Open Access

A weighted W 2 , p -a priori bound for a class of elliptic operators

Journal of Inequalities and Applications20132013:263

https://doi.org/10.1186/1029-242X-2013-263

Received: 28 February 2013

Accepted: 7 May 2013

Published: 27 May 2013

Abstract

We prove a weighted W 2 , p -a priori bound, p > 1 , 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.

Keywords

elliptic equations discontinuous coefficients a priori bounds weighted Sobolev spaces

1 Introduction

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 R n , n 2 , the uniformly elliptic second-order linear differential operator with discontinuous coefficients:
L = i , j = 1 n a i j 2 x i x j + i = 1 n a i x i + a ,
and the related Dirichlet problem
(1.1)
where p > 1 , s R , and W s 2 , p ( Ω ) , and L s p ( Ω ) are certain weighted Sobolev and Lebesgue spaces recently introduced in [1]. To be more precise, the considered weight ρ s is a power of a function ρ of class C 2 ( Ω ¯ ) such that ρ : Ω ¯ R + and
sup x Ω | α ρ ( x ) | ρ ( x ) < + , | α | 2 , lim | x | + ( ρ ( x ) + 1 ρ ( x ) ) = + and lim | x | + ρ x ( x ) + ρ x x ( x ) ρ ( x ) = 0 .
To fix the ideas, one can think of the function
ρ ( x ) = ( 1 + | x | 2 ) t , t R { 0 } .

Among the various hypotheses involving discontinuous coefficients, we consider here those of Miranda type, referring to the classical paper [2] where the a i j have derivatives in L n ( Ω ) . Namely, we suppose that the ( a i j ) x h 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 [2] have been studied. We quote here, for instance, [36] for p = 2 and [7] for p > 1 . A very general case, where the a i j have vanishing mean oscillation (VMO), has been taken into account in [8] (for the pioneer works considering VMO assumptions in the framework of bounded domains, we refer to [911]) and in [12, 13] in a weighted contest. Variational problems can be found in [1416]. Quasilinear elliptic equations with quadratic growth have been considered in [17].

The main result of this work consists in a weighted W 2 , p -bound, p > 1 , having the only term L u L s p ( Ω ) on the right-hand side,
(1.2)

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 p > 1 a previous result of [1] where a no-weighted and a weighted case have been analyzed for p = 2 . In [18] we considered an analogous problem, with p > 1 , but without weight. Related variational results were studied in [1921].

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 [1], where a detailed description and the proofs of all the properties below can be found.

Let Ω be an open subset of R n , not necessarily bounded, n 2 . We consider a weight ρ : Ω ¯ R + such that ρ C 2 ( Ω ¯ ) and
sup x Ω | α ρ ( x ) | ρ ( x ) < + , | α | 2 .
(2.1)
An example is given by
ρ ( x ) = ( 1 + | x | 2 ) t , t R .
For k N 0 , p [ 1 , + [ and s R , and given a function ρ satisfying (2.1), we define W s k , p ( Ω ) as the space of distributions u on Ω such that
u W s k , p ( Ω ) = | α | k ρ s α u L p ( Ω ) < + ,
(2.2)

equipped with the norm given in (2.2). Furthermore, we denote the closure of C ( Ω ) in W s k , p ( Ω ) by , and put W s 0 , p ( Ω ) = L s p ( Ω ) .

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 k N 0 , p [ 1 , + [ and s R . If assumption (2.1) is satisfied, then there exist two constants c 1 , c 2 R + such that
c 1 u W s k , p ( Ω ) ρ t u W s t k , p ( Ω ) c 2 u W s k , p ( Ω ) , t R , u W s k , p ( Ω ) ,
(2.3)

with c 1 = c 1 ( t ) and c 2 = c 2 ( t ) .

Moreover, if Ω has the segment property, then the map
u ρ s u

defines a topological isomorphism from W s k , p ( Ω ) to W k , p ( Ω ) and from to .

