- Open Access
A weighted -a priori bound for a class of elliptic operators
Journal of Inequalities and Applications volume 2013, Article number: 263 (2013)
We prove a weighted -a priori bound, , for a class of uniformly elliptic second-order differential operators on unbounded domains. We deduce a uniqueness and existence result for the solution of the related Dirichlet problem.
MSC: 35J25, 35B45, 35R05.
In the last years the interest in weighted Sobolev spaces has been rapidly increasing, both for what concerns the structure of the spaces themselves as well as for their suitability in the theory of PDEs with prescribed boundary conditions. In particular, these spaces find a natural field of application in the case of unbounded domains. Indeed, in this setting, there is the need to impose not only conditions on the boundary of the set, but also conditions that control the behavior of the solution at infinity.
In this paper, we study, in an unbounded open subset Ω of , , the uniformly elliptic second-order linear differential operator with discontinuous coefficients:
and the related Dirichlet problem
where , , and , and are certain weighted Sobolev and Lebesgue spaces recently introduced in . To be more precise, the considered weight is a power of a function ρ of class such that and
To fix the ideas, one can think of the function
Among the various hypotheses involving discontinuous coefficients, we consider here those of Miranda type, referring to the classical paper  where the have derivatives in . Namely, we suppose that the belong to suitable Morrey-type spaces that extend to unbounded domains the classical notion of Morrey spaces (see Section 3 for the details).
Always in the framework of unbounded domains, no-weighted problems weakening the hypotheses of  have been studied. We quote here, for instance, [3–6] for and  for . A very general case, where the have vanishing mean oscillation (VMO), has been taken into account in  (for the pioneer works considering VMO assumptions in the framework of bounded domains, we refer to [9–11]) and in [12, 13] in a weighted contest. Variational problems can be found in [14–16]. Quasilinear elliptic equations with quadratic growth have been considered in .
The main result of this work consists in a weighted -bound, , having the only term on the right-hand side,
where the dependence of the constant c is completely described. Estimate (1.2) allows us to deduce the solvability of the related Dirichlet problem. This work generalizes to all a previous result of  where a no-weighted and a weighted case have been analyzed for . In  we considered an analogous problem, with , but without weight. Related variational results were studied in [19–21].
2 Weight functions and weighted spaces
We start recalling the definitions of our specific weight functions. Then we will focus on certain classes of related weighted Sobolev spaces recently introduced in , where a detailed description and the proofs of all the properties below can be found.
Let Ω be an open subset of , not necessarily bounded, . We consider a weight such that and
An example is given by
For , and , and given a function ρ satisfying (2.1), we define as the space of distributions u on Ω such that
equipped with the norm given in (2.2). Furthermore, we denote the closure of in by , and put .
Let us quote the following fundamental tool, which will allow us to exploit no-weighted results in order to pass to the weighted case.
Lemma 2.1 Let , and . If assumption (2.1) is satisfied, then there exist two constants such that
with and .
Moreover, if Ω has the segment property, then the map
defines a topological isomorphism from to and from to .
From now on, we also suppose that the weight ρ satisfies the further assumptions
As an example, we can then consider
Let us associate to ρ the function σ defined by
It is easily seen that σ verifies (2.1) too, and, moreover,
Now, we fix a cutoff function such that
and we set
Let us finally introduce the sequence
One has that, for any , σ and are equivalent, namely
Furthermore, concerning the derivatives, we have also, for any ,
3 A class of spaces of Morrey type
Here we recall the definitions and the main properties of a class of spaces of Morrey type where the coefficients of our operator will be chosen. These spaces are a generalization to unbounded domains of the classical Morrey spaces and were introduced for the first time in ; see also  for more details.
Thus, in the sequel let Ω be an unbounded open subset of , . The σ-algebra of all Lebesgue measurable subsets of Ω is denoted by . Given , is its Lebesgue measure, its characteristic function and (, ), where is the open ball centered in x and with radius τ.
For , , the space of Morrey type () is the set of all functions g in such that
endowed with the norm defined in (3.1). One can easily check that, for any arbitrary , a function g belongs to if and only if it belongs to and the norms of g in these two spaces are equivalent. Hence, we limit our attention to the space .
The closures of and in are denoted by and , respectively.
The following inclusions (algebraic and topological) hold true:
Moreover, one has
We put , and .
