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Linear and nonlinear abstract elliptic equations with VMO coefficients and applications
Fixed Point Theory and Applications volume 2013, Article number: 6 (2013)
In this paper, maximal regularity properties for linear and nonlinear elliptic differential-operator equations with VMO (vanishing mean oscillation) coefficients are studied. For linear case, the uniform separability properties for parameter dependent boundary value problems is obtained in spaces. Then the existence and uniqueness of a strong solution of the boundary value problem for a nonlinear equation is established. In application, the maximal regularity properties of the anisotropic elliptic equation and the system of equations with VMO coefficients are derived.
MSC: 58I10, 58I20, 35Bxx, 35Dxx, 47Hxx, 47Dxx.
The goal of the present paper is to study the nonlocal boundary value problems (BVPs) for parameter dependent linear differential-operator equations (DOEs) with discontinuous top-order coefficients,
and the nonlinear equation
where a is a complex valued function, s is a positive, λ is a complex parameter, and , are linear and B is a nonlinear operator in a Banach space E. Here, the principal coefficients a and A may be discontinuous. More precisely, we assume that a and belong to the operator-valued Sarason class VMO. The Sarason class VMO was first defined in . In the recent years, there has been considerable interest in elliptic and parabolic equations with VMO coefficients. This is mainly due to the fact that the VMO class contains as a subspace that ensures the extension of -theory of operators with continuous coefficients to discontinuous coefficients (see, e.g., [2–7]). On the other hand, the Sobolev spaces and , , are also contained in VMO. Global regularity of the Dirichlet problem for elliptic equations with VMO coefficients has been studied in [2–4]. We refer to the survey , where an excellent presentation and relations with another similar results can be found concerning the regularizing properties of these operators in the framework of Sobolev spaces.
It is known that many classes of PDEs (partial differential equations), pseudo DEs and integro DEs can be expressed in the form of DOEs. Many researchers (see, e.g., [8–24]) investigated similar spaces of functions and classes of PDEs under a single DOE. Moreover, the maximal regularity properties of DOEs with continuous coefficients were studied, e.g., in [10, 19, 20].
Here, the equation with top-order VMO-operator coefficients is considered in abstract spaces. We shall prove the uniform separability of the problem (1), i.e., we show that, for each , there exists a unique strong solution u of the problem (1) and a positive constant C depending only on p, γ, E and A such that
Note that the principal part of a corresponding differential operator is nonselfadjoint. Nevertheless, the sharp uniform coercive estimate for the resolvent and Fredholmness are established. Then the existence and uniqueness of the above nonlinear problem is derived. In application, we study maximal regularity properties of anisotropic elliptic equations in mixed spaces and the systems (finite or infinite) of differential equations with VMO coefficients in the scalar space.
Since (1) involves unbounded operators, it is not easy to get representation for the Green function and estimate of solutions. Therefore, we use the modern harmonic analysis elements, e.g., the Hilbert operators and the commutator estimates in E-valued spaces, embedding theorems of Sobolev-Lions spaces and some semigroups estimates to overcome these difficulties. Moreover, we also use our previous results on equations with continuous leading coefficients and the perturbation theory of linear operators to obtain our main assertions.
2 Notations and background
Throughout the paper, we set E to be a Banach space and . denotes the space of all strongly measurable E-valued functions that are defined on Ω with the norm
where B ranges in the class of the balls in and is the average .
For and , we set
where B ranges in the class of balls with radius ρ.
We will say that a function is in if . We will call the a VMO modulus of f.
Note that, if , where C is the set of complex numbers, then and coincide with the John-Nirenberg class BMO and Sarason class VMO, respectively.
The Banach space E is called a UMD (unconditional martingale difference)-space if the Hilbert operator
A linear operator A is said to be a φ-positive (or positive) in a Banach space E with the bound if is dense on E and
for , , I is an identity operator in E and is the space of bounded linear operators in E. Sometimes instead of , it will be written as and denoted by . It is known [, §1.15.1] that there exist fractional powers of the positive operator A. Let denote the space with the graphical norm
where is a sequence of independent symmetric -valued random variables on .
Let denote the Schwarz class, i.e., the space of all E-valued rapidly decreasing smooth functions on . Let F denote the Fourier transformation. A function is called a Fourier multiplier from to if the map , is well defined and extends to a bounded linear operator
The set of all multipliers from to will be denoted by . For , it will be denoted by .
Definition 1 A Banach space E is said to be a space satisfying the multiplier condition if, for any , the R-boundedness of the set implies that Ψ is a Fourier multiplier in , i.e., for any .
Definition 2 The φ-positive operator A is said to be an R-positive in a Banach space E if there exists such that the set is R-bounded.
