Abstract
We consider the twodimensional differential operator defined on functions on the halfplane 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 wellposedness in a Hölder space of elliptic problems are obtained.
MSC: 35J25, 47E05, 34B27.
Keywords:
positive operator; fractional spaces; Green’s function; Hölder spaces1 Introduction
It is well known that (see, for example, [13] and the references therein) various classical and nonclassical 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, [47] and the references therein). Several authors have investigated the positivity of a wider class of differential and difference operators in Banach spaces (see [818] 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.
The operator A is said to be positive in E if its spectrum lies inside of the sector S of the angle ϕ, , symmetric with respect to the real axis, and the following estimate (see, for example, [6,19])
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 .
With the help of the positive operator A, we introduce the fractional space (), consisting of all elements for which the norm
is finite.
Theorem 1[20]
Let, , and letbe any two nonnegative integrable functions such thatand. Then the following Hilbert’s inequality holds:
Danelich in [12] considered the positivity of a difference analog of the 2mthorder multidimensional elliptic operator with dependent coefficients on semispaces .
The structure of fractional spaces generated by positive multidimensional differential and difference operators on the space in Banach spaces has been well investigated (see [2123] and the references therein).
In papers [19,2427] the structure of fractional spaces generated by positive onedimensional differential and difference operators in Banach spaces was studied. Note that the structure of fractional spaces generated by positive multidimensional 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 twodimensional differential operator
defined over the region with the boundary condition , . Here, the coefficients , , are continuously differentiable and satisfy the uniform ellipticity
Following the paper [12], passing limit when in the special case and , we get that there exists the inverse operator for all and the following formula
holds, where is the Green function of differential operator (1). Moreover, the following estimates
and
Next, to formulate our result, we need to introduce the Hölder space of all continuous bounded functions φ defined on satisfying a Hölder condition with the indicator with the norm
Here, denotes the Banach space of all continuous bounded functions φ defined on with the norm
Clearly, from estimates (3) and (4) it follows that A is a positive operator in . Namely, we have the following.
Theorem 2Let. Then the following estimate
is valid.
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 wellposedness in a Hölder space of parabolic and elliptic problems are presented. Finally, the conclusion is given.
2 Positivity of A in Hölder spaces
Theorem 3Let. For, we have the following estimate:
Proof Applying formula (2), the triangle inequality, the definition of norm, estimate (3), and Hilbert’s inequality, we get
Then from that it follows
Without loss of generality, we can put . Using formula (2) and the triangle inequality, we get
for . Now, we will estimate the righthand 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 , ,
Second, we consider the case . Using formula (2), the triangle inequality, estimate (3), the definition of norm, and estimate (6), we get
Estimates (7) and (8) yield that
Combining estimates (5) and (9), we obtain
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.
Theorem 4Let. Then the following estimate holds:
3 The structure of fractional spaces
Suppose β, . Consider the fractional space and the Hölder space . In this section, we prove the following structure theorem.
Theorem 5The norms of the spacesandare equivalent.
Proof Assume that . Let and be fixed. From formula (2) it follows that
Using equation (10), the triangle inequality, the following inequalities
estimates (10), (11), (12), and the definition of norm, we obtain
Thus, it follows from estimate (13) that
Let and be fixed. Using equation (10), we can write
Now, we will estimate the righthand 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
We will estimate , , separately.
First, let us estimate . Clearly, by the assumption , we have
From estimates (3), (11), (12), and the assumption , it follows that
Estimates (3), (11), (12), and the assumption yield that
Combining estimates (17)(19), we get
Now, let us consider the case . Then, using equation (10), we can write
From equation (21), the triangle inequality, the Lagrange theorem, the definition of norm, and the assumption , it follows that for some between , , and between , ,
We will estimate , , separately. Let us start with . Clearly, we have
Using estimates (3), (4), (11), (12), we obtain
Note that by the triangle inequality and the fact that between , , we get
By inequalities (11), (12), estimates (4), (24), (25), and Hilbert’s inequality, we have
Similarly, we get
Combining estimates (22), (23), (26), (27), we obtain that for , ,
It follows from estimates (20) and (28) that for , ,
Next, let us assume that . By equation (10), we have
Equation (30), estimates (3), (11), Hilbert’s inequality, the triangle inequality, the definition of norm, and the assumption yield that for ,
From estimates (29) and (31) it follows that
Combining estimates (14) and (32), we obtain
Now, we will prove that . By Theorem 4, A is a positive operator in the Banach space . Hence, for , we have
Let . It follows from formula (2) and equation (33) that
Using the triangle inequality, equation (34), estimate (3), and the definition of norm, we obtain
Thus,
Let be fixed. From equation (34) it follows that
Now, we will estimate the righthand 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
We will estimate , , separately. Using the triangle inequality, estimates (4), (11), (25), (24), (12), the Lagrange theorem, and the assumption , we obtain
From the triangle inequality, estimates (4), (11), (12), (24), (25), the Lagrange theorem, and the assumption , it follows that
Using the triangle inequality, estimates (3), (24), (25), (12), (11), the assumption , and the following estimate
we obtain
Next, let us assume that . Using formula (36), estimate (3), the triangle inequality, Hilbert’s inequality, and the assumption , we get
Combining estimates (37) and (38), we get
Estimates (35) and (39) yield that
This is the end of the proof of Theorem 5. □
4 Applications
In this section, we consider some applications of Theorem 5. First, we consider the boundary value problem for the elliptic equation
Here, , , , , and are given smooth functions and they satisfy every compatibility condition and
and , which guarantees that problem (40) has a smooth solution .
Theorem 6For the solution of boundary value problem (40), we have the following estimate:
whereis independent ofφ, ψ, andf.
Proof We introduce the Banach space of all continuous abstract functions defined on with values in E, equipped with the norm
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 [[5], Chapter 5] and it is based on estimates (3) and (4).
Theorem 8 ([[5], Theorem 5.2.48])
The spacesandcoincide for any, and their norms are equivalent.
Theorem 9 ([[28], Theorem 3.1])
LetAbe a positive operator in a Banach spaceEand (). Then, for the solution of nonlocal boundary value problem (42), the coercive inequality
holds, whereMdoes not depend onα, φ, ψ, andf.
Second, we consider the nonlocal boundary value problem for the elliptic equation under assumption (41)
Here, , , and are given smooth functions and they satisfy every compatibility condition and (41), which guarantees that problem (43) has a smooth solution .
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 ([[28], Theorem 3.1])
LetAbe a positive operator in a Banach spaceEand (). Then, for the solution of nonlocal boundary value problem (40)
in a Banach space E with a positive operatorA, the coercive inequality
holds, whereMdoes not depend onαandf.
5 Conclusion
In the present article, the structure of the fractional spaces generated by the twodimensional 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
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
Some of the results of the present article were announced in the conference proceeding [11] 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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