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
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 -norm of the right-hand side.
If Ω is a bounded domain in ( ) and
is a uniformly elliptic operator in Ω, the classical result of AD Aleksandrov states that if , with in ∂Ω, verifies , where ( ), then
where depends only on 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,  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 -norms of the right-hand side ( ) under different conditions on the leading coefficients.
For instance, if Ω is an arbitrary open subset of and , a bound of type (1.2) and a consequent uniqueness result can be found in . In fact, it has been proved that, if the coefficients are bounded and locally VMO, the coefficients , a satisfy suitable summability conditions, and , then for any solution u of the problem
there exist a ball and a constant such that
where is the negative part of f,
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 , 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 of the boundary of an arbitrary domain Ω, not necessarily bounded and regular (see Section 2 for the definition of such weight function), in  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 , , the coefficients are bounded and locally VMO, the coefficients , a belong to suitable weighted spaces , in  the author has proved that the solution u of the problem
verifies the estimate
where is an open ball and the constant 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  under the more general hypothesis , but for an operator L with coefficients .
The aim of this paper is to improve the above quoted results in  by obtaining a similar estimate under much weaker assumptions. In particular, the main difference lies in the hypotheses on the coefficients , a which are not supposed to belong to weighted spaces but just to appropriate weighted Sobolev spaces (see Section 2 for the definition of such weighted spaces), which strictly contain the weighted spaces . Moreover, as in , we consider the more general hypothesis .
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 and let be the collection of all Lebesgue measurable subsets of A. If , we denote by the Lebesgue measure of F and by the class of restrictions to F of functions with . Moreover, if is a space of functions defined on F, we denote by the class of all functions such that for all . Furthermore, for ( ), we put
Since is a decreasing function and , we can refer to as the modulus of continuity of g in .
Let Ω be an open subset of , . We denote by the class of measurable weight functions such that
where is independent of x and y, and is the open ball of radius centered at y.
We remark that contains the class of all functions which are Lipschitz continuous in Ω with Lipschitz constant less than 1.
Typical examples of functions are the function
if and, if and S is a nonempty subset of ∂Ω, the function
For any we put
We recall that the set is a closed subset of ∂Ω and
It is well known that
Let . For , and , we denote by the space of distributions u on Ω such that for . We observe that is a Banach space with the norm defined by
For any , we put
Let . For and , we denote by the class of functions such that
Obviously is a Banach space with the norm defined by (2.6). It is easy to prove that the space is a subset of (see ). Thus, we can define a new space of functions as the closure of in .
We recall the following characterization of the above defined space (see ):
where denotes the characteristic function of the set E.
Therefore, we define modulus of continuity of g in as a map such that 
If Ω has the property
where and A is a positive constant independent of x and r, it is possible to consider the space ( ) composed by all functions such that
If , with
we will say that if for . A function is called a modulus of continuity of g in if
We say that if for any , where denotes the zero extension of ζg outside of Ω. A more detailed account of properties of the above defined spaces and can be found in .
We conclude this section introducing a class of applications needed in the sequel.
From now on we consider and we suppose that the following condition on ρ holds:
(h0) there exists a function which is equivalent to ρ and such that
where is independent of x (see ).
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 (see ).
Let us fix satisfying the conditions
For each , we put
where , . Obviously, for any and
Moreover, for , it is easy to prove that
where depends on k and σ, and depend only on n. Furthermore, for any , we have
where depends on s and n.
3 Hypotheses and preliminary results
Suppose that Ω has the property (2.8) and let . Consider in Ω the differential operator defined by
with the following assumptions on the coefficients:
where if , and if .
Fixing and such that , we put and .
We observe that under assumptions (h1) and (i1), the operator from into is bounded and the following estimate holds:
where depends on n, p, r, ρ, , , , , .
Let v be a solution of the problem
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 ) it is necessary to make an appropriate change of variables which allows to transform the operator into a differential operator 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 . To this aim, let us consider the map defined by
For any function g defined on B, we set
Using the equivalence between ρ and σ it is easy to prove that for any and ; moreover,
where depends on n, ρ, r and depends on n, ρ, p.
On the other hand, for any and , we have if and only if where . Thus, we obtain
Using again the equivalence between ρ and σ and (2.7), from (3.6) we also deduce
where depends on ρ, r and depends on ρ, p.
We are now able to prove the requested a priori bound.
Lemma 3.1Suppose that the conditions (h1) and (i1) hold. Letvbe a solution of the problem (3.1). Then there exists such that
where depends onn, p, r, ρ, ν, , , , , , , , , and where are the extensions of to in for any .
Proof Let . Taking into account the definitions (3.2) and (3.3), it is easily seen that
Let us denote by the extensions of to such that
(for the existence of such functions see Theorem 5.1 in ). Since
Moreover, from assumptions (h1), (i1), and (3.4), (3.5) it follows that
where r and p are as in hypothesis (i1).
