- Open Access
The structure of fractional spaces generated by a two-dimensional elliptic differential operator and its applications
© Ashyralyev et al.; licensee Springer. 2014
- Received: 30 October 2013
- Accepted: 3 December 2013
- Published: 2 January 2014
We consider the two-dimensional differential operator defined on functions on the half-plane with the boundary conditions , , where , , are continuously differentiable and satisfy the uniform ellipticity condition , . The structure of the fractional spaces generated by the operator A is investigated. The positivity of A in Hölder spaces is established. In applications, theorems on well-posedness in a Hölder space of elliptic problems are obtained.
MSC: 35J25, 47E05, 34B27.
- positive operator
- fractional spaces
- Green’s function
- Hölder spaces
It is well known that (see, for example, [1–3] and the references therein) various classical and non-classical boundary value problems for partial differential equations can be considered as an abstract boundary value problem for an ordinary differential equation in a Banach space with a densely defined unbounded operator. The importance of the positivity property of the differential operators in a Banach space in the study of various properties for partial differential equations is well known (see, for example, [4–7] and the references therein). Several authors have investigated the positivity of a wider class of differential and difference operators in Banach spaces (see [8–18] and the references therein).
Let us give the definition of positive operators and introduce the fractional spaces and preliminary facts that will be needed in the sequel.
holds on the edges , of S, and outside of the sector S. The infimum of all such angles ϕ is called the spectral angle of the positive operator A and is denoted by . We say that A is a strongly positive operator in E if .
Throughout the article, M indicates positive constants which may differ from time to time, and we are not interested to precise. If the constant depends only on , then we will write .
Theorem 1 
Danelich in  considered the positivity of a difference analog of the 2m th-order multi-dimensional elliptic operator with dependent coefficients on semi-spaces .
The structure of fractional spaces generated by positive multi-dimensional differential and difference operators on the space in Banach spaces has been well investigated (see [21–23] and the references therein).
In papers [19, 24–27] the structure of fractional spaces generated by positive one-dimensional differential and difference operators in Banach spaces was studied. Note that the structure of fractional spaces generated by positive multi-dimensional differential and difference operators with local and nonlocal conditions on in Banach spaces has not been well studied.
hold. Here .
Clearly, from estimates (3) and (4) it follows that A is a positive operator in . Namely, we have the following.
Here, the structure of fractional spaces generated by the operator A is investigated. The positivity of A in Hölder spaces is studied. The organization of the present paper is as follows. In Section 2, the positivity of A in Hölder spaces is established. In Section 3, the main theorem on the structure of fractional spaces generated by A is investigated. In Section 4, applications on theorems on well-posedness in a Hölder space of parabolic and elliptic problems are presented. Finally, the conclusion is given.
This finishes the proof of Theorem 3. □
Note that from the commutativity of A and its resolvent , and Theorem 3, we have the following theorem.
Suppose β, . Consider the fractional space and the Hölder space . In this section, we prove the following structure theorem.
Theorem 5 The norms of the spaces and are equivalent.
We will estimate , , separately.
This is the end of the proof of Theorem 5. □
and , which guarantees that problem (40) has a smooth solution .
where is independent of φ, ψ, and f.
in a Banach space with a positive operator A defined by (1). Here is the given abstract function defined on with values in E, , are elements of . Therefore, the proof of Theorem 6 is based on Theorem 5 on the structure of the fractional spaces , Theorem 4 on the positivity of the operator A, on the following theorems on coercive stability of elliptic problems, nonlocal boundary value for the abstract elliptic equation and on the structure of the fractional space . This is the end of the proof of Theorem 6. □
where () does not depend on g.
The proof of Theorem 7 uses the techniques introduced in [, Chapter 5] and it is based on estimates (3) and (4).
Theorem 8 ([, Theorem 5.2.48])
The spaces and coincide for any , and their norms are equivalent.
Theorem 9 ([, Theorem 3.1])
holds, where M does not depend on α, φ, ψ, and f.
Here, , , and are given smooth functions and they satisfy every compatibility condition and (41), which guarantees that problem (43) has a smooth solution .
where is independent of f.
The proof of Theorem 10 is based on Theorem 5 on the structure of the fractional spaces , Theorem 4 on the positivity of the operator A, Theorem 7 on coercive stability of the elliptic problem, Theorem 8 on the structure of the fractional space , and the following theorem on coercive stability of the nonlocal boundary value for the abstract elliptic equation.
Theorem 11 ([, Theorem 3.1])
holds, where M does not depend on α and f.
Some of the results of the present article were announced in the conference proceeding  as an extended abstract without proofs. The authors would like to thank Prof. Pavel Sobolevskii (Universidade Federal do Ceará, Brasil). The second author would also like to thank The Scientific and Technological Research Council of Turkey (TUBİTAK) for the financial support.
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