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.
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
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
It is well known 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 ):
Therefore, we define modulus of continuity of g in as a map such that 
If Ω has the property
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.
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 ).
3 Hypotheses and preliminary results
with the following assumptions on the coefficients:
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 again the equivalence between ρ and σ and (2.7), from (3.6) we also deduce
We are now able to prove the requested a priori bound.
(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:
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
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
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.
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 us set
It is easily seen that
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.
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
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
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
Finally, if we choose
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.