From now on, we also suppose that the weight ρ satisfies the further assumptions
lim | x | + ( ρ ( x ) + 1 ρ ( x ) ) = + and lim | x | + ρ x ( x ) + ρ x x ( x ) ρ ( x ) = 0 .
(2.4)
As an example, we can then consider
ρ ( x ) = ( 1 + | x | 2 ) t , t R { 0 } .
Let us associate to ρ the function σ defined by
{ σ = ρ if  ρ +  for  | x | + , σ = 1 ρ if  ρ 0  for  | x | + .
(2.5)
It is easily seen that σ verifies (2.1) too, and, moreover,
lim | x | + σ ( x ) = + , lim | x | + σ x ( x ) + σ x x ( x ) σ ( x ) = 0 .
(2.6)
Now, we fix a cutoff function f C ( R ¯ + ) such that
0 f 1 , f ( t ) = 1 if  t [ 0 , 1 ] , f ( t ) = 0 if  t [ 2 , + [ ,
and we set
ζ k : x Ω ¯ f ( σ ( x ) k ) , k N ,
and
Ω k = { x Ω : σ ( x ) < k } , k N .
(2.7)
Let us finally introduce the sequence
η k : x Ω ¯ 2 k ζ k ( x ) + ( 1 ζ k ( x ) ) σ ( x ) , k N .
One has that, for any k N , σ and η k are equivalent, namely
c 1 σ η k c 2 σ in  Ω ¯ .
(2.8)
Furthermore, concerning the derivatives, we have also, for any k N ,
(2.9)
(2.10)
Moreover,
lim k + sup Ω Ω ¯ k σ x ( x ) + σ x x ( x ) σ ( x ) = 0 .
(2.11)

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 [22]; see also [23] for more details.

Thus, in the sequel let Ω be an unbounded open subset of R n , n 2 . The σ-algebra of all Lebesgue measurable subsets of Ω is denoted by Σ ( Ω ) . Given E Σ ( Ω ) , | E | is its Lebesgue measure, χ E its characteristic function and E ( x , τ ) = E B ( x , τ ) ( x R n , τ R + ), where B ( x , τ ) is the open ball centered in x and with radius τ.

For λ [ 0 , n [ , q [ 1 , + [ , the space of Morrey type M q , λ ( Ω , t ) ( t R + ) is the set of all functions g in L loc q ( Ω ¯ ) such that
g M q , λ ( Ω , t ) = sup τ ] 0 , t ] x Ω τ λ / q g L q ( Ω ( x , τ ) ) < + ,
(3.1)

endowed with the norm defined in (3.1). One can easily check that, for any arbitrary t 1 , t 2 R + , a function g belongs to M q , λ ( Ω , t 1 ) if and only if it belongs to M q , λ ( Ω , t 2 ) and the norms of g in these two spaces are equivalent. Hence, we limit our attention to the space M q , λ ( Ω ) = M q , λ ( Ω , 1 ) .

The closures of C ( Ω ) and L ( Ω ) in M q , λ ( Ω ) are denoted by M q , λ ( Ω ) and M ˜ q , λ ( Ω ) , respectively.

The following inclusions (algebraic and topological) hold true:
M q , λ ( Ω ) M ˜ q , λ ( Ω ) .
Moreover, one has
M q , λ ( Ω ) M q 0 , λ 0 ( Ω ) if  q 0 q  and  λ 0 n q 0 λ n q .
(3.2)

We put M q ( Ω ) = M q , 0 ( Ω ) , M ˜ q ( Ω ) = M ˜ q , 0 ( Ω ) and M q ( Ω ) = M q , 0 ( Ω ) .

Now, let us define the moduli of continuity of functions belonging to M ˜ q , λ ( Ω ) or M q , λ ( Ω ) . For h R + and g M q , λ ( Ω ) , we set
F [ g ] ( h ) = sup E Σ ( Ω ) sup x Ω | E ( x , 1 ) | 1 h g χ E M q , λ ( Ω ) .
Given a function g M q , λ ( Ω ) , the following characterization holds:
g M ˜ q , λ ( Ω ) lim h + F [ g ] ( h ) = 0 ,
while
g M q , λ ( Ω ) lim h + ( F [ g ] ( h ) + ( 1 ζ h ) g M q , λ ( Ω ) ) = 0 ,
where ζ h denotes a function of class C ( R n ) such that
0 ζ h 1 , ζ h | B ( 0 , h ) ¯ = 1 , supp ζ h B ( 0 , 2 h ) .
Thus, if g is a function in M ˜ q , λ ( Ω ) , a modulus of continuity of g in M ˜ q , λ ( Ω ) is a map σ ˜ q , λ [ g ] : R + R + such that
F [ g ] ( h ) σ ˜ q , λ [ g ] ( h ) , lim h + σ ˜ q , λ [ g ] ( h ) = 0 .
While if g belongs to M q , λ ( Ω ) , a modulus of continuity of g in M q , λ ( Ω ) is an application σ q , λ [ g ] : R + R + such that
F [ g ] ( h ) + ( 1 ζ h ) g M q , λ ( Ω ) σ q , λ [ g ] ( h ) , lim h + σ q , λ [ g ] ( h ) = 0 .

