Maximal regularity properties of Agranovich-Vishik type abstract elliptic operators in the half-plane
© Ozer and Shakhmurov; licensee Springer. 2014
Received: 20 December 2013
Accepted: 27 May 2014
Published: 5 June 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.
depending polynomially on a complex parameter q, of order l with constant complex coefficients where .
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 .
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 ).
For any fixed q, the norms and are clearly equivalent.
2 Notation and background
is bounded in for .
The term ‘UMD’ is an abbreviation for ‘unconditional martingale differences’. UMD spaces include spaces such as , for .
The definition of is independent of and .
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)
are bounded, linear, and surjective from into .
will be linear, bounded, and surjective from into .
This completes the proof. □
Theorem 2 (Extension theorem)
Let E be a UMD-space. Then there exists a bounded linear extension operator from to .
with . Thus, the proof is finished. □
By virtue of  we state the following theorems.
Theorem 3 (Embedding theorem)
Theorem 4 (Fourier multiplier theorem)
The constants depend only on , n, and p (see ).
3 Elliptic problem in
We begin our analysis by proving Proposition 1 with the help of Theorem 3.
where the constants and are independent of q and u.
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.
for where the constant is independent of q and u.
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.
which implies that . □
4 Elliptic problem in the half-space
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.
First, by using the embedding theorem (Theorem 3) and the trace theorem (Theorem 1) in the space , we obtain the following.
where the constant does not depend on q or u.
depending on the parameters and q.
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 , .
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). □
where the constant C does not depend on q or u.
By Theorem 5, this solution belongs to ; so its restriction belongs to .
such that, by virtue of the trace theorem (Theorem 1), .
with constants , independent of q and u.
has a unique solution and that estimate (36) holds.
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. □
where , , are complex coefficients.
where , and and are complex coefficients.
From Theorem 6 we obtain the following.
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