory, it is natural to assume that the lower order coefficients and the right-hand side of the equation belong to some weighted spaces $L^{p}$ , where the weight is usually a power of the distance function from the ‘singular set’ on the boundary of domain. In these cases, the estimates on the solutions are obtained in terms of such weight function.

For instance, if ρ is a bounded weight function related to the distance function from a non-empty subset

S_{ρ}

of the boundary of an arbitrary domain Ω, not necessarily bounded and regular (see Section 2 for the definition of such weight function), in [3] has been studied a problem similar to the problem (1.3) with boundary conditions and data related to the weight function ρ. In particular, if

s \in R

,

S_{ρ} = \partial Ω

, the coefficients

a_{i j}

are bounded and locally VMO, the coefficients

a_{i}

, a belong to suitable weighted spaces

L^{\infty}

, in [3] the author has proved that the solution u of the problem

{\begin{cases} u \in W_{loc}^{2, p} (Ω), \\ L u \geq f, f \in L_{loc}^{p} (Ω), \\ {lim sup}_{x \to x_{o}} ρ^{s} (x) u (x) \leq 0, \forall x_{o} \in \partial Ω, \\ {lim sup}_{| x | \to + \infty} ρ^{s} (x) u (x) \leq 0 if Ω is unbounded, \end{cases}

(1.5)

verifies the estimate

sup_{x \in Ω} ρ^{s} (x) u (x) \leq c {(⨏_{B} {| ρ^{s + 2} f^{-} |}^{p} d x)}^{\frac{1}{p}},

(1.6)

where $B \subset \subset Ω$ is an open ball and the constant $c \in R_{+}$ depends on n, p, s, ρ, on the ellipticity constant and on the regularity of the coefficients of L. As a consequence, some uniqueness results are also obtained. Results of this type are also established in [4] under the more general hypothesis $\emptyset \neq S_{ρ} \subset \partial Ω$ , but for an operator L with coefficients $a_{i} = 0$ .

The aim of this paper is to improve the above quoted results in [3] by obtaining a similar estimate under much weaker assumptions. In particular, the main difference lies in the hypotheses on the coefficients $a_{i}$ , a which are not supposed to belong to weighted spaces $L^{\infty}$ but just to appropriate weighted Sobolev spaces $K_{t}^{r} (Ω)$ (see Section 2 for the definition of such weighted spaces), which strictly contain the weighted spaces $L^{\infty}$ . Moreover, as in [4], we consider the more general hypothesis $\emptyset \neq S_{ρ} \subset \partial Ω$ .

2 Notation

In this section we introduce some notation used throughout this paper. Moreover, we recall the definitions of a class of weight functions and of some function spaces in which the coefficients of our operator will be chosen.

Let A be a Lebesgue measurable subset of

R^{n}

and let

Σ (A)

be the collection of all Lebesgue measurable subsets of A. If

F \in Σ (A)

, we denote by

| F |

the Lebesgue measure of F and by

D (F)

the class of restrictions to F of functions

ζ \in C_{0}^{\infty} (R^{n})

with

\bar{F} \cap supp ζ \subseteq F

. Moreover, if

X (F)

is a space of functions defined on F, we denote by

X_{loc} (F)

the class of all functions

g : F \to R

such that

ζ g \in X (F)

for all

ζ \in D (F)

. Furthermore, for

g \in L^{p} (A)

(

p \geq 1

), we put

ω^{p} [g, A] (t) = \underset{| E | \leq t}{sup_{E \in Σ (A)}} {∥ g ∥}_{L^{p} (E)}, t \in R_{+} .

Since $ω^{p} [g, A] (t)$ is a decreasing function and ${lim}_{t \to 0} ω^{p} [g, A] (t) = 0$ , we can refer to $ω^{p} [g, A]$ as the modulus of continuity of g in $L^{p} (A)$ .

Let Ω be an open subset of

R^{n}

,

n \geq 2

. We denote by

A (Ω)

the class of measurable weight functions

ρ : Ω \to R_{+}

such that

γ^{- 1} ρ (y) \leq ρ (x) \leq γ ρ (y), \forall y \in Ω, \forall x \in Ω \cap B (y, ρ (y)),

(2.1)

where $γ \in R_{+}$ is independent of x and y, and $B (y, ρ (y))$ is the open ball of radius $ρ (y)$ centered at y.

We remark that $A (Ω)$ contains the class of all functions $ρ : Ω \to R_{+}$ which are Lipschitz continuous in Ω with Lipschitz constant less than 1.

Typical examples of functions

ρ \in A (Ω)

are the function

x \in Ω \to 1 + a | x |, a \in] 0, 1 [,

if

Ω = R^{n}

and, if

Ω \neq R^{n}

and S is a nonempty subset of ∂Ω, the function

x \in Ω \to a \cdot dist (x, S), a \in] 0, 1 [.

For any

ρ \in A (Ω)

we put

S_{ρ} = {z \in \partial Ω ∣ ρ (x) \leq | x - z | \forall x \in Ω} .

(2.2)

We recall that the set

S_{ρ}

is a closed subset of ∂Ω and

z \in S_{ρ} ⟺ lim_{x \to z} ρ (x) = 0

(see [5]).

It is well known that

ρ \in L_{loc}^{\infty} (\bar{Ω}), ρ^{- 1} \in L_{loc}^{\infty} (\bar{Ω} ∖ S_{ρ}),

(2.3)

and, if

S_{ρ} \neq \emptyset

[5, 6],

ρ (x) \leq dist (x, S_{ρ}), \forall x \in Ω .

(2.4)

Let

ρ \in A (Ω)

. For

k \in N_{0}

,

1 \leq p \leq + \infty

and

s \in R

, we denote by

W_{s}^{k, p} (Ω)

the space of distributions u on Ω such that

ρ^{s + | α | - k} \partial^{α} u \in L^{p} (Ω)

for

| α | \leq k

. We observe that

W_{s}^{k, p} (Ω)

is a Banach space with the norm defined by

{∥ u ∥}_{W_{s}^{k, p} (Ω)} = \sum_{| α | \leq k} {∥ ρ^{s + | α | - k} \partial^{α} u ∥}_{L^{p} (Ω)} .

Moreover, it is separable if $1 \leq p < + \infty$ , reflexive if $1 < p < + \infty$ , and, in particular, $W_{s}^{k, 2} (Ω)$ is an Hilbert space. We put $W_{s}^{0, p} (Ω) = L_{s}^{p} (Ω)$ , and we observe that the space $C_{0}^{\infty} (Ω)$ is dense in $L_{s}^{p} (Ω)$ (see [7, 8]).

