- Open Access
Optimal regular differential operators with variable coefficients and applications
© Shakhmurov; licensee Springer 2013
Received: 23 November 2012
Accepted: 10 February 2013
Published: 27 February 2013
In this paper, maximal regularity properties for linear and nonlinear high-order elliptic differential-operator equations with VMO coefficients are studied. For the linear case, the uniform coercivity property of 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 high-order nonlinear equation are established. In application, the maximal regularity properties of the anisotropic elliptic equation and the system of equations with VMO coefficients are derived.
AMS Subject Classification:58I10, 58I20, 35Bxx, 35Dxx, 47Hxx, 47Dxx.
where a is a complex-valued function, s is a positive and λ is a complex parameter; , 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 (vanishing mean oscillation). Sarason class VMO was at first defined in . In the recent years, there has been considerable interest to elliptic and parabolic equations with VMO coefficients. This is mainly due to the fact that VMO spaces contain as a subspace that ensures the extension of -theory of operators with continuous coefficients to discontinuous coefficients (see, e.g., [2–11]). 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 excellent presentation and relations with 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 (differential equations) and integro DEs can be expressed in the form of DOEs. Many researchers (see, e.g., [12–24]) 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 [12, 14, 18, 19].
Note that the principal part of a corresponding differential operator is non self-adjoint. Nevertheless, the sharp uniform coercive estimate for the resolvent and Fredholmness are established. Then, the existence and uniqueness of the above nonlinear problem are derived. In application, we study maximal regularity properties of anisotropic elliptic equations in mixed spaces and 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 the 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 semigroup 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 main assertions.
2 Notations and background
where B ranges in the class of the balls in , is the Lebesgue measure of B and is the average .
where B ranges in the class of balls with radius ρ.
We will say that a function is in if . We will call the VMO modulus of f.
Note that if , where C is the set of complex numbers, then and coincide with John-Nirenberg class BMO and Sarason class VMO, respectively.
is bounded in , (see, e.g., ). UMD spaces include, e.g., , spaces and Lorentz spaces , .
where is a sequence of independent symmetric -valued random variables on .
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 a multiplier condition if for any , the R-boundedness of the set implies that Ψ is a Fourier multiplier in , i.e., for any .
for all , .
Function satisfying equation (1) a.e. on is said to be a solution of the problem (1) on .
From  we have the following theorem.
E is a Banach space satisfying the multiplier condition with respect to and A is an R-positive operator in E;
- (2)are n-tuples of nonnegative integer numbers such that
is a region such that there exists a bounded linear extension operator from to .
for all and .
In a similar way as in [, Theorem 2.1], we have the following result.
f is in the BMO closure of the set of uniformly continuous functions which belong to VMO;
Proof Indeed, we observe that if with VMO modulus η, there exists a constant C such that for so that the E-valued usual mollifiers converge to f in the BMO norm. More precisely, given with VMO modulus , we can find a sequence of E-valued functions converging to f in E-valued BMO spaces as with VMO moduli such that . In a similar way, other cases are derived. □
Theorem A3 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:
Note that singular integral operators in E-valued spaces were studied, e.g., in .
Theorem A6 Assume all conditions of Theorem A5 are satisfied and η is a VMO modulus of .
E is a Banach space satisfying the multiplier condition with respect to ;
A is an R-positive operator in E for and ;
, , , and .
- (1)for , , and for sufficiently large , the problem (2) has a unique solution . Moreover, the following coercive uniform estimate holds:
- (2)For , the solution is represented as(3)
where is a complex-valued function, , , are complex numbers, s is a positive and λ is a complex parameter, and is a linear operator in E.
In the next theorem, we consider the case when principal coefficients are continuous. The well-posedness of this problem occurs in studying of equations with VMO coefficients. From [, Theorem 3] and [, Theorem 3.2], we get the following.
E is a Banach space satisfying the multiplier condition with respect to ;
, , , and a.e. ;
, and for a.e. ;
- (4)is a uniformly R-positive operator in E and
3 DOEs with VMO coefficients
E is a UMD space, ;
, is a VMO modulus of a;
, , , and for , a.e. ;
- (4)is a uniformly R-positive operator in E and
, and is a VMO modulus of .
First, we obtain an integral representation formula for solutions.
and the expression is a scalar multiple of .
for , with large enough .
where the expression differs from only by a constant.
Hence the estimates (13)-(15) imply (9). □
Proof This fact is shown by 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 [18, 19]), we obtain the assertion.
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 that the rank space coincides with the all 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 [18, 19]. More precisely, by virtue of [, Theorem 3.4], UMD spaces satisfy the multiplier condition. Moreover, by part (2) of Lemma A1, given with VMO modules , we can find a sequence of mollifiers functions converging to a in BMO as with VMO modulus such that . In a similar way, it can be derived for the operator function . □
for , and .
The estimate (20) particularly implies that the operator Q is uniformly positive in and generates an analytic semigroup for (see, e.g., [, §1.14.5]).
Remark 1 Conditions , arise due to nonlocality of the boundary conditions (4). If 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 the differential operator generated by the problem (1). We will show the separability and Fredholmness of (1).
Condition 1 holds;
- (2)for any , there is such that for a.e. and
Then, by the above relation and by virtue of Theorem 3, we get the assertion. □
Theorem 4 implies the following result.
for , and .
Theorem 5 Assume all conditions of Theorem 4 hold and . Then the problem (23) 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
where a is a complex-valued function and , are linear operators in a Banach space E, where is a positive continuous function independent of u.
Theorem 4 implies the following.
Then, by virtue of Theorem 4, we obtain the assertion. □
where , , are complex numbers, , where b is a positive number in .
E is a UMD space, ;
, , , and for , , a.e. ;
- (3)is a measurable function for each , ; is continuous with respect to and . Moreover, for each , there exists such that
for , the operator is R-positive in E uniformly with respect to ; , , where 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 2 hold. Then there is such that the problem (26) has a unique solution belonging to space .
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
The Fredholm property of BVPs for elliptic equations with parameters in smooth domains were studied, e.g., in [14, 24, 28]; also, for non-smooth domains, these questions were investigated, e.g., in .
Analogously, denotes the anisotropic Sobolev space with the corresponding mixed norm [, §10].
, , , 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 ;
has a unique solution for all , and for with .
- (a)for all , and sufficiently large , the problem (28)-(30) has a unique solution u belonging to and the following coercive uniform estimate holds:
for the problem (28)-(30) is Fredholm in .
has a unique solution for and , , and 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 to see that 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
where s is a positive parameter, a, , are complex-valued functions, N is finite or infinite natural number, and are complex numbers, .
From Theorem 4, we obtain the following.
, , , and for , a.e. ;
and , , .
It is clear to see that the operator A is R-positive in . Therefore, by Theorem 4, the problem (31)-(32) has a unique solution for all , and the estimate (33) 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 Theorems 4 and 5, we can obtain a different class of maximal regular BVPs for partial differential or pseudo-differential equations or its finite and infinite systems with VMO coefficients.
Dedicated to Professor Hari M Srivastava.
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