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Maximal regularity properties of Agranovich-Vishik type abstract elliptic operators in the half-plane
Journal of Inequalities and Applications volume 2014, Article number: 233 (2014)
In this work, Agranovich-Vishik type abstract elliptic operators in the half-plane are studied. We derive maximal regularity properties of these operators in UMD-valued Sobolev spaces. Our main aim is to prove existence and uniqueness theorems for the solution of abstract elliptic equation with regular boundary conditions on these function spaces. First, by applying the Fourier multiplier, we prove the separability properties of this differential operator in . By using the embedding theorem and the trace theorem, we obtain the main result.
Boundary value problems (BVPs) for differential-operator equations (DOEs) in abstract spaces have been studied extensively by many researchers [1–11]. The maximal regularity properties for partial differential equations (PDEs), and, particularly, for elliptic equations have been studied in [2–4, 12–15]. The main objective of the present paper is to discuss the BVP for a general elliptic equation with complex parameter in a Banach space-valued Sobolev class. Regularity properties in parameter dependent elliptic equations were derived in  for a polynomial dependence, and in [3, 13, 14] for the case of linear dependence of a complex parameter. Here, the complex parameter is included polynomially in the principal part of the equation.
Consider, on , the following differential operator:
depending polynomially on a complex parameter q, of order l with constant complex coefficients where .
We firstly consider the following equation in the whole space:
then we prove the separability properties of differential operator (1) in the Bochner space (i.e., E-valued spaces where E is a Banach space). In particular, the existence and uniqueness of maximal regular solution is derived. In addition, we derive the uniformly coercive estimation of the solution in the space .
Let and . We denote . Consider the following BVP:
are trace functions defined on . The boundary operators are subject to an algebraic condition which we call Condition II (see Section 3). Then we prove the isomorphism theorem (algebraic and topological) of problem (3)-(4) between E-valued Sobolev type spaces and (). Since E is an arbitrary UMD space, the maximal regularity properties of various class of elliptic BVPs is obtained by choosing a different E. This condition, when , becomes the well-known condition of Shapiro-Lopatinskii [16, 17], which is often also called the ellipticity condition for problem (3)-(4). When , the BVP (3)-(4) is considered in a bounded domain with sufficiently smooth boundary, satisfying the complementing condition for all properly elliptic differential operators in [18, 19]. In addition, Agranovich-Vishik worked out problem (3)-(4) in a half-plane and domain with sufficiently smooth boundary . Extensive references can be found in  (see also ).
While studying the elliptic operator depending on a parameter q, it will be convenient for us to use norms depending on the parameter. We put
For any fixed q, the norms and are clearly equivalent.
2 Notation and background
The notation follows the usual standard. Let E be a Banach space and denotes the space of strongly measurable E-valued functions that are defined on the measurable subset with the norm
The Banach space E is often called a UMD space if the Hilbert operator
is bounded in for .
The term ‘UMD’ is an abbreviation for ‘unconditional martingale differences’. UMD spaces include spaces such as , for .
Let l be an integer ≥1. The E-valued Sobolev space, , of order l on Ω is defined by
We shall set
By , we denote the Schwartz space of rapidly decreasing smooth functions. Let F denote the E-valued Fourier transform and let . A function is called a multiplier from to if the mapping for is well defined and provided that there is a constant C so that
The norm of ψ in is defined by
Let be positive integers, be non-negative integers, be positive numbers and , , , , , . The E-valued Besov spaces are defined as
for . For , recall that
The definition of is independent of and .
Let Ω be a domain in . A linear operator T mapping into is called an extension for Ω provided that, for every , the equality holds a.e. in Ω, and for each m there is a constant K such that
Now, we shall give some theorems which will be used to prove the maximal regularity properties of elliptic differential equations in E-valued Sobolev spaces.
Theorem 1 (Trace theorem)
Let E be a UMD space. Then the transformations
are bounded, linear, and surjective from into .
Proof It is clear that
Then by the virtue of the trace theorem of , the operator
will be linear, bounded, and surjective from into .