A linear operator is said to be positive in E uniformly in x if is independent of x, is dense in E and
for all , .
Let denote the space of all compact operators from to . For , it is denoted by .
Assume and E are two Banach spaces and is continuously and densely embedded into E. Let m be a natural number. (the so-called Sobolev-Lions type space) denotes a space of all functions possessing the generalized derivatives such that endowed with the norm
For , the space will be denoted by . It is clear that
Let s be a positive parameter. We define in the following parameter-dependent norm:
The space is defined as a trace space for as in a scalar case (see, e.g., [, §3.6.1]).
Function satisfying the equation (1) a.e. on is said to be a solution of the problem (1) on .
From  we have
Theorem A 1 Suppose the following conditions are satisfied:
E is a Banach space satisfying the multiplier condition with respect to and A is an R-positive operator in E;
are n tuples of nonnegative integer numbers such that and ;
is a region such that there exists a bounded linear extension operator from to .
Then the embedding
is continuous and there exists a positive constant such that
for all and .
Theorem A 2 Suppose all the conditions of Theorem A1 are satisfied. Assume Ω is a bounded region in and . Then, for , the embedding
In a similar way as in [, Theorem 2.1], we have the following result.
Lemma A 1 Let E be a Banach space and . The following conditions are equivalent:
f is in the BMO closure of the set of uniformly continuous functions which belong to VMO;
For , , , consider the commutator operator
Theorem A 3 Let E be a UMD space and . Then is a bounded operator in , .
From Theorem A3 and the property (2) of Lemma A1 we obtain, respectively, the following.
Theorem A 4 Assume all the conditions of Theorem A3 are satisfied. Also, let and let η be the VMO modulus of a. Then, for any , there exists a positive number such that
Theorem A 5 Let E be a UMD space and uniformly R-positive in E. Moreover, let , . Then the commutator operator
is bounded in , .
Theorem A 6 Assume all the conditions of Theorem A5 are satisfied and η is a VMO modulus of .
Then, for any , there exists a positive number such that
Consider the nonlocal BVP for a parameter dependent DOE with constant coefficients
where , a, , are complex numbers, s is a positive parameter, λ is a complex spectral parameter, , , and A is a linear operator in E. Let , be roots of the equation and let
It is known that if the operator A is φ-positive in E, then operators , generate analytic semigroups
Theorem A 7 Assume the following conditions are satisfied:
E is a Banach space satisfying the multiplier condition with respect to ;
A is an R-positive operator in E for and ;
and for , .
for , , and for sufficiently large , the problem (2) has a unique solution . Moreover, the coercive uniform estimate holds
For , the solution is represented as(3)
where and are uniformly bounded operators in E and
Consider the BVP for a DOE with variable coefficients
where is a complex valued function, s is a positive parameter, , , are complex numbers, λ is a spectral parameter, and is a linear operator in E.
Let , be roots of the equation and let
In the next theorem, let us consider the case that principal coefficients are continuous. The well-posedness of this problem occurs in the studying of equations with VMO coefficients. From [, Theorem 3] and [, Theorem 5.1], we get
Theorem A 8 Suppose the following conditions are satisfied:
E is a Banach space satisfying the multiplier condition with respect to ;
, , , and for a.e. ;
and for , . a.e. ;
is a uniformly R-positive operator in E and
Then, for all , and for sufficiently large , there is a unique solution of the problem (4). Moreover, the following coercive uniform estimate holds:
3 DOEs with VMO coefficients
Consider the principal part of the problem (1)
Condition 1 Assume the following conditions are satisfied:
E is a UMD space, ;
, is a VMO modulus of a, , ;
and for , , . a.e. ;
is a uniformly R-positive operator in E and
, and is a VMO modulus of .
First, we obtain an integral representation formula for solutions.
Lemma 1 Let Condition 1 hold and . Then, for all solutions u of the problem (5) belonging to , we have
here , are uniformly bounded operators,
and the expression is a scalar multiple of .
Proof Consider the problem (5) for and , i.e.,
Let be a solution of the problem (7). Taking into account the equality and Theorem A7, we get
Setting in above, we get (6) for . Then the density argument, using Theorem A3, gives the conclusion for
Consider the problem (5) on , i.e.,
Theorem 1 Suppose Condition 1 is satisfied. Then there exists a number such that the uniform coercive estimate
holds for large enough and , , , where C is a positive constant depending only on p, , , .
Proof By Lemma 1, for any solution of the problem (8), we have
here , are uniformly bounded operators,
Moreover, from (10) and (11), clearly, we get
where the expression differs from only by a constant.
