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Optimal regular differential operators with variable coefficients and applications
Fixed Point Theory and Applications volume 2013, Article number: 42 (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.
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 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].
Here the equation with top-order VMO-operator coefficients is considered in abstract spaces. We will 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 p, E, m and A such that
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
Throughout the paper, we set E a Banach space and . denotes the space of all strongly measurable E-valued functions that are defined on Ω with the norm
(bounded mean oscillation, see [25, 26]) is the space of all E-valued local integrable functions with the norm
where B ranges in the class of the balls in , is the Lebesgue measure of B 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 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.
The Banach space E is called a UMD-space if the Hilbert operator
is bounded in , (see, e.g., ). UMD spaces include, e.g., , spaces and Lorentz spaces , .
A linear operator A is said to be φ-positive (or positive) in a Banach space E with 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 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
Let and be two Banach spaces. A set is called R-bounded (see [14, 23]) if there is a positive constant C such that for all and , ,
where is a sequence of independent symmetric -valued random variables on .
Let denote the Schwartz 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 a 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
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 is endowed with the norm
For the space will be denoted by . It is clear to see that
Let s be a positive parameter. We define in the following parameterized norm:
Function satisfying equation (1) a.e. on is said to be a solution of the problem (1) on .
From  we have the following theorem.
Theorem A1 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
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 A2 Suppose all 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 A1 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
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. □
From [, Theorem 1] and [, Corollary 2.7], we have the following.
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:
Theorem A4 Assume all 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 A5 Let E be a UMD space, and uniformly R-positive in E. Moreover, let , . Then the following commutator operator is bounded in :
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 .
Then, for any , there exists a positive number such that
Consider the nonlocal BVP for parameter-dependent DOE with constant coefficients
where , a, , are complex numbers, , , s is a positive and λ is a complex parameter; and A is a linear operator in E. Let be roots of the equation , be a 2m-dimensional matrix and be a determinant of the matrix , where
It is known that (see, e.g., [, §1.15]) if the operator A is φ-positive in E, then operators , generate the following analytic semigroups:
From [, Theorem 1] and [, Theorem 3.2], we obtain the following.
Theorem A7 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 , , and for sufficiently large , the problem (2) has a unique solution . Moreover, the following coercive uniform estimate holds:
For , the solution is represented as(3)
where are uniformly bounded operators in E and
Consider the BVP for DOE with variable coefficients
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.
Let , be roots of the equation , be a 2m-dimensional matrix and be a determinant of the function matrix , where
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.
Theorem A8 Suppose the following conditions are satisfied:
E is a Banach space satisfying the multiplier condition with respect to ;
, , , and a.e. ;
, and for a.e. ;
is a uniformly R-positive operator in E and
Then, for , 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
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 the above, we get (6) for . Then a density argument and Theorem A3 give 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 following uniform coercive estimate holds:
for , with large enough .
Proof By Lemma 1, for any solution of the problem (8), we have
here are uniformly bounded operators, and
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 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.
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 to see 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
In view of (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 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 . □
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 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).
Theorem 4 Assume the following:
Condition 1 holds;
for any , there is such that for a.e. and
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
By Theorem 3, the operator has a bounded inverse from to for sufficiently large . So, (22) implies the following uniform estimate:
It is clear to see that
Then, by the above relation and by virtue of Theorem 3, we get the assertion. □
Theorem 4 implies the following result.
Result 2 Suppose all 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 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
Let at first consider the linear BVP in a moving domain
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.
Result 3 Let all conditions of Theorem 4 be satisfied. Then the problem (24) has a unique solution for , , with large enough , and the following coercive uniform estimate holds:
Proof Really, under the substitution , the moving boundary problem (24) maps to the following BVP with parameter in the fixed domain :
Then, by virtue of Theorem 4, we obtain the assertion. □
Consider the following 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 (25). 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 a constant such that for , , ,
Condition 2 Assume the following are satisfied:
E is a UMD space, ;
, , , and for , , a.e. ;
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 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 .
Consider the linear problem
By virtue of Result 3, the problem (26) 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 (25) locally by means of maximal regularity of the linear problem (26) 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 (27). 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
Bear 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
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 .
Let be an open connected set with 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,
For , , 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:
, , , 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 , the local BVP in local coordinates corresponding to
has a unique solution for all , and for with .
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 .
Proof Let . Then by virtue of , part (1) of Condition 1 is satisfied. Consider the operator A acting in defined by
For also consider operators in
The problem (28)-(30) can be rewritten in the form (1), where , are functions with values in . By virtue of [, Theorem 8.2], the problem
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
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 finite or infinite natural number, and are complex numbers, .
Let be a real function and
From Theorem 4, we obtain the following.
Theorem 8 Suppose the following conditions are satisfied:
, , , and for , a.e. ;
and , , .
Then, for all , and for sufficiently large , problem (32)-(33) has a unique solution belonging to and the following coercive estimate holds:
Proof Really, let , A and be matrices such that
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.
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Dedicated to Professor Hari M Srivastava.
The author declares that they have no competing interests.
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Shakhmurov, V. Optimal regular differential operators with variable coefficients and applications. Fixed Point Theory Appl 2013, 42 (2013). https://doi.org/10.1186/1687-1812-2013-42
- differential equations with VMO coefficients
- boundary value problems
- differential-operator equations
- maximal regularity
- abstract function spaces
- nonlinear elliptic equations