A more detailed account of properties of the above defined weighted Sobolev spaces can be found in [7, 9] and [8].

For any

x \in Ω

, we put

Ω (x) = Ω \cap B (x, ρ (x)) .

(2.5)

Let

ρ \in A (Ω)

. For

1 \leq p < + \infty

and

s \in R

, we denote by

K_{s}^{p} (Ω)

the class of functions

g \in L_{loc}^{p} (\bar{Ω} ∖ S_{ρ})

such that

{∥ g ∥}_{K_{s}^{p} (Ω)} = sup_{x \in Ω} (ρ^{s - \frac{n}{p}} (x) {∥ g ∥}_{L^{p} (Ω (x))}) < + \infty .

(2.6)

Obviously $K_{s}^{p} (Ω)$ is a Banach space with the norm defined by (2.6). It is easy to prove that the space $L_{s}^{\infty} (Ω)$ is a subset of $K_{s}^{p} (Ω)$ (see [10]). Thus, we can define a new space of functions ${\tilde{K}}_{s}^{p} (Ω)$ as the closure of $L_{s}^{\infty} (Ω)$ in $K_{s}^{p} (Ω)$ .

We recall the following characterization of the above defined space (see [10]):

g \in {\tilde{K}}_{s}^{p} (Ω) ⟺ g \in K_{s}^{p} (Ω) and lim_{t \to 0} (sup_{\begin{matrix} E \in Σ (Ω) \\ {sup}_{x \in Ω} \frac{| Ω (x) \cap E |}{ρ^{n} (x)} \leq t \end{matrix}} {∥ g χ_{E} ∥}_{K_{s}^{p} (Ω)}) = 0,

where $χ_{E}$ denotes the characteristic function of the set E.

Therefore, we define modulus of continuity of g in

{\tilde{K}}_{s}^{p} (Ω)

as a map

{\tilde{ω}}_{s}^{p} [g] : R_{+} \to R_{+}

such that [11]

\begin{aligned} \underset{{sup}_{x \in Ω} \frac{| Ω (x) \cap E |}{ρ^{n} (x)} \leq t}{sup_{E \in Σ (Ω)}} {∥ g χ_{E} ∥}_{K_{s}^{p} (Ω)} \leq {\tilde{ω}}_{s}^{p} [g] (t), \\ lim_{t \to 0} {\tilde{ω}}_{s}^{p} [g] (t) = 0 . \end{aligned}

(2.7)

Further properties of above mentioned function spaces can be found in [5, 10], and [11].

If Ω has the property

| Ω (x, r) | \geq A r^{n}, \forall x \in Ω, \forall r \in] 0, 1],

(2.8)

where

Ω (x, r) = B (x, r) \cap Ω

and A is a positive constant independent of x and r, it is possible to consider the space

B M O (Ω, t)

(

t \in R_{+}

) composed by all functions

g \in L_{loc}^{1} (\bar{Ω})

such that

{[g]}_{B M O (Ω, t)} = sup_{\begin{matrix} x \in Ω \\ r \in] 0, t] \end{matrix}} ⨏_{Ω (x, r)} | g - ⨏_{Ω (x, r)} g | < + \infty,

where

⨏_{Ω (x, r)} g = {| Ω (x, r) |}^{- 1} \int_{Ω (x, r)} g .

If

g \in B M O (Ω) = B M O (Ω, t_{A})

, with

t_{A} = sup_{t \in R_{+}} (\underset{r \in] 0, t]}{sup_{x \in Ω}} \frac{r^{n}}{| Ω (x, r) |} \leq \frac{1}{A}),

we will say that

g \in V M O (Ω)

if

{[g]}_{B M O (Ω, t)} \to 0

for

t \to 0^{+}

. A function

η [g] : R_{+} \to R_{+}

is called a modulus of continuity of g in

V M O (Ω)

if

{[g]}_{B M O (Ω, t)} \leq η [g] (t), \forall t \in R_{+}, lim_{t \to 0^{+}} η [g] (t) = 0 .

We say that $g \in V M O_{loc} (Ω)$ if ${(ζ g)}_{o} \in V M O (R^{n})$ for any $ζ \in C_{0}^{\infty} (Ω)$ , where ${(ζ g)}_{o}$ denotes the zero extension of ζg outside of Ω. A more detailed account of properties of the above defined spaces $B M O (Ω)$ and $V M O (Ω)$ can be found in [12].

We conclude this section introducing a class of applications needed in the sequel.

From now on we consider $ρ \in A (Ω) \cap L^{\infty} (Ω)$ and we suppose that the following condition on ρ holds:

(h₀) there exists a function

σ \in A (Ω) \cap C^{\infty} (Ω) \cap C^{0, 1} (\bar{Ω})

which is equivalent to ρ and such that

| \partial^{α} σ (x) | \leq c_{α} σ^{1 - | α |} (x), \forall x \in Ω, \forall α \in N_{0}^{n},

where $c_{α}$ is independent of x (see [6]).

We observe that the condition (h₀) holds, for example, if Ω is an unbounded open set with the cone property, or if the open set Ω has not the cone property but the weight function ρ is equivalent to the function $dist (\cdot, \partial Ω)$ (see [6]).

Let us fix

g \in C_{0}^{\infty} ({\bar{R}}_{+})

satisfying the conditions

0 \leq g \leq 1, g (t) = 1 if t \geq 1, g (t) = 0 if t \leq \frac{1}{2} .

(2.9)

For each

k \in N

, we put

η_{k} (x) = \frac{1}{k} ζ_{k} (x) + (1 - ζ_{k} (x)) \cdot σ (x), x \in Ω,

where

ζ_{k} (x) = g (k σ (x))

,

x \in Ω

. Obviously,

η_{k} \in C^{\infty} (Ω)

for any

k \in N

and

η_{k} (x) = {\begin{cases} \frac{1}{k} & if x \in {\bar{Ω}}_{k}, \\ σ (x) & if x \in Ω ∖ Ω_{2 k}, \end{cases}

where

Ω_{k} = {x \in Ω | σ (x) > \frac{1}{k}} .