Consider the operators
Since the operators and are regular on , by using the positivity properties of A and the analyticity of semigroups in a similar way as in [, Theorem 3.1], we get
Since the Hilbert operator is bounded in for a UMD space E, we have
Thus, by virtue of Theorems A4, A6 and in view of (10)-(12) for any , there exists a positive number such that
Hence, the estimates (13)-(15) imply (9). □
Theorem 2 Assume Condition 1 holds. Let be a solution of (4). Then, for sufficiently large , , the following coercive uniform estimate holds:
Proof This fact is shown by a covering and flattening argument, in a similar way as in Theorem A8. Particularly, by partition of unity, the problem is localized. Choosing diameters of supports for corresponding finite functions, by using Theorem 1, Theorems A4, A6, A7 and embedding Theorem A1 (see the same technique for DOEs with continuous coefficients [19, 20]), we obtain the assertion.
Let denote the operator in generated by the problem (4) for , i.e.,
Theorem 3 Assume Condition 1 holds. Then, for all , and for large enough , the problem (5) has a unique solution . Moreover, the following coercive uniform estimate holds:
Proof First, let us show that the operator has a left inverse. Really, it is clear that
By Theorem A1 for , we have
Then, by virtue of (16) and in view of the above relations, we infer, for all and sufficiently large , that there is a small ε and such that
From the estimate (18) for , we get
The estimate (19) implies that (4) has a unique solution and the operator has a bounded inverse in its rank space. We need to show the rank space coincides with all the space . It suffices to prove that there is a solution for all . This fact can be derived in a standard way, approximating the equation with a similar one with smooth coefficients [19, 20]. More precisely, by virtue of [, Theorem 3.4], UMD spaces satisfy the multiplier condition. Moreover, by the part (2) of Lemma A1, given with VMO modules , we can find a sequence of mollifiers functions converging to a in BMO as with a VMO modulus such that . In a similar way, it can be derived for the operator function . □
Result 1 Theorem 3 implies that the resolvent satisfies the following sharp uniform estimate:
for , and .
The estimate (20) particularly implies that the operator Q is uniformly positive in and generates analytic semigroups for (see, e.g., [, §1.14.5]).
Remark 1 Conditions , arise due to nonlocality of the boundary conditions (4). If the boundary conditions are local, then the conditions mentioned above are not required any more.
Consider the problem (1), where is the same boundary condition as in (4). Let denote a differential operator generated by the problem (1). We will show the separability and Fredholmness of (1).
Theorem 4 Assume
Condition 1 holds;
for any , there is such that for a.e. and for , ,
Then, for all and for large enough , , there is a unique solution of the problem (1) and the following coercive uniform estimate holds:
Proof It is sufficient to show that the operator has a bounded inverse from to . Put , where
By the second assumption and Theorem A1, there is a small ε and such that
In view of estimates (17) and (22), we have
for and . By Theorem 3, the operator has a bounded inverse from to for sufficiently large . So, (23) implies the following uniform estimate:
It is clear that
Then, by above relation and by virtue of Theorem 3, we get the assertion. □
Theorem 4 implies the following result.
Result 2 Suppose all the conditions of Theorem 4 are satisfied. Then the resolvent of the operator satisfies the following sharp uniform estimate:
for , and .
Consider the problem (1) for , i.e.,
Theorem 5 Assume all the conditions of Theorem 4 hold and . Then the problem (24) is Fredholm from into .
Proof Theorem 4 implies that the operator has a bounded inverse from to for large enough ; that is, the operator is Fredholm from into . Then, by virtue of Theorem A2 and by perturbation theory of linear operators, we obtain the assertion. □
4 Nonlinear DOEs with VMO coefficients
Let, at first, consider the linear BVP in a moving domain ,
where a is a complex valued function, and , are possible operators in a Banach space E, where is a positive continuous independent of u.
Theorem 4 implies the following result.
Result 3 Let all the conditions of Theorem 4 be satisfied. Then the problem (25), for , , and for large enough , has a unique solution and the following coercive uniform estimate holds:
Proof Really, under the substitution , the moving boundary problem (25) maps to the following BVP with a parameter in a fixed domain :
Then, by virtue of Theorem 4, we obtain the assertion. □
Consider the nonlinear problem
where , , are complex numbers, , where b is a positive number in .
In this section, we will prove the existence and uniqueness of a maximal regular solution of the nonlinear problem (26). Assume A is a φ-positive operator in a Banach space E. Let
Remark 2 By using [, §1.8], we obtain that the embedding is continuous and there exists the constant such that for , , , ,
Condition 2 Assume the following are satisfied:
and is a positive continuous function on , ;
E is a UMD space and ;
is a measurable function for each , ; is continuous with respect to and . Moreover, for each , there exists such that
where and for a.a. , and
for , the operator is R-positive in E uniformly with respect to ; , where the domain definition does not depend on x and U; is continuous, where for fixed ;
for each , there is a positive constant such that for , , and and .