It is well known that (see, for example, )
This completes the proof. □
Theorem 2 (Extension theorem)
Let E be a UMD-space. Then there exists a bounded linear extension operator from to .
Proof By virtue of , the restriction to of a function in is dense in for any m and p. So, we define the extension operator T only for such functions. Let be a real-valued function satisfying if , if . Let . Then we set
where is a sequence defined as follows:
It is clear that Tu has compact support in . If , we have
which we write
Since when , it follows from (10) that the above series converges absolutely and uniformly as tends to zero. Using (11), we get
Hence, we say that . Moreover, if ,
where depends on m, p, n, and f. Integrating over ,
It follows from (10)-(11) that
It is obvious that
Combining these, we obtain
with . Thus, the proof is finished. □
By virtue of  we state the following theorems.
Theorem 3 (Embedding theorem)
Let E be a UMD space and , , . Then the embedding
is continuous, and for all and , the following estimate holds:
Theorem 4 (Fourier multiplier theorem)
Let E be a UMD space and ψ be a function defined on . Assume there is a constant such that for all multi-indices α satisfying we have
Then for the operator , which is defined as , has an extension to which satisfies
The constants depend only on , n, and p (see ).
3 Elliptic problem in
We shall consider the equation
in the whole space , where is a differential operator with constant complex coefficients depending polynomially on a complex parameter q in such a way that, after replacing by , we get a homogeneous polynomial of degree s. Here, s is a non-negative integer. We symbolize it as
and the symbol of the operator is as follows:
The parameter varies among the limits of a closed sector Q of the complex plane, with vertex at the origin of coordinates. That is,
We begin our analysis by proving Proposition 1 with the help of Theorem 3.
Proposition 1 Let E be a UMD space. For , the operator is a bounded operator from to . More precisely, we have the following estimate:
where the constants and are independent of q and u.
Proof Let be any function. By taking the norm of , and using (5) and interpolation, we obtain
Since the coefficients of the operator A are constants, by using Theorem 3, we obtain
where C is chosen such that . This completes the proof. □
We require the following conditions.
If , then for all and ().
We suppose holds for all and (), where M is a constant.
The main conclusion of this section is the following result.
Theorem 5 Suppose that Condition I is satisfied, , and . Then for there exists one and only one solution of problem (16). Moreover, the coercive uniform estimate holds:
for where the constant is independent of q and u.
Proof First of all, we will prove that there is a solution . Consider the equation
Applying the Fourier transformation F to both sides of (20), we get
By Condition I, for a non-vanishing , we have . Hence, from (21) we obtain
Now, we rewrite the inequality (19) as follows:
Moreover, by using the Fourier transformation, we see that the above estimate is equivalent to
Replacing with we obtain
We have to verify the following inequality to finish the proof:
Let us rewrite the inequalities as follows:
Define the following functions:
In order to prove the above estimate, we have to show that the functions and are Fourier multipliers in . By applying the multiplier theorem (see, for example, [3, 20]), we will show that , , for . That is,
For every there exists a constant such that
Let . Using the inequality (23) and Condition I(a) we find that
where is a constant depending on . Similarly, we can apply the same process to to obtain
where is also a constant depending on . If we choose , we obtain the following inequality from (24)-(25):
Because ξ and q are not zero at the same time, and is bounded as ξ and q tend to infinity, we can write
where C is a constant that does not dependent on ξ or q. That is, we get and . Let , and . Then, by using the boundedness of , we obtain
where and .
In a similar way, we obtain the above estimate for all α with . Then we have . Moreover, applying the same operations we obtain . That is, the functions , are multipliers in . Hence, we find that there is a solution for and the coercive estimate (19) holds.
Finally, to show that the solution is unique, we use the inequality (19). Suppose that there are two solutions and satisfying and . If we subtract from we get
Hence, by the estimate (19) we obtain
which implies that . □
4 Elliptic problem in the half-space
In this section, we consider the following boundary value problem:
Here ; and denote the differential operator with constant complex coefficients depending on a complex parameter q. By replacing D by ξ we obtain homogeneous polynomials and in of degree 2m and , respectively. The parameter q is the same as before.
