- Open Access
On the maximum principle for elliptic operators in weighted spaces
© Caso and D'Ambrosio; licensee Springer. 2014
- Received: 16 January 2014
- Accepted: 15 April 2014
- Published: 6 May 2014
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.
- elliptic operators
- 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.
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.
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.
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.
Since is a decreasing function and , we can refer to as the modulus of continuity of g in .
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.
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 .
where denotes the characteristic function of the set E.
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:
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 ).
where depends on s and n.
where if , and if .
Fixing and such that , we put and .
where depends on n, p, r, ρ, , , , , .
where depends on n, ρ, r and depends on n, ρ, p.
where depends on ρ, r and depends on ρ, p.
We are now able to prove the requested a priori bound.
where depends on n, p, r, ρ, ν, , , , , , , , , and where are the extensions of to in for any .
where r and p are as in hypothesis (i1).
where depends on n, p, ν, , , , , , .
From (3.21), converting back to the x-variables (), we easily deduce the estimate (3.9). □
In this section we use the previous result to prove a bound for the solution of our main problem.
where is defined in (2.10). For an example of function ρ whose regularizing function σ satisfy (h3) we can refer to .
for t small enough.
Now we are able to prove our main result.
where c depends on n, p, r, ρ, ν, , , (), , , , .
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 .
For simplicity of notation, for each , we denote by the open ball .
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.
where depends on , n and s.
where depends on the same parameters as and on .
where is such that , the estimate (4.2) follows from (4.46), (4.47), and Remark 4.1. □
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