Theorem 6 Let Condition 1 hold. Then there is such that the problem (26) has a unique solution that belongs to the space .
Proof Consider the linear problem
By virtue of Result 3, the problem (27) has a unique solution for all and for sufficiently large that satisfies the following
where the constant C does not depend on and . We want to solve the problem (26) locally by means of maximal regularity of the linear problem (27) via the contraction mapping theorem. For this purpose, let w be a solution of the linear BVP (27). Consider a ball
For , consider the linear problem
Define a map Q on by , where u is a solution of the problem (28). We want to show that and that Q is a contraction operator provided b is sufficiently small and r is chosen properly. For this aim, by using maximal regularity properties of the problem (28), we have
By assumption (5), we have
Bearing in mind
where is a fixed number. In view of the above estimates, by a suitable choice of , and for sufficiently small , we have
Moreover, in a similar way, we obtain
By a suitable choice of , and for sufficiently small , we obtain , , i.e., Q is a contraction operator. Eventually, the contraction mapping principle implies a unique fixed point of Q in which is the unique strong solution . □
5 Boundary value problems for anisotropic elliptic equations with VMO coefficients
Let be an open connected set with a compact -boundary ∂ Ω. Let us consider the nonlocal boundary value problems on a cylindrical domain for the following anisotropic elliptic equation with VMO top-order coefficients:
where s is a positive parameter, a, are complex valued functions, and are complex numbers,
If , , will denote the space of all p-summable scalar-valued functions with a mixed norm (see, e.g., [, §1]), i.e., the space of all measurable functions f defined on G, for which
Analogously, denotes the anisotropic Sobolev space with the corresponding mixed norm [, §10].
Theorem 7 Let the following conditions be satisfied;
, , , , a.e. ;
and for , , a.e. ;
, for a.e. and ;
for each and for each with and ;
for each j, β and , , for , , where is a normal to ∂G;
for , , , , let ;
for each , a local BVP in local coordinates corresponding to
has a unique solution for all , and for with .
for all , and sufficiently large , the problem (29)-(31) has a unique solution u belonging to and the following coercive uniform estimate holds:
for , the problem (29)-(31) is Fredholm in .
Proof Let . Then by virtue of , the part (1) of Condition 1 is satisfied. Consider the operator A acting in defined by
For , also consider operators in
The problem (29)-(31) can be rewritten in the form (1), where , are functions with values in . By virtue of , the problem
has a unique solution for and arg , . Moreover, in view of [, Theorem 8.2], the operator A is R-positive in , i.e., Condition 1 holds. Moreover, it is known that the embedding is compact (see, e.g., [, Theorem 3.2.5]). Then, by using interpolation properties of Sobolev spaces (see, e.g., [, §4]), it is clear that the condition (2) of Theorem 4 is fulfilled too. Then from Theorems 4, 5, the assertions are obtained. □
6 Systems of differential equations with VMO coefficients
Consider the nonlocal BVPs for infinity systems of parameter-differential equations with principal VMO coefficients,
where s is a positive parameter, a, , are complex valued functions, N is a finite or infinite natural number, and are complex numbers, .
Let Q denote the operator in generated by the problem (32)-(33). Let
Theorem 8 Suppose the following conditions are satisfied:
, , , a.e. ;
and for , a.e. , , ;
, , ;
there are , such that
Then, for all , and for sufficiently large , the problem (32)-(33) has a unique solution belonging to and the following coercive estimate holds:
Proof Really, let , A and be infinite matrices such that
It is clear that the operator A is R-positive in . Therefore, by Theorem 4, the problem (32)-(33) has a unique solution for all , the estimate (34) holds. □
Remark 3 There are many positive operators in different concrete Banach spaces. Therefore, putting concrete Banach spaces and concrete positive operators (i.e., pseudo-differential operators or finite or infinite matrices for instance) instead of E and A, respectively, by virtue of Theorem 4, 5, we can obtain a different class of maximal regular BVPs for partial differential or pseudo-differential equations or their finite and infinite systems with VMO coefficients.
Sarason D: On functions of vanishing mean oscillation. Trans. Am. Math. Soc. 1975, 207: 391–405.
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(2):841–853.
Chiarenza F: regularity for systems of PDEs with coefficients in VMO. 5. In Nonlinear Analysis, Function Spaces and Applications. Prometheus, Prague; 1994.