Now, let us define the moduli of continuity of functions belonging to or . For and , we set
Given a function , the following characterization holds:
where denotes a function of class such that
Thus, if g is a function in , a modulus of continuity of g in is a map such that
While if g belongs to , a modulus of continuity of g in is an application such that
Lemma 3.1 Let and . If Ω is an open subset of having the cone property and , with if , then
is a bounded operator from to . Moreover, there exists a constant such that
If , with if , then the operator in (3.3) is bounded from to . Moreover, there exists a constant such that
4 An a priori bound
Let and assume that
(we refer the reader, for instance, to  for the definition).
Consider the differential operator
with the following conditions on the leading coefficients:
For lower-order terms, we suppose that
Let us start observing that, in view of Lemma 3.1, under the assumptions ()-(), the operator is bounded.
Then let us recall some known results contained in Theorem 3.2 and Corollary 3.3 of .
Theorem 4.1 Let L be defined in (4.1). If hypotheses ()-() are satisfied, then there exists a constant such that
Moreover, the problem
is uniquely solvable.
Now we prove the claimed weighted -bound.
Theorem 4.2 Let L be defined in (4.1). Under hypotheses ()-(), there exists a constant such that
Proof Fix . If for , . Thus, in view of the isomorphism of Lemma 2.1, one has that .
Now if we write, , for a fixed , since η and σ are equivalent, one also has that . Hence, the estimate in Theorem 4.1 applies giving that there exists such that
Simple computations give then
Using (4.5) and (4.6), we deduce that
where depends on the same parameters as and on s.
On the other hand, from Lemma 3.1 and (2.9), we get
Putting together (2.9), (2.10), (4.7) and (4.8), we obtain the bound
where depends on the same parameters as and on .
Observe that by (2.11) it follows that there exists such that
Therefore, if we still denote by η the function , combining (4.9) and (4.10), we obtain
This together with the fact that σ and η are equivalent and in view of Lemma 2.1 (applied with and considering σ as weight function) gives
with depending on the same parameters as and on , that is (4.4).
If for , then , thus, always in view of the isomorphism of Lemma 2.1, one has that . Therefore arguing as to get (4.12), one obtains
This concludes the proof of Theorem 4.2. □
5 Some uniqueness and existence results
In this last section we exploit our weighted estimate in order to deduce the solvability of the related Dirichlet problem.
A preliminary result is needed.
Lemma 5.1 If hypothesis () is satisfied, then the Dirichlet problem
is uniquely solvable, with
if for , or
if for .
Proof Let us first consider the case for . Since , the function u is a solution of problem (5.1) if and only if is a solution of
with b given by (5.2).
Clearly, for any , one has
hence (5.4) is equivalent to the problem
Since σ verifies (2.1) and (2.6), by (1.6) of  (which gives that both and are in every , ) one has that we are in the hypotheses of Theorem 4.1, therefore (5.5) is uniquely solvable, and then problem (5.1) is uniquely solvable too.
Now assume that for . Since in this case , now the function u solves problem (5.1) if and only if is a solution of
with b given by (5.3).
Thus, problem (5.6) is equivalent to
The thesis follows then arguing as in the previous case. □
Theorem 5.2 Let L be defined in (4.1). Under hypotheses ()-(), the problem
is uniquely solvable.
Proof For each we put
with b given by (5.2) if for , or by (5.3) if for .
By Theorem 4.2 one obtains
Thus, taking into account the result of Lemma 5.1 and using the method of continuity along a parameter (see, e.g., Theorem 5.2 of ), we obtain the claimed result. □
Monsurrò S, Salvato M, Transirico M: A priori bounds for a class of elliptic operators . Int. J. Differ. Equ. 2011. doi:10.1155/2011/572824
Miranda C: Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui . Ann. Mat. Pura Appl. 1963, 63: 353–386. 10.1007/BF02412185
Transirico M, Troisi M: Equazioni ellittiche del secondo ordine di tipo non variazionale in aperti non limitati . Ann. Mat. Pura Appl. 1988, 152: 209–226. 10.1007/BF01766150
Transirico M, Troisi M: Ulteriori contributi allo studio delle equazioni ellittiche del secondo ordine in aperti non limitati . Boll. Unione Mat. Ital., B 1990, 4: 679–691.