We finally recall a result of [18], obtained adapting to our framework a more general embedding theorem proved in [24].

Lemma 3.1 Let p > 1 and r , t [ p , + [ . If Ω is an open subset of R n having the cone property and g M r ( Ω ) , with r > p if p = n , then
u g u
(3.3)
is a bounded operator from W 1 , p ( Ω ) to L p ( Ω ) . Moreover, there exists a constant c R + such that
g u L p ( Ω ) c g M r ( Ω ) u W 1 , p ( Ω ) ,
(3.4)

with c = c ( Ω , n , p , r ) .

If g M t ( Ω ) , with t > p if p = n / 2 , then the operator in (3.3) is bounded from W 2 , p ( Ω ) to L p ( Ω ) . Moreover, there exists a constant c R + such that
g u L p ( Ω ) c g M t ( Ω ) u W 2 , p ( Ω ) ,
(3.5)

with c = c ( Ω , n , p , t ) .

4 An a priori bound

Let p > 1 and assume that

(we refer the reader, for instance, to [25] for the definition).

Consider the differential operator
L = i , j = 1 n a i j 2 x i x j + i = 1 n a i x i + a ,
(4.1)
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 ( h 0 )-( h 3 ), the operator L : W 2 , p ( Ω ) L p ( Ω ) is bounded.

Then let us recall some known results contained in Theorem 3.2 and Corollary 3.3 of [18].

Theorem 4.1 Let L be defined in (4.1). If hypotheses ( h 0 )-( h 3 ) are satisfied, then there exists a constant c R + such that
(4.2)

with c = c ( Ω , n , ν , p , r , t , a i j L ( Ω ) , σ o q , λ [ ( a i j ) x h ] , σ o r [ a i ] , σ ˜ t [ a ] , a 0 ) .

Moreover, the problem
(4.3)

is uniquely solvable.

Now we prove the claimed weighted W 2 , p -bound.

Theorem 4.2 Let L be defined in (4.1). Under hypotheses ( h 0 )-( h 3 ), there exists a constant c R + such that
(4.4)

with c = c ( Ω , n , s , ν , p , r , t , a i j L ( Ω ) , a i M r ( Ω ) , σ o q , λ [ ( a i j ) x h ] , σ o r [ a i ] , σ ˜ t [ a ] , a 0 ) .

Proof Fix . If ρ + for | x | + , σ = ρ . Thus, in view of the isomorphism of Lemma 2.1, one has that .

Now if we write, η k = η , for a fixed k N , since η and σ are equivalent, one also has that . Hence, the estimate in Theorem 4.1 applies giving that there exists c 0 R + such that
η s u W 2 , p ( Ω ) c 0 L ( η s u ) L p ( Ω ) ,
(4.5)

with c 0 = c 0 ( Ω , n , ν , p , r , t , a i j L ( Ω ) , σ o q , λ [ ( a i j ) x h ] , σ o r [ a i ] , σ ˜ t [ a ] , a 0 ) .

Simple computations give then
L ( η s u ) = η s L u s i , j = 1 n a i j ( ( s 1 ) η s 2 η x i η x j u + η s 1 η x i x j u + 2 η s 1 η x i u x j ) + s i = 1 n a i η s 1 η x i u .
(4.6)
Using (4.5) and (4.6), we deduce that
η s u W 2 , p ( Ω ) c 1 ( η s L u L p ( Ω ) + i , j = 1 n ( η s 2 η x i η x j u L p ( Ω ) + η s 1 η x i x j u L p ( Ω ) + η s 1 η x i u x j L p ( Ω ) ) + i = 1 n a i η s 1 η x i u L p ( Ω ) ) ,
(4.7)

where c 1 R + depends on the same parameters as c 0 and on s.