The operators in (28) and (29) can be connected as follows:
First, by using the embedding theorem (Theorem 3) and the trace theorem (Theorem 1) in the space , we obtain the following.
Proposition 2 Let E be a UMD space and be an integer. Then N is a bounded linear operator from to (). Moreover, we have the estimate
where the constant does not depend on q or u.
Let denote the Fourier transform with respect to . By applying the Fourier transformation in problem (28)-(29) with respect to (), supposing that , and by replacing with y, we get the problem on the half-line
depending on the parameters and q.
We denote by the roots of with positive imaginary part and we set
where the coefficients are analytic functions of and , and they are homogeneous of degree k. Moreover, for every rectifiable Jordan curve γ in the complex plane which encircles all the roots (see ) we have
where is defined as follows:
Condition II For , and , the polynomials
in λ are linearly independent modulo
Condition II is equivalent to the fact that, if is given by
the determinant of the matrix is not equal to zero for all such that , (see ).
For this condition is the same as the Shapiro-Lopatinskii condition.
Proposition 3 Assume that Condition I holds. Then Condition II is equivalent to the fact that problem (32)-(33) admits a solution belonging to , for all , .
Proof Assume that Condition II is satisfied and let , , be m given E-valued functions and . Then, since the determinant of the matrix does not vanish, the system
has unique E-valued solutions which depend on and q. We set, in a similar way to ,
where γ is a rectifiable Jordan curve which encircles the roots of (see , p.130). Here, the function is also an E-valued function, and it satisfies (32); furthermore, in a similar way to , we find that it also satisfies the boundary conditions (33).
Hence, u is a unique solution of (32)-(33). □
Theorem 6 Suppose that Conditions I and II are satisfied. Let l be an integer greater than 2m and . Assume E is a UMD-space. Then, with a non-vanishing , for any functions and there is a unique solution of problem (28)-(29). In addition, for the following uniformly coercive estimate holds:
where the constant C does not depend on q or u.
Proof Let T be the extension operator from to . By the extension theorem (Theorem 2), T is a bounded linear operator from to . First, we consider the equation
By using the Fourier transformation, we find that (37) has a solution expressed as
The following estimate holds:
By Theorem 5, this solution belongs to ; so its restriction belongs to .
Then the following estimate holds:
Now, we consider the problem
such that, by virtue of the trace theorem (Theorem 1), .
First, we have to show that problem (41)-(42) has a unique solution and that the following estimate holds:
with a constant independent of q or of the functions considered. Then it is clear that is a solution of (36), and by Proposition 2 and (40) the following estimate is satisfied:
Moreover, by using the trace theorem (Theorem 1) and estimate (45), we get
with constants , independent of q and u.
Thus, it suffices to prove that the problem
has a unique solution and that estimate (36) holds.
Applying the Fourier transformation with respect to to problem (46)-(47), we obtain
Now we apply Proposition 3 to solve problem (48)-(49). We find that problem (48)-(49) has a solution , and it is obvious that this solution is unique. This completes the proof of the theorem. □
Let , where
Consider the BVP for a system of elliptic equations in
where , , are complex coefficients.
Theorem 7 Let Condition I hold. Then for , there is a unique solution of problem (51) and the following coercive estimate holds:
Now, consider the BVP for system of elliptic equations in as follows:
where , and and are complex coefficients.
From Theorem 6 we obtain the following.
Theorem 8 Let Conditions I and II hold. Then for , and , problem (52) has a unique solution and the uniform coercive estimate
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The authors declare that they have no competing interests.
The ideas and methods in this paper have been consulted by VBS for accuracy and appropriateness. The paper is written by AO. Calculations and proof have been conducted by AO. All authors read and approved the final manuscript.
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Ozer, A., Shakhmurov, V.B. Maximal regularity properties of Agranovich-Vishik type abstract elliptic operators in the half-plane. J Inequal Appl 2014, 233 (2014). https://doi.org/10.1186/1029-242X-2014-233
- maximal regularity
- elliptic operators
- Fourier multiplier
- embedding in Sobolev spaces
- trace in Sobolev spaces