Caso L, Cavaliere P, Transirico M: A priori bounds for elliptic equations . Ric. Mat. 2002, 51: 381–396.
Caso L, Cavaliere P, Transirico M: Existence results for elliptic equations . J. Math. Anal. Appl. 2002, 274: 554–563. 10.1016/S0022-247X(02)00287-1
Caso L, Cavaliere P, Transirico M:Solvability of the Dirichlet problem in for elliptic equations with discontinuous coefficients in unbounded domains . Matematiche 2002, 57: 287–302.
Caso L, Cavaliere P, Transirico M:An existence result for elliptic equations with -coefficients . J. Math. Anal. Appl. 2007, 325: 1095–1102. 10.1016/j.jmaa.2006.02.048
Chiarenza F, Frasca M, Longo P:Interior estimates for non divergence elliptic equations with discontinuous coefficients . Ric. Mat. 1991, 40: 149–168.
Chiarenza F, Frasca M, Longo P:-Solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients . Trans. Am. Math. Soc. 1993, 336: 841–853.
Ragusa MA: Cauchy-Dirichlet problem associated to divergence form parabolic equations . Commun. Contemp. Math. 2004, 6(3):377–393. 10.1142/S0219199704001392
Boccia S, Monsurrò S, Transirico M: Elliptic equations in weighted Sobolev spaces on unbounded domains . Int. J. Math. Math. Sci. 2008. doi:10.1155/2008/582435
Boccia S, Monsurrò S, Transirico M: Solvability of the Dirichlet problem for elliptic equations in weighted Sobolev spaces on unbounded domains . Bound. Value Probl. 2008. doi:10.1155/2008/901503
Bottaro G, Marina ME: Problema di Dirichlet per equazioni ellittiche di tipo varia-zionale su insiemi non limitati . Boll. Unione Mat. Ital. 1973, 8(4):46–56.
Lions PL: Remarques sur les équations linéaires elliptiques du second ordre sous forme divergence dans les domaines non bornés . Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 1985, 78(8):205–212.
Lions PL: Remarques sur les équations linéaires elliptiques du second ordre sous forme divergence dans les domaines non bornés II . Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 1985, 79(8):178–183.
Donato P, Giachetti D: Quasilinear elliptic equations with quadratic growth on unbounded domains . Nonlinear Anal. 1986, 8(10):791–804.
Monsurrò S, Transirico M:A -estimate for a class of elliptic operators . Int. J. Pure Appl. Math. 2013, 83(3):489–499. doi:10.12732/ijpam.v83i3.9
Monsurrò S, Transirico M:A -estimate for weak solutions of elliptic equations . Abstr. Appl. Anal. 2012. doi:10.1155/2012/376179
Monsurrò S, Transirico M: Dirichlet problem for divergence form elliptic equations with discontinuous coefficients . Bound. Value Probl. 2012. doi:10.1186/1687–2770–2012–67
Monsurrò S, Transirico M:A priori bounds in for solutions of elliptic equations in divergence form . Bull. Sci. Math. 2013. (in press). doi:10.1016/j.bulsci.2013.02.002
Transirico M, Troisi M, Vitolo A: Spaces of Morrey type and elliptic equations in divergence form on unbounded domains . Boll. Unione Mat. Ital., B 1995, 9(7):153–174.
Caso L, D’Ambrosio R, Monsurrò S: Some remarks on spaces of Morrey type . Abstr. Appl. Anal. 2010. doi:10.1155/2010/242079
Cavaliere P, Longobardi M, Vitolo A: Imbedding estimates and elliptic equations with discontinuous coefficients in unbounded domains . Matematiche 1996, 51: 87–104.
Adams RA: Sobolev Spaces. Academic Press, New York; 1975.
Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order. 2nd edition. Springer, Berlin; 1983.
The authors declare that they have no competing interests.
The authors conceived and wrote this article in collaboration and with the same responsibility. Both of them read and approved the final manuscript.
About this article
Cite this article
Monsurrò, S., Transirico, M. A weighted -a priori bound for a class of elliptic operators. J Inequal Appl 2013, 263 (2013). https://doi.org/10.1186/1029-242X-2013-263
- elliptic equations
- discontinuous coefficients
- a priori bounds
- weighted Sobolev spaces