(2.10)

Moreover, for

k \in N

, it is easy to prove that

σ (x) \leq η_{k} (x) \leq 2 σ (x), x \in Ω ∖ {\bar{Ω}}_{k},

(2.11)

{c^{'}}_{k} σ (x) \leq η_{k} (x) \leq σ (x), x \in Ω_{k},

(2.12)

{(η_{k} (x))}_{x} \leq c_{1} {(σ (x))}_{x}, x \in Ω,

(2.13)

{(η_{k} (x))}_{x x} \leq c_{2} \frac{{(σ (x))}_{x}^{2} + σ (x) \cdot {(σ (x))}_{x x}}{σ (x)}, x \in Ω,

(2.14)

where

{c^{'}}_{k} \in R_{+}

depends on k and σ, and

c_{1}, c_{2} \in R_{+}

depend only on n. Furthermore, for any

s \in R

, we have

\frac{{(η_{k}^{s} (x))}_{x}}{η_{k}^{s} (x)} \leq c_{3} \frac{{(η_{k} (x))}_{x}}{σ (x)}, x \in Ω,

(2.15)

\frac{{(η_{k}^{s} (x))}_{x x}}{η_{k}^{s} (x)} \leq c_{3} \frac{{(η_{k} (x))}_{x}^{2} + η_{k} (x) \cdot {(η_{k} (x))}_{x x}}{σ^{2} (x)}, x \in Ω,

(2.16)

where $c_{3} \in R_{+}$ depends on s and n.

3 Hypotheses and preliminary results

Suppose that Ω has the property (2.8) and let

p > n / 2

. Consider in Ω the differential operator

\tilde{L}

defined by

\tilde{L} = \sum_{i, j = 1}^{n} a_{i j} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} + \sum_{i = 1}^{n} d_{i} \frac{\partial}{\partial x_{i}} + d,

with the following assumptions on the coefficients:

\begin{matrix} (h_{1}) {\begin{cases} a_{i j} = a_{j i} \in L^{\infty} (Ω) \cap V M O_{loc} (Ω), i, j = 1, \dots, n, \\ \exists ν_{o}, ν \in R_{+} : \sum_{i, j = 1}^{n} {∥ a_{i j} ∥}_{L^{\infty} (Ω)} \leq ν_{o}, \\ \sum_{i, j = 1}^{n} ξ_{i} ξ_{j} a_{i j} \geq ν {| ξ |}^{2} a.e. in Ω, \forall ξ \in R^{n}, \end{cases} \\ (i_{1}) {\begin{cases} d_{i} = a_{i} + {\tilde{d}}_{i}, a_{i} \in {\tilde{K}}_{1}^{r} (Ω), {\tilde{d}}_{i} \in L_{1}^{\infty} (Ω), i = 1, \dots, n, \\ d = a + \tilde{d}, a \in {\tilde{K}}_{2}^{p} (Ω), \tilde{d} \in L_{2}^{\infty} (Ω), \\ d \leq 0 a.e. in Ω, \end{cases} \end{matrix}

where $r > n$ if $p \leq n$ , and $r = p$ if $p > n$ .

Fixing $x_{o} \in Ω$ and $τ \in R_{+}$ such that $τ \leq σ (x_{o})$ , we put $B = B (x_{o}, τ)$ and $B^{*} = B (x_{o}, 1)$ .

We observe that under assumptions (h₁) and (i₁), the operator

\tilde{L}

from

W^{2, p} (B)

into

L^{p} (B)

is bounded and the following estimate holds:

{∥ \tilde{L} u ∥}_{L^{p} (B)} \leq c^{'} {∥ u ∥}_{W^{2, p} (B)}, \forall u \in W^{2, p} (B),

where $c^{'} \in R_{+}$ depends on n, p, r, ρ, $ν_{o}$ , ${∥ a_{i} ∥}_{L^{r} (B)}$ , ${∥ {\tilde{d}}_{i} ∥}_{L^{\infty} (B)}$ , ${∥ a ∥}_{L^{p} (B)}$ , ${∥ \tilde{d} ∥}_{L^{\infty} (B)}$ .

Let v be a solution of the problem

{\begin{cases} v \in W^{2, p} (B), \\ \tilde{L} v \geq h, h \in L^{p} (B), \\ v_{|_{\partial B}} \leq 0 . \end{cases}

(3.1)

We want to prove a bound for the solution v of the above problem (see Lemma 3.1 below), which will be the primary technical tool in the proof of our main result (see the next Section). In order to use a classical result of Vitanza (see, Theorem 2.1 in [13]) it is necessary to make an appropriate change of variables which allows to transform the operator

\tilde{L}

into a differential operator

{\tilde{L}}^{*}

whose lower order coefficients, in particular, belonging to Lebesgue spaces and their moduli of continuity can be estimated by moduli of continuity of the corresponding coefficients of

\tilde{L}

. To this aim, let us consider the map

T : B \to B^{*}

defined by

T (x) = x_{o} + \frac{x - x_{o}}{τ} .

(3.2)

Clearly

z = T (x) \Leftrightarrow x = x_{o} + τ (z - x_{o}) = T^{- 1} (z) .

For any function g defined on B, we set

g^{*} = g \circ T^{- 1} .

(3.3)

Using the equivalence between ρ and σ it is easy to prove that

τ d_{i}^{*} \in L^{r} (B^{*})

for any

i = 1, \dots, n

and

τ^{2} d^{*} \in L^{p} (B^{*})

; moreover,

{∥ τ d_{i}^{*} ∥}_{L^{r} (B^{*})} \leq c_{1} ({∥ a_{i} ∥}_{K_{1}^{r} (Ω)} + {∥ {\tilde{d}}_{i} ∥}_{L_{1}^{\infty} (Ω)}^{\frac{1}{r}}), i = 1, \dots, n,

(3.4)

and

{∥ τ^{2} d^{*} ∥}_{L^{p} (B^{*})} \leq c_{2} ({∥ a ∥}_{K_{2}^{p} (Ω)} + {∥ \tilde{d} ∥}_{L_{2}^{\infty} (Ω)}^{\frac{1}{p}}),

(3.5)

where $c_{1} \in R_{+}$ depends on n, ρ, r and $c_{2} \in R_{+}$ depends on n, ρ, p.