On the other hand, from Lemma 3.1 and (2.9), we get
a i η s 1 η x i u L p ( Ω ) c 2 sup Ω Ω ¯ k σ x σ a i M r ( Ω ) η s u W 1 , p ( Ω ) ,
(4.8)

with c 2 = c 2 ( Ω , n , p , r ) .

Putting together (2.9), (2.10), (4.7) and (4.8), we obtain the bound
η s u W 2 , p ( Ω ) c 3 [ η s L u L p ( Ω ) + ( sup Ω Ω ¯ k σ x 2 + σ σ x x σ 2 + sup Ω Ω ¯ k σ x σ ) η s u W 2 , p ( Ω ) ] ,
(4.9)

where c 3 depends on the same parameters as c 1 and on a i M r ( Ω ) .

Observe that by (2.11) it follows that there exists k o N such that
( sup Ω Ω ¯ k o σ x 2 + σ σ x x σ 2 + sup Ω Ω ¯ k o σ x σ ) 1 2 c 3 .
(4.10)
Therefore, if we still denote by η the function η k o , combining (4.9) and (4.10), we obtain
η s u W 2 , p ( Ω ) 2 c 3 η s L u L p ( Ω ) .
(4.11)
This together with the fact that σ and η are equivalent and in view of Lemma 2.1 (applied with t = s and considering σ as weight function) gives
| α | 2 σ s α u L p ( Ω ) c 4 σ s L u L p ( Ω ) ,
(4.12)

with c 4 depending on the same parameters as c 3 and on k o , that is (4.4).

If ρ 0 for | x | + , then σ = ρ 1 , thus, always in view of the isomorphism of Lemma 2.1, one has that . Therefore arguing as to get (4.12), one obtains
| α | 2 σ s α u L p ( Ω ) c 5 σ s L u L p ( Ω ) .
(4.13)

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 ( h 0 ) is satisfied, then the Dirichlet problem
(5.1)
is uniquely solvable, with
b = 1 + | s ( s + 1 ) i = 1 n σ x i 2 σ 2 + s i = 1 n σ x i x i σ | ,
(5.2)
if ρ + for | x | + , or
b = 1 + | s ( s 1 ) i = 1 n σ x i 2 σ 2 s i = 1 n σ x i x i σ | ,
(5.3)

if ρ 0 for | x | + .

Proof Let us first consider the case ρ + for | x | + . Since σ = ρ , the function u is a solution of problem (5.1) if and only if w = σ s u is a solution of
(5.4)

with b given by (5.2).

Clearly, for any i { 1 , , n } , one has
2 x i 2 ( σ s w ) = σ s w x i x i 2 s σ s 1 σ x i w x i + s ( s + 1 ) σ s 2 σ x i 2 w s σ s 1 σ x i x i w ,
hence (5.4) is equivalent to the problem
(5.5)
where
α i = 2 s σ x i σ , i = 1 , , n , α = b s ( s + 1 ) i = 1 n σ x i 2 σ 2 + s i = 1 n σ x i x i σ , g = σ s f .

Since σ verifies (2.1) and (2.6), by (1.6) of [3] (which gives that both σ x i / σ and σ x i x i / σ are in every M τ ( Ω ) , τ > 1 ) 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 ρ 0 for | x | + . Since in this case σ = ρ 1 , now the function u solves problem (5.1) if and only if w = σ s u is a solution of
(5.6)

with b given by (5.3).

Thus, problem (5.6) is equivalent to
(5.7)
where
α i = 2 s σ x i σ , i = 1 , , n , α = b s ( s 1 ) i = 1 n σ x i 2 σ 2 s i = 1 n σ x i x i σ , g = σ s f .

The thesis follows then arguing as in the previous case. □

Theorem 5.2 Let L be defined in (4.1). Under hypotheses ( h 0 )-( h 3 ), the problem
(5.8)

is uniquely solvable.

Proof For each τ [ 0 , 1 ] we put
L τ = τ ( L ) + ( 1 τ ) ( Δ + b ) ,

with b given by (5.2) if ρ + for | x | + , or by (5.3) if ρ 0 for | x | + .

