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.
Keywords: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 .
Danelich in  considered the positivity of a difference analog of the 2mth-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.
In the present paper, we study the structure of fractional spaces generated by the two-dimensional differential operator
Following the paper , passing limit when in the special case and , we get that there exists the inverse operator for all and the following formula
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.
Then from that it follows
for . Now, we will estimate the right-hand side of inequality (6). Let us consider two cases and separately. First, we consider the case . Using the triangle inequality, estimate (4), the definition of -norm, Hilbert’s inequality, and the Lagrange theorem, it follows that for some between , , and between , ,
Estimates (7) and (8) yield that
Combining estimates (5) and (9), we obtain
This finishes the proof of Theorem 3. □
Using equation (10), the triangle inequality, the following inequalities
Thus, it follows from estimate (13) that
Now, we will estimate the right-hand side of equation (15). We consider two cases and , respectively. Let us first assume that . Furthermore, this situation will be considered in two cases: and . Let . From equation (15), the triangle inequality, the definition of -norm, the assumptions and , it follows that
Combining estimates (17)-(19), we get
Using estimates (3), (4), (11), (12), we obtain
By inequalities (11), (12), estimates (4), (24), (25), and Hilbert’s inequality, we have
Similarly, we get
From estimates (29) and (31) it follows that
Combining estimates (14) and (32), we obtain
Now, we will estimate the right-hand side of equation (36). We consider two cases and . Let us first assume that . By equation (36), the triangle inequality, the Lagrange theorem, the definition of -norm, and the assumption , we have, for some between , , and between , , that
Combining estimates (37) and (38), we get
Estimates (35) and (39) yield that
This is the end of the proof of Theorem 5. □
In this section, we consider some applications of Theorem 5. First, we consider the boundary value problem for the elliptic equation
Theorem 6For the solution of boundary value problem (40), we have the following estimate:
Note that problem (40) can be written in the form of the abstract boundary value problem
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. □
Theorem 7Under assumption (41) for the solution of elliptic problem
the following coercive inequality holds:
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])
Theorem 9 ([, Theorem 3.1])
holds, whereMdoes not depend onα, φ, ψ, andf.
Second, we consider the nonlocal boundary value problem for the elliptic equation under assumption (41)
Theorem 10For the solution of initial boundary value problem (43), we have the following estimate:
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])
in a Banach space E with a positive operatorA, the coercive inequality
holds, whereMdoes not depend onαandf.
In the present article, the structure of the fractional spaces generated by the two-dimensional elliptic differential operator A is investigated. The positivity of this operator A in a Hölder space is established. Of course, the Banach fixed point theorem and the method of the present paper enable us to establish the existence and uniqueness results which hold under some sufficient conditions on the nonlinear term for the solution of the mixed problem
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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.
Alibekov, KhA: Investigations in C and Lp of difference schemes of high order accuracy for approximate solutions of multidimensional parabolic boundary value problems. Dissertation. Voronezh State University, Voronezh (1978)
Stewart, HB: Generation of analytic semigroups by strongly elliptic operators under general boundary conditions. Trans. Am. Math. Soc.. 259, 299–310 (1980). Publisher Full Text
Ashyralyev, A: Fractional spaces generated by the positive differential and difference operators in a Banach space. In: Tas K, Tenreiro Machado JA, Baleanu D (eds.) Mathematical Methods in Engineering, Springer, Dordrecht (2007)
Ashyralyev, A: Method of positive operators of investigations of the high order of accuracy difference schemes for parabolic and elliptic equations. Dissertation, Inst. of Math. of Acad. Sci. Kiev (1992)
Ashyralyev, A, Yaz, N: On structure of fractional spaces generated by positive operators with the nonlocal boundary value conditions. In: Agarwal RP (ed.) Proceedings of the Conference Differential and Difference Equations and Applications, Hindawi Publishing Corporation, New York (2006)
Bazarov, MA: On the structure of fractional spaces. Proceedings of the XXVII All-Union Scientific Student Conference ‘The Student and Scientific-Technological Progress’, pp. 3–7. Novosibirsk. Gos. Univ., Novosibirsk (1989) (in Russian)
Ashyralyev, A, Nalbant, N, Sozen, Y: Structure of fractional spaces generated by second order difference operators. J. Franklin Inst. (2013). Publisher Full Text
Ashyralyev, A: On well-posedness of the nonlocal boundary value problems for elliptic equations. Numer. Funct. Anal. Optim.. 24, 1–15 (2003). Publisher Full Text