On the other hand, for any

E^{*} \in Σ (B^{*})

and

t \in R_{+}

, we have

| E^{*} | \leq t

if and only if

\frac{| E |}{τ^{n}} \leq t

where

E = {x \in B | x = T^{- 1} (z), z \in E^{*}}

. Thus, we obtain

\begin{aligned} ω^{r} [τ d_{i}^{*}, B^{*}] (t) \leq sup_{\begin{matrix} E \in Σ (B) \\ \frac{| E |}{σ^{n} (x_{o})} \leq t \end{matrix}} {∥ τ^{1 - \frac{n}{r}} d_{i} ∥}_{L^{r} (E)}, i = 1, \dots, n, \\ ω^{p} [τ^{2} d^{*}, B^{*}] (t) \leq sup_{\begin{matrix} E \in Σ (B) \\ \frac{| E |}{σ^{n} (x_{o})} \leq t \end{matrix}} {∥ τ^{2 - \frac{n}{p}} d ∥}_{L^{p} (E)} . \end{aligned}

(3.6)

Using again the equivalence between ρ and σ and (2.7), from (3.6) we also deduce

ω^{r} [τ d_{i}^{*}, B^{*}] (t) \leq c_{3} ({\tilde{ω}}_{1}^{r} [a_{i}] (t) + {∥ {\tilde{d}}_{i} ∥}_{L_{1}^{\infty} (Ω)}^{\frac{1}{r}} \cdot t^{\frac{1}{r}}), i = 1, \dots, n,

(3.7)

and

ω^{p} [τ^{2} d^{*}, B^{*}] (t) \leq c_{4} ({\tilde{ω}}_{2}^{p} [a] (t) + {∥ \tilde{d} ∥}_{L_{2}^{\infty} (Ω)}^{\frac{1}{p}} \cdot t^{\frac{1}{p}}),

(3.8)

where $c_{3} \in R_{+}$ depends on ρ, r and $c_{4} \in R_{+}$ depends on ρ, p.

We are now able to prove the requested a priori bound.

Lemma 3.1 Suppose that the conditions (h₁) and (i₁) hold. Let v be a solution of the problem (3.1). Then there exists

c_{o} \in R_{+}

such that

sup_{B} v \leq c_{o} \cdot τ^{2 - \frac{n}{p}} {∥ h^{-} ∥}_{L^{p} (B)},

(3.9)

where $c_{o}$ depends on n, p, r, ρ, ν, $ν_{0}$ , ${[p (a_{i j})]}_{B M O (R^{n}, \cdot)}$ , ${∥ a_{i} ∥}_{K_{1}^{r} (Ω)}$ , ${∥ a ∥}_{K_{2}^{p} (Ω)}$ , ${∥ {\tilde{d}}_{i} ∥}_{L_{1}^{\infty} (Ω)}$ , ${∥ \tilde{d} ∥}_{L_{2}^{\infty} (Ω)}$ , ${\tilde{ω}}_{1}^{r} [a_{i}]$ , ${\tilde{ω}}_{2}^{p} [a]$ , and where $p (a_{i j})$ are the extensions of $a_{i j}$ to $R^{n}$ in $L^{\infty} (R^{n}) \cap V M O (R^{n})$ for any $i, j = 1, \dots, n$ .

Proof Let

v \in W^{2, p} (B)

. Taking into account the definitions (3.2) and (3.3), it is easily seen that

\begin{aligned} {(\tilde{L} v)}^{*} & = \sum_{i, j = 1}^{n} a_{i j}^{*} {(v_{x_{i} x_{j}})}^{*} + \sum_{i = 1}^{n} d_{i}^{*} {(v_{x_{i}})}^{*} + d^{*} v^{*} \\ = τ^{- 2} \sum_{i, j = 1}^{n} a_{i j}^{*} v_{z_{i} z_{j}}^{*} + τ^{- 1} \sum_{i = 1}^{n} d_{i}^{*} v_{z_{i}}^{*} + d^{*} v^{*}, \end{aligned}

and hence

τ^{2} {(\tilde{L} v)}^{*} = {\tilde{L}}^{*} v^{*},

where

{\tilde{L}}^{*} = \sum_{i, j = 1}^{n} a_{i j}^{*} \frac{\partial^{2}}{\partial z_{i} \partial z_{j}} + τ \sum_{i = 1}^{n} d_{i}^{*} \frac{\partial}{\partial z_{i}} + τ^{2} d^{*} .

Let us denote by

p (a_{i j})

the extensions of

a_{i j}

to

R^{n}

such that

p (a_{i j}) \in L^{\infty} (R^{n}) \cap V M O (R^{n}), i, j = 1, \dots, n

(3.10)

(for the existence of such functions see Theorem 5.1 in [12]). Since

p {(a_{i j})}^{*} \in L^{\infty} (R^{n}) \cap V M O (R^{n}), p {(a_{i j})}_{|_{B^{*}}}^{*} = a_{i j}^{*}, i, j = 1, \dots, n,

(3.11)

we have

a_{i j}^{*} \in L^{\infty} (B^{*}) \cap V M O (B^{*}), i, j = 1, \dots, n .

(3.12)

Moreover, from assumptions (h₁), (i₁), and (3.4), (3.5) it follows that

{\begin{cases} a_{i j}^{*} = a_{j i}^{*}, i, j = 1, \dots, n, \\ \sum_{i, j = 1}^{n} ξ_{i} ξ_{j} a_{i j}^{*} \geq ν {| ξ |}^{2} a.e. in B^{*}, \forall ξ \in R^{n}, \\ τ d_{i}^{*} \in L^{r} (B^{*}), i = 1, \dots, n, τ^{2} d^{*} \in L^{p} (B^{*}), d^{*} \leq 0 a.e. in B^{*}, \end{cases}

(3.13)

where r and p are as in hypothesis (i₁).

Consider now the following problem:

{\begin{cases} {\tilde{L}}^{*} w = g, g \in L^{p} (B^{*}), \\ w \in W^{2, p} (B^{*}) \cap {\overset{\circ}{W}}^{1, p} (B^{*}) . \end{cases}

(3.14)

Putting together (3.11) and (3.13) with Theorem 2.1 of [13] if

n \geq 3

or with Theorem 3.5 of [14] if

n = 2

, it follows that there exists a unique solution w of (3.14) satisfying the estimate

{∥ w ∥}_{W^{2, p} (B^{*})} \leq K {∥ g ∥}_{L^{p} (B^{*})},

(3.15)

where $K \in R_{+}$ depends on n, p, ν, $ν_{0}$ , ${[p {(a_{i j})}^{*}]}_{B M O (R^{n}, \cdot)}$ , ${∥ τ \cdot d_{i}^{*} ∥}_{L^{r} (B^{*})}$ , ${∥ τ^{2} \cdot d^{*} ∥}_{L^{p} (B^{*})}$ , $ω^{r} [τ d_{i}^{*}, B^{*}]$ , $ω^{p} [τ^{2} d^{*}, B^{*}]$ .