Miranda C: Partial Differential Equations of Elliptic Type. Springer, Berlin; 1970.
Maugeri A, Palagachev DK, Softova L: Elliptic and Parabolic Equations with Discontinuous Coefficients. Wiley, Berlin; 2000.
Krylov NV: Parabolic and elliptic equations with VMO coefficients. Commun. Partial Differ. Equ. 2007, 32(3):453–475. 10.1080/03605300600781626
Amann H 1. In Linear and Quasi-Linear Problems. Birkhäuser, Boston; 1995.
Ashyralyev A: On well-posedeness of the nonlocal boundary value problem for elliptic equations. Numer. Funct. Anal. Optim. 2003, 24(1–2):1–15. 10.1081/NFA-120020240
Denk R, Hieber M, Prüss J: R -boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 2003., 166: Article ID 788
Favini A, Shakhmurov V, Yakubov Y: Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. Semigroup Forum 2009, 79(1):22–54. 10.1007/s00233-009-9138-0
Gorbachuk VI, Gorbachuk ML: Boundary Value Problems for Differential-Operator Equations. Naukova Dumka, Kiev; 1984.
Heck H, Hieber M: Maximal -regularity for elliptic operators with VMO -coefficients. J. Evol. Equ. 2003, 3: 62–88.
Lions JL, Peetre J: Sur une classe d’espaces d’interpolation. Publ. Math. 1964, 19: 5–68.
Ragusa MA: Necessary and sufficient condition for VMO function. Appl. Math. Comput. 2012, 218(24):11952–11958. 10.1016/j.amc.2012.06.005
Ragusa MA: Embeddings for Morrey-Lorentz spaces. J. Optim. Theory Appl. 2012, 154(2):491–499. 10.1007/s10957-012-0012-y
Sobolevskii PE: Inequalities coerciveness for abstract parabolic equations. Dokl. Akad. Nauk SSSR 1964, 57(1):27–40.
Shakhmurov VB: Coercive boundary value problems for regular degenerate differential-operator equations. J. Math. Anal. Appl. 2004, 292(2):605–620. 10.1016/j.jmaa.2003.12.032
Shakhmurov VB: Separable anisotropic differential operators and applications. J. Math. Anal. Appl. 2007, 327(2):1182–1201. 10.1016/j.jmaa.2006.05.007
Shakhmurov VB: Degenerate differential operators with parameters. Abstr. Appl. Anal. 2007, 2006: 1–27.
Segovia C, Torrea JL: Vector-valued commutators and applications. Indiana Univ. Math. J. 1989, 38(4):959–971. 10.1512/iumj.1989.38.38044
Triebel H: Interpolation Theory. Function Spaces. Differential Operators. North-Holland, Amsterdam; 1978.
Weis L: Operator-valued Fourier multiplier theorems and maximal regularity. Math. Ann. 2001, 319: 735–758. 10.1007/PL00004457
Yakubov S, Yakubov Ya: Differential-Operator Equations. Ordinary and Partial Differential Equations. Chapman & Hall/CRC, Boca Raton; 2000.
John F, Nirenberg L: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 1961, 14: 415–476. 10.1002/cpa.3160140317
Burkholder DL: A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions. In Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II. Wadsworth, Belmont; 1983:270–286. Chicago, Ill., 1981
Hytönen T, Weis L: A T 1 theorem for integral transformations with operator-valued kernel. J. Reine Angew. Math. 2006, 599: 155–200.
Shakhmurov VB: Embedding theorems and maximal regular differential operator equations in Banach-valued function spaces. J. Inequal. Appl. 2005, 4: 605–620.
Blasco O: Operator valued BMO commutators. Publ. Mat. 2009, 53(1):231–244.
Shakhmurov VB, Shahmurova A: Nonlinear abstract boundary value problems atmospheric dispersion of pollutants. Nonlinear Anal., Real World Appl. 2010, 11: 932–951. 10.1016/j.nonrwa.2009.01.037
Grisvard P: Elliptic Problems in Nonsmooth Domains. Pitman, London; 1985.
Besov OV, Ilin VP, Nikolskii SM: Integral Representations of Functions and Embedding Theorems. Nauka, Moscow; 1975.
The author declares that he has no competing interests.
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Cite this article
Shakhmurov, V.B. Linear and nonlinear abstract elliptic equations with VMO coefficients and applications. Fixed Point Theory Appl 2013, 6 (2013). https://doi.org/10.1186/1687-1812-2013-6
- differential equations with VMO coefficients
- boundary value problems
- differential-operator equations
- maximal regularity
- abstract function spaces
- nonlinear elliptic equations