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 [26]), we obtain the claimed result. □

Declarations

Authors’ Affiliations

(1)
Dipartimento di Matematica, Università di Salerno

References

  1. Monsurrò S, Salvato M, Transirico M: W 2 , 2 A priori bounds for a class of elliptic operators . Int. J. Differ. Equ. 2011. doi:10.1155/2011/572824Google Scholar
  2. 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/BF02412185MathSciNetView ArticleMATHGoogle Scholar
  3. 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/BF01766150MathSciNetView ArticleGoogle Scholar
  4. 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.MathSciNetGoogle Scholar
  5. Caso L, Cavaliere P, Transirico M: A priori bounds for elliptic equations . Ric. Mat. 2002, 51: 381–396.MathSciNetMATHGoogle Scholar
  6. Caso L, Cavaliere P, Transirico M: Existence results for elliptic equations . J. Math. Anal. Appl. 2002, 274: 554–563. 10.1016/S0022-247X(02)00287-1MathSciNetView ArticleMATHGoogle Scholar
  7. Caso L, Cavaliere P, Transirico M:Solvability of the Dirichlet problem in W 2 , p for elliptic equations with discontinuous coefficients in unbounded domains . Matematiche 2002, 57: 287–302.MathSciNetMATHGoogle Scholar
  8. Caso L, Cavaliere P, Transirico M:An existence result for elliptic equations with V M O -coefficients . J. Math. Anal. Appl. 2007, 325: 1095–1102. 10.1016/j.jmaa.2006.02.048MathSciNetView ArticleMATHGoogle Scholar
  9. Chiarenza F, Frasca M, Longo P:Interior W 2 , p estimates for non divergence elliptic equations with discontinuous coefficients . Ric. Mat. 1991, 40: 149–168.MathSciNetMATHGoogle Scholar
  10. Chiarenza F, Frasca M, Longo P: W 2 , p -Solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients . Trans. Am. Math. Soc. 1993, 336: 841–853.MathSciNetMATHGoogle Scholar
  11. Ragusa MA: Cauchy-Dirichlet problem associated to divergence form parabolic equations . Commun. Contemp. Math. 2004, 6(3):377–393. 10.1142/S0219199704001392MathSciNetView ArticleMATHGoogle Scholar
  12. Boccia S, Monsurrò S, Transirico M: Elliptic equations in weighted Sobolev spaces on unbounded domains . Int. J. Math. Math. Sci. 2008. doi:10.1155/2008/582435Google Scholar
  13. 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:10.1155/2008/901503Google Scholar
  14. 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.MathSciNetMATHGoogle Scholar
  15. 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.MATHGoogle Scholar
  16. 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.MATHGoogle Scholar
  17. Donato P, Giachetti D: Quasilinear elliptic equations with quadratic growth on unbounded domains . Nonlinear Anal. 1986, 8(10):791–804.MathSciNetView ArticleMATHGoogle Scholar
  18. Monsurrò S, Transirico M:A W 2 , p -estimate for a class of elliptic operators . Int. J. Pure Appl. Math. 2013, 83(3):489–499. doi:10.12732/ijpam.v83i3.9View ArticleMATHGoogle Scholar
  19. Monsurrò S, Transirico M:A L p -estimate for weak solutions of elliptic equations . Abstr. Appl. Anal. 2012. doi:10.1155/2012/376179Google Scholar
  20. Monsurrò S, Transirico M: Dirichlet problem for divergence form elliptic equations with discontinuous coefficients . Bound. Value Probl. 2012. doi:10.1186/1687–2770–2012–67Google Scholar
  21. Monsurrò S, Transirico M:A priori bounds in L p for solutions of elliptic equations in divergence form . Bull. Sci. Math. 2013. (in press). doi:10.1016/j.bulsci.2013.02.002Google Scholar
  22. 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.MathSciNetMATHGoogle Scholar
  23. Caso L, D’Ambrosio R, Monsurrò S: Some remarks on spaces of Morrey type . Abstr. Appl. Anal. 2010. doi:10.1155/2010/242079Google Scholar
  24. Cavaliere P, Longobardi M, Vitolo A: Imbedding estimates and elliptic equations with discontinuous coefficients in unbounded domains . Matematiche 1996, 51: 87–104.MathSciNetMATHGoogle Scholar
  25. Adams RA: Sobolev Spaces. Academic Press, New York; 1975.MATHGoogle Scholar
  26. Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order. 2nd edition. Springer, Berlin; 1983.View ArticleMATHGoogle Scholar

Copyright

© Monsurrò and Transirico; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.