Thus from (3.15) and classical Sobolev embedding theorems (see Lemma 5.15 in [15]) we deduce that there exists

K_{1} \in R_{+}

, depending on the same parameters as K, such that

max_{{\bar{B}}^{*}} | w | \leq K_{1} {∥ g ∥}_{L^{p} (B^{*})},

(3.16)

and hence for each

z \in B^{*}

there is a function

G (z, \cdot) \in L^{p^{'}} (B^{*})

(

1 / p + 1 / p^{'} = 1

) such that

w (z) = - \int_{B^{*}} G (z, y) \cdot g (y) d y .

(3.17)

The map

G (z, \cdot)

is the Green function for the operator

{\tilde{L}}^{*}

in

B^{*}

and it has the following properties:

\int_{B^{*}} G (z, y) \cdot \tilde{g} (y) d y \geq 0, \forall \tilde{g} \in L^{p} (B^{*}), \tilde{g} \geq 0,

(3.18)

{∥ G (z, \cdot) ∥}_{L^{p^{'}} (B^{*})} \leq K_{1} .

(3.19)

Setting

g = {\tilde{L}}^{*} v^{*}

in (3.14), we find that the function

w - v^{*}

, belonging to

W^{2, p} (B^{*})

, is a solution of the following problem:

{\begin{cases} {\tilde{L}}^{*} (w - v^{*}) = 0 in B^{*}, \\ {(w - v^{*})}_{|_{\partial B^{*}}} = - v_{|_{\partial B^{*}}}^{*} \geq 0 . \end{cases}

(3.20)

Moreover, from (3.12), (3.13) and Lemma 3.1 of [16] (see also Lemma 3.1 of [2] for the case

n \geq 3

) it follows that

w - v^{*} \geq 0

in

B^{*}

. Finally, applying (3.17) with

g = {\tilde{L}}^{*} v^{*} = τ^{2} {(\tilde{L} v)}^{*}

and using (3.18) and (3.19) we obtain

\begin{aligned} v^{*} (z) & \leq - \int_{B^{*}} G (z, y) \cdot τ^{2} {(\tilde{L} v)}^{*} (y) d y \\ \leq - τ^{2} \int_{B^{*}} G (z, y) \cdot h^{*} (y) d y \leq - 2 τ^{2} \int_{B^{*}} G (z, y) \cdot {(h^{*})}^{-} (y) d y \\ \leq 2 τ^{2} {∥ G (z, \cdot) ∥}_{L^{p^{'}} (B^{*})} \cdot {∥ {(h^{*})}^{-} ∥}_{L^{p} (B^{*})} \leq 2 τ^{2} \cdot K_{1} {∥ {(h^{*})}^{-} ∥}_{L^{p} (B^{*})}, \forall z \in B^{*} . \end{aligned}

(3.21)

From (3.21), converting back to the x-variables ( $z = T (x)$ ), we easily deduce the estimate (3.9). □

4 Main results

In this section we use the previous result to prove a bound for the solution of our main problem.

Consider in Ω the differential operator L defined by

L = \sum_{i, j = 1}^{n} a_{i j} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} + \sum_{i = 1}^{n} a_{i} \frac{\partial}{\partial x_{i}} + a,

and put

L_{o} = \sum_{i, j = 1}^{n} a_{i j} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} .

Suppose that the leading coefficients of operator L satisfy the assumption (h₁) while the lower order coefficients verify the following condition:

(h_{2}) {\begin{cases} a_{i} \in {\tilde{K}}_{1}^{r} (Ω), i = 1, \dots, n, \\ a \in {\tilde{K}}_{2}^{p} (Ω), \\ \exists a_{o} \in R_{+} : {ess sup}_{Ω} σ^{2} a = - a_{o}, \end{cases}

where r and p are as in hypothesis (i₁). Moreover, assume that the following condition on ρ holds:

(h_{3}) lim_{k \to + \infty} (sup_{Ω ∖ Ω_{k}} ({(σ (x))}_{x} + σ (x) {(σ (x))}_{x x})) = 0,

where $Ω_{k}$ is defined in (2.10). For an example of function ρ whose regularizing function σ satisfy (h₃) we can refer to [17].

We introduce now a class of mappings needed in the sequel. Let us fix a function

α \in C^{\infty} (Ω) \cap C^{0, 1} (\bar{Ω})

which is equivalent to

dist (\cdot, \partial Ω)

(for more details on the existence of such an α see, for instance, Theorem 2, Chapter IV in [18] and Lemma 3.6.1 in [19]). Hence, for any

m \in N

we define the functions

ψ_{m} : x \in \bar{Ω} \to g (m α (x)) (1 - g (\frac{| x |}{2 m})),

where

g \in C^{\infty} ({\bar{R}}_{+})

verifies (2.9). It is easy to prove that each

ψ_{m}

belongs to

C_{0}^{\infty} (Ω)

and

0 \leq ψ_{m} \leq 1, supp ψ_{m} \subseteq E_{2 m}, {(ψ_{m})}_{|_{{\bar{E}}_{m}}} = 1,

where

E_{m} = {x \in Ω : | x | < m, α (x) > \frac{1}{m}} .

Remark 4.1 From hypothesis (h₁) and Lemma 4.2 in [12] it follows that for any

m \in N

the functions

{(ψ_{m} a_{i j})}_{o}

(obtained as extensions of

ψ_{m} a_{i j}

to

R^{n}

with zero values out of Ω) belong to

V M O (R^{n})

and

{[{(ψ_{m} a_{i j})}_{o}]}_{B M O (R^{n}, t)} \leq {[ψ_{m} a_{i j}]}_{B M O (Ω, t)},

for t small enough.

Now we are able to prove our main result.

Theorem 4.2 Suppose that conditions (h₁), (h₂), (h₃) hold. Fixing

s \in R

, let u be a solution of the problem

{\begin{cases} u \in W_{loc}^{2, p} (Ω), \\ L u \geq f, f \in L_{loc}^{p} (Ω), \\ {lim sup}_{x \to x_{o}} σ^{s} (x) u (x) \leq 0, \forall x_{o} \in \partial Ω, \\ {lim sup}_{| x | \to + \infty} σ^{s} (x) u (x) \leq 0 if Ω is unbounded . \end{cases}

(4.1)

Then there exist an open ball

B \subset \subset Ω

and a constant

c \in R_{+}

such that

sup_{x \in Ω} σ^{s} (x) u (x) \leq c {(⨏_{B} {| σ^{s + 2} f^{-} |}^{p} d x)}^{\frac{1}{p}},

(4.2)

where c depends on n, p, r, ρ, ν, $ν_{0}$ , $a_{o}$ , $η [ψ_{m} a_{i j}]$ ( $m \in N$ ), ${∥ a_{i} ∥}_{K_{1}^{r} (Ω)}$ , ${∥ a ∥}_{K_{2}^{p} (Ω)}$ , ${\tilde{ω}}_{1}^{r} [a_{i}]$ , ${\tilde{ω}}_{2}^{p} [a]$ .