Consider now the following problem:
where depends on n, p, ν, , , , , , .
Thus from (3.15) and classical Sobolev embedding theorems (see Lemma 5.15 in ) we deduce that there exists , depending on the same parameters as K, such that
and hence for each there is a function ( ) such that
The map is the Green function for the operator in and it has the following properties:
Setting in (3.14), we find that the function , belonging to , is a solution of the following problem:
From (3.21), converting back to the x-variables ( ), 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
Suppose that the leading coefficients of operator L satisfy the assumption (h1) while the lower order coefficients verify the following condition:
where r and p are as in hypothesis (i1). Moreover, assume that the following condition on ρ holds:
where is defined in (2.10). For an example of function ρ whose regularizing function σ satisfy (h3) we can refer to .
We introduce now a class of mappings needed in the sequel. Let us fix a function which is equivalent to (for more details on the existence of such an α see, for instance, Theorem 2, Chapter IV in  and Lemma 3.6.1 in ). Hence, for any we define the functions
where verifies (2.9). It is easy to prove that each belongs to and
Remark 4.1 From hypothesis (h1) and Lemma 4.2 in  it follows that for any the functions (obtained as extensions of to with zero values out of Ω) belong to and
for t small enough.
Now we are able to prove our main result.
Theorem 4.2Suppose that conditions (h1), (h2), (h3) hold. Fixing , letube a solution of the problem
Then there exist an open ball and a constant such that
wherecdepends onn, p, r, ρ, ν, , , ( ), , , , .
Proof Without loss of generality it can be assumed that . For any , we put
Thus, from the last two conditions of (4.1) and from (2.11), (2.12) and (4.3) it follows that there exists such that . Moreover, taking into account the classical Sobolev embedding theorem (see Theorem 5.4 in ), there exists such that for all .
Let , with (which will be suitably chosen later), such that
For simplicity of notation, for each , we denote by the open ball .
Let us set
It is easily seen that
Consider now the function defined by
The first step of the proof is to show that there exists such that, for any , each function 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 , it is easy to prove
and u is a solution of problem (4.1), from (4.11) we deduce
where we have put
We observe that using the hypotheses (h0), (h1), (h2), the equivalence between ρ and σ, and (2.11)-(2.16), we easily get
Using now the estimate (4.13), it is easily seen that
This last inequality can be rewritten as
where we have set
Hence, putting together (4.9) with (4.19) we get
Observe that using the hypotheses (h1), (h2), and (4.17), (4.5)-(4.8), it is easy to prove that, for any , the coefficients (for ) and satisfy the first two conditions of assumption (i1) and the function . We show now that, for a suitable choice of the constant , there exists such that for any the coefficients 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
where depends on , n and s.
Thus, from (4.4), (2.11)-(2.14) and hypothesis (h3) it follows that there exists such that for any we get
Now, for , putting together (4.25) with (4.21) and using the assumption (h1), the properties (4.6)-(4.8) and (4.4), we obtain
Hence, fixing such that
from (4.26) it follows that for each
Putting together (4.28) with (4.25) and observing that in , we deduce that a.e. in Ω. The above considerations together with (4.4), (4.9), (4.10), and (4.22) show that for any the problem
satisfy the assumptions of Lemma 3.1. Therefore, there exists a constant depending on n, p, r, ρ, ν, , , , , , such that
By (4.10), the last bound with becomes
Now, in order to obtain the estimate (4.2), we have to provide a lower bound for the function 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
On the other hand, by assumption (h1), and by (2.15), (4.7), and (4.4) we easily obtain
where depends on , n, s, ρ, . Thus, using (2.13) and hypothesis (h3) it follows that there exists , with , such that for any
Putting together (4.34) and (4.28) with (4.32) we obtain
Taking into account (4.35), from (4.31) we get
where we have put
To end the proof, we give some upper bounds for the functions and (with ). First of all, observe that using (4.7) and Hölder’s inequality in (4.37) we obtain
where depends on the same parameters as . Using now (4.4), the equivalence on ρ and σ, we get
where depends on the same parameters as . If we choose λ such that
from (4.40), for , we get
Arguing similarly we obtain, for each , the following bound on the function :
where depends on the same parameters as and on s. Thus, using again (2.13) and assumption (h3), we see that there exists , with , such that for we get
Finally, chosen , putting together (4.42) and (4.44) with (4.36) and using (4.4), (2.11), and (2.12) it follows that
where depends on the same parameters as and on . Taking into account (4.3) and using again (2.11) and (2.12), from (4.45) we get
where depends on the same parameters as and on .
Finally, if we choose
where is such that , the estimate (4.2) follows from (4.46), (4.47), and Remark 4.1. □
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.