Proof Without loss of generality it can be assumed that

{sup}_{Ω} σ^{s} (x) u (x) > 0

. For any

k \in N

, we put

w_{k} (x) = η_{k}^{s} (x) u (x), x \in Ω .

(4.3)

Thus, from the last two conditions of (4.1) and from (2.11), (2.12) and (4.3) it follows that there exists $y_{k} \in Ω$ such that ${sup}_{Ω} w_{k} (x) = w_{k} (y_{k})$ . Moreover, taking into account the classical Sobolev embedding theorem (see Theorem 5.4 in [15]), there exists $R_{k} \in] 0, dist (y_{k}, \partial Ω) [$ such that $w_{k} (x) > 0$ for all $x \in B (y_{k}, R_{k})$ .

Let

λ, α_{k}, α_{o} \in R_{+}

, with

α_{o} > 1

(which will be suitably chosen later), such that

α_{k} = α_{o} σ (y_{k}), λ \leq 1, λ α_{k} \leq min {R_{k}, σ (y_{k})} .

(4.4)

For simplicity of notation, for each $k \in N$ , we denote by $B_{k}$ the open ball $B (y_{k}, λ α_{k})$ .

Let us set

φ_{k} (x) = {\begin{cases} 1 + λ^{2} - \frac{{| x - y_{k} |}^{2}}{α_{k}^{2}}, & x \in B_{k}, \\ 1, & x \in Ω ∖ B_{k} . \end{cases}

(4.5)

It is easily seen that

1 \leq φ_{k} \leq 1 + λ^{2} \leq 2 .

(4.6)

Moreover, for

x \in B_{k}

{(φ_{k})}_{x_{i}} \leq \frac{2 λ}{α_{k}}, {(φ_{k})}_{x_{i}} \cdot {(φ_{k})}_{x_{j}} \leq \frac{4 λ^{2}}{α_{k}^{2}}, i, j = 1, \dots, n,

(4.7)

{(φ_{k})}_{x_{i} x_{j}} = 0 if i \neq j, {(φ_{k})}_{x_{i} x_{j}} = - \frac{2}{α_{k}^{2}} if i = j .

(4.8)

Consider now the function

v_{k}

defined by

v_{k} (x) = φ_{k} (x) w_{k} (x) - w_{k} (y_{k}), x \in B_{k} .

(4.9)

Clearly

{(v_{k})}_{|_{\partial B_{k}}} = {(w_{k})}_{|_{\partial B_{k}}} - w_{k} (y_{k}) \leq 0, v_{k} (y_{k}) = λ^{2} w_{k} (y_{k}) .

(4.10)

The first step of the proof is to show that there exists $k_{o} \in N$ such that, for any $k \geq k_{o}$ , each function $v_{k}$ is a solution of a problem of type (3.1), where the coefficients of associated differential operator verify the assumptions of Lemma 3.1.

For any

k \in N

, it is easy to prove

\begin{aligned} L_{o} w_{k} - u L_{o} η_{k}^{s} - 2 \sum_{i, j = 1}^{n} a_{i j} {(η_{k}^{s})}_{x_{j}} u_{x_{i}} + \sum_{i = 1}^{n} a_{i} {(η_{k}^{s} u)}_{x_{i}} \\ - u \sum_{i = 1}^{n} a_{i} {(η_{k}^{s})}_{x_{i}} + a η_{k}^{s} u = η_{k}^{s} L u, x \in Ω . \end{aligned}

(4.11)

Since

{(η_{k}^{s})}_{x_{j}} u_{x_{i}} = {(η_{k}^{s} u)}_{x_{i}} \cdot \frac{{(η_{k}^{s})}_{x_{j}}}{η_{k}^{s}} - \frac{{(η_{k}^{s})}_{x_{i}} \cdot {(η_{k}^{s})}_{x_{j}}}{{(η_{k}^{s})}^{2}} \cdot (η_{k}^{s} u), i, j = 1, \dots, n,

(4.12)

and u is a solution of problem (4.1), from (4.11) we deduce

L_{o} w_{k} + \sum_{i = 1}^{n} b_{i}^{k} {(w_{k})}_{x_{i}} + b^{k} w_{k} \geq g^{k} in Ω,

(4.13)

where we have put

b_{i}^{k} = a_{i} - 2 \sum_{j = 1}^{n} a_{i j} \frac{{(η_{k}^{s})}_{x_{j}}}{η_{k}^{s}}, i = 1, \dots, n,

(4.14)

b^{k} = a + 2 \sum_{i, j = 1}^{n} a_{i j} \frac{{(η_{k}^{s})}_{x_{i}} {(η_{k}^{s})}_{x_{j}}}{{(η_{k}^{s})}^{2}} - \sum_{i, j = 1}^{n} a_{i j} \frac{{(η_{k}^{s})}_{x_{i} x_{j}}}{η_{k}^{s}},

(4.15)

g^{k} = η_{k}^{s} f + w_{k} \sum_{i = 1}^{n} a_{i} \frac{{(η_{k}^{s})}_{x_{i}}}{η_{k}^{s}} .

(4.16)

We observe that using the hypotheses (h₀), (h₁), (h₂), the equivalence between ρ and σ, and (2.11)-(2.16), we easily get

{\begin{cases} (a_{i j} \cdot \frac{{(η_{k}^{s})}_{x_{j}}}{η_{k}^{s}}) \in L_{1}^{\infty} (Ω), i, j = 1, \dots, n, \\ a_{i j} \cdot \frac{{(η_{k}^{s})}_{x_{i}} {(η_{k}^{s})}_{x_{j}}}{{(η_{k}^{s})}^{2}}, a_{i j} \cdot \frac{{(η_{k}^{s})}_{x_{i} x_{j}}}{η_{k}^{s}} \in L_{2}^{\infty} (Ω), i, j = 1, \dots, n, \\ g^{k} \in L_{loc}^{p} (Ω) . \end{cases}

(4.17)

Using now the estimate (4.13), it is easily seen that

\begin{aligned} L_{o} (φ_{k} w_{k}) - w_{k} L_{o} φ_{k} - 2 \sum_{i, j = 1}^{n} a_{i j} {(φ_{k})}_{x_{j}} {(w_{k})}_{x_{i}} \\ + \sum_{i = 1}^{n} b_{i}^{k} {(φ_{k} w_{k})}_{x_{i}} - \sum_{i = 1}^{n} b_{i}^{k} {(φ_{k})}_{x_{i}} w_{k} + b^{k} φ_{k} w_{k} \\ = φ_{k} (L_{o} w_{k} + \sum_{i = 1}^{n} b_{i}^{k} {(w_{k})}_{x_{i}} + b^{k} w_{k}) \geq φ_{k} g^{k} in B_{k} . \end{aligned}

(4.18)

This last inequality can be rewritten as

\begin{aligned} L_{o} (φ_{k} w_{k}) + \sum_{i = 1}^{n} d_{i}^{k} {(φ_{k} w_{k})}_{x_{i}} + d^{k} φ_{k} w_{k} \\ \geq φ_{k} g^{k} + \sum_{i = 1}^{n} b_{i}^{k} {(φ_{k})}_{x_{i}} w_{k} in B_{k}, \end{aligned}

(4.19)

where we have set

d_{i}^{k} = b_{i}^{k} - 2 \sum_{j = 1}^{n} a_{i j} \frac{{(φ_{k})}_{x_{j}}}{φ_{k}}, i = 1, \dots, n,

(4.20)

d^{k} = b^{k} + 2 \sum_{i, j = 1}^{n} a_{i j} \frac{{(φ_{k})}_{x_{i}} {(φ_{k})}_{x_{j}}}{{(φ_{k})}^{2}} - \sum_{i, j = 1}^{n} a_{i j} \frac{{(φ_{k})}_{x_{i} x_{j}}}{φ_{k}} .

(4.21)

Hence, putting together (4.9) with (4.19) we get

L_{o} v_{k} + \sum_{i = 1}^{n} d_{i}^{k} {(v_{k})}_{x_{i}} + d^{k} v_{k} \geq h^{k} in B_{k},

(4.22)

where

h^{k} = φ_{k} g^{k} + w_{k} \sum_{i = 1}^{n} b_{i}^{k} {(φ_{k})}_{x_{i}} - d^{k} w_{k} (y_{k}) .

(4.23)

Observe that using the hypotheses (h₁), (h₂), and (4.17), (4.5)-(4.8), it is easy to prove that, for any

k \in N

, the coefficients

d_{i}^{k}

(for

i = 1, \dots, n

) and

d^{k}

satisfy the first two conditions of assumption (i₁) and the function

h^{k} \in L^{p} (B_{k})

. We show now that, for a suitable choice of the constant

α_{o}

, there exists

k_{o} \in N

such that for any

k \geq k_{o}

the coefficients

d^{k}

verify also the last condition of (i₁). To this aim, we firstly observe that using again hypotheses (h₁), (h₂), and (2.15), (2.16), from (4.15) we obtain

b^{k} \leq - \frac{a_{o}}{σ^{2}} + \frac{c_{1}}{σ^{2}} [{(η_{k})}_{x}^{2} + η_{k} {(η_{k})}_{x x}], a.e. in Ω,

(4.24)

where $c_{1} \in R_{+}$ depends on $ν_{o}$ , n and s.

Thus, from (4.4), (2.11)-(2.14) and hypothesis (h₃) it follows that there exists

k_{o} \in N

such that for any

k \geq k_{o}

we get

b^{k} \leq - \frac{a_{o}}{2 σ^{2}}, a.e. in Ω .

(4.25)

Now, for

k \geq k_{o}

, putting together (4.25) with (4.21) and using the assumption (h₁), the properties (4.6)-(4.8) and (4.4), we obtain

d^{k} \leq - \frac{a_{o}}{2 σ^{2}} + \frac{8 ν_{o} λ^{2}}{α_{k}^{2}} + \frac{2 ν_{o}}{α_{k}^{2}} \leq [- \frac{a_{o}}{2 \cdot γ^{2}} + \frac{10 ν_{o}}{α_{o}^{2}}] σ^{- 2} (y_{k}), a.e. in B_{k} .

(4.26)

Hence, fixing

α_{o}

such that

\frac{1}{α_{o}^{2}} \leq \frac{a_{o}}{40 ν_{o} \cdot γ^{2}}

(4.27)

from (4.26) it follows that for each

k \geq k_{o}

d^{k} \leq - \frac{a_{o}}{4 γ^{4} σ^{2} (x)}, a.e. in B_{k} .

(4.28)

Putting together (4.28) with (4.25) and observing that

d^{k} = b^{k}

in

Ω ∖ B_{k}

, we deduce that

d^{k} \leq 0

a.e. in Ω. The above considerations together with (4.4), (4.9), (4.10), and (4.22) show that for any

k \geq k_{o}

the problem

{\begin{cases} v_{k} \in W^{2, p} (B_{k}), \\ L_{o} v_{k} + \sum_{i = 1}^{n} d_{i}^{k} {(v_{k})}_{x_{i}} + d^{k} v_{k} \geq h^{k}, h^{k} \in L^{p} (B_{k}), \\ {v_{k}}_{|_{\partial B_{k}}} \leq 0 \end{cases}

(4.29)

satisfy the assumptions of Lemma 3.1. Therefore, there exists a constant

c_{1} \in R_{+}

depending on n, p, r, ρ, ν,

ν_{o}

,

{[p (a_{i j})]}_{B M O (R^{n}, \cdot)}

,

{∥ a_{i} ∥}_{K_{1}^{r} (Ω)}

,

{∥ a ∥}_{K_{2}^{p} (Ω)}

,

{\tilde{ω}}_{1}^{r} [a_{i}]

,

{\tilde{ω}}_{2}^{p} [a]

such that

sup_{B_{k}} v_{k} \leq c_{1} {(λ α_{k})}^{2 - \frac{n}{p}} {∥ {(h^{k})}^{-} ∥}_{L^{p} (B_{k})} .

(4.30)

By (4.10), the last bound with

x = y_{k}

becomes

λ^{2} w_{k} (y_{k}) \leq c_{1} {(λ α_{k})}^{2 - \frac{n}{p}} {∥ {(h^{k})}^{-} ∥}_{L^{p} (B_{k})} .

(4.31)

Now, in order to obtain the estimate (4.2), we have to provide a lower bound for the function

h^{k}

in terms of the data f. First of all, we observe that, using the definitions (4.14) and (4.16), we can rewrite (4.23) as

\begin{aligned} h^{k} = & φ_{k} η_{k}^{s} f + w_{k} \sum_{i = 1}^{n} a_{i} (\frac{{(η_{k}^{s})}_{x_{i}}}{η_{k}^{s}} \cdot φ_{k} + {(φ_{k})}_{x_{i}}) \\ - 2 w_{k} \sum_{i, j = 1}^{n} a_{i j} \frac{{(η_{k}^{s})}_{x_{j}}}{η_{k}^{s}} \cdot {(φ_{k})}_{x_{i}} - d^{k} w_{k} (y_{k}) . \end{aligned}

(4.32)

On the other hand, by assumption (h₁), and by (2.15), (4.7), and (4.4) we easily obtain

| 2 \cdot \sum_{i, j = 1}^{n} a_{i j} \frac{{(η_{k}^{s})}_{x_{j}}}{η_{k}^{s}} \cdot {(φ_{k})}_{x_{i}} | \leq \frac{c_{2}}{σ^{2} (x)} \cdot {(η_{k})}_{x},

(4.33)

where

c_{2} \in R_{+}

depends on

ν_{o}

, n, s, ρ,

a_{o}

. Thus, using (2.13) and hypothesis (h₃) it follows that there exists

k_{1} \geq k_{o}

, with

k_{1} \in N

, such that for any

k \geq k_{1}

| 2 \cdot \sum_{i, j = 1}^{n} a_{i j} \frac{{(η_{k}^{s})}_{x_{j}}}{η_{k}^{s}} \cdot {(φ_{k})}_{x_{i}} | \leq \frac{a_{o}}{4 γ^{4} σ^{2} (x)} .

(4.34)

Putting together (4.34) and (4.28) with (4.32) we obtain

h^{k} \geq φ_{k} η_{k}^{s} f + w_{k} \sum_{i = 1}^{n} a_{i} (\frac{{(η_{k}^{s})}_{x_{i}}}{η_{k}^{s}} \cdot φ_{k} + {(φ_{k})}_{x_{i}}) .

(4.35)

Taking into account (4.35), from (4.31) we get

w_{k} (y_{k}) \leq c_{1} λ^{- \frac{n}{p}} α_{k}^{2 - \frac{n}{p}} {∥ {(η_{k}^{s} f)}^{-} ∥}_{L^{p} (B_{k})} + Ψ_{1}^{k} + Ψ_{2}^{k},

(4.36)

where we have put

Ψ_{1}^{k} = c_{1} λ^{- \frac{n}{p}} α_{k}^{2 - \frac{n}{p}} {∥ w_{k} \sum_{i = 1}^{n} a_{i} {(φ_{k})}_{x_{i}} ∥}_{L^{p} (B_{k})},

(4.37)

and

Ψ_{2}^{k} = c_{1} λ^{- \frac{n}{p}} α_{k}^{2 - \frac{n}{p}} {∥ w_{k} \sum_{i = 1}^{n} a_{i} \frac{{(η_{k}^{s})}_{x_{i}}}{η_{k}^{s}} \cdot φ_{k} ∥}_{L^{p} (B_{k})} .

(4.38)

To end the proof, we give some upper bounds for the functions

Ψ_{1}^{k}

and

Ψ_{2}^{k}

(with

k \geq k_{1}

). First of all, observe that using (4.7) and Hölder’s inequality in (4.37) we obtain

Ψ_{1}^{k} \leq c_{3} λ^{1 - \frac{n}{r}} α_{k}^{1 - \frac{n}{r}} w_{k} (y_{k}) {∥ \sum_{i = 1}^{n} a_{i} ∥}_{L^{r} (B_{k})},

(4.39)

where

c_{3} \in R_{+}

depends on the same parameters as

c_{1}

. Using now (4.4), the equivalence on ρ and σ, we get

Ψ_{1}^{k} \leq c_{4} λ^{1 - \frac{n}{r}} α_{o}^{1 - \frac{n}{r}} w_{k} (y_{k}) {∥ \sum_{i = 1}^{n} a_{i} ∥}_{K_{1}^{r} (Ω)},

(4.40)

where

c_{4} \in R_{+}

depends on the same parameters as

c_{1}

. If we choose λ such that

λ^{1 - \frac{n}{r}} \leq \frac{1}{4 c_{4} α_{o}^{1 - n / r} {∥ \sum_{i = 1}^{n} a_{i} ∥}_{K_{1}^{r} (Ω)}}

(4.41)

from (4.40), for

k \geq k_{1}

, we get

Ψ_{1}^{k} \leq \frac{w_{k} (y_{k})}{4} .

(4.42)

Arguing similarly we obtain, for each

k \geq k_{1}

, the following bound on the function

Ψ_{2}^{k}

:

Ψ_{2}^{k} \leq c_{5} λ^{- \frac{n}{r}} α_{o}^{2 - \frac{n}{r}} w_{k} (y_{k}) {∥ \sum_{i = 1}^{n} a_{i} ∥}_{K_{1}^{r} (Ω)} sup_{Ω ∖ Ω_{k}} {(η_{k})}_{x},

(4.43)

where

c_{5} \in R_{+}

depends on the same parameters as

c_{1}

and on s. Thus, using again (2.13) and assumption (h₃), we see that there exists

k_{2} \geq k_{1}

, with

k_{2} \in N

, such that for

k \geq k_{2}

we get

Ψ_{2}^{k} \leq \frac{w_{k} (y_{k})}{4} .

(4.44)

Finally, chosen

k = k_{2}

, putting together (4.42) and (4.44) with (4.36) and using (4.4), (2.11), and (2.12) it follows that

w_{k_{2}} (y_{k_{2}}) \leq c_{6} {(λ α_{k_{2}})}^{- \frac{n}{p}} {∥ σ^{2 + s} f^{-} ∥}_{L^{p} (B_{k_{2}})},

(4.45)

where

c_{6} \in R_{+}

depends on the same parameters as

c_{1}

and on

a_{o}

. Taking into account (4.3) and using again (2.11) and (2.12), from (4.45) we get

sup_{Ω} σ^{s} (x) u (x) \leq c_{7} {(⨏_{B_{k_{2}}} {| σ^{2 + s} f^{-} |}^{p})}^{1 / p},

(4.46)

where $c_{7} \in R_{+}$ depends on the same parameters as $c_{1}$ and on $a_{o}$ .

Finally, if we choose

p ({a_{i j}}_{|_{B_{k_{2}}}}) = {(ψ_{m_{o}} a_{i j})}_{o},

(4.47)

where $m_{o} \in N$ is such that ${ψ_{m_{o}}}_{|_{B_{k_{2}}}} = 1$ , the estimate (4.2) follows from (4.46), (4.47), and Remark 4.1. □

Declarations

Authors’ Affiliations

(1)

Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, I - 84084 Fisciano, Italy

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