Positivity of TwoDimensional Elliptic Differential ...

Positivity of TwoDimensional Elliptic Differential Operators with Nonlocal Conditions Allaberen Ashyralyev, Sema Kaplan, and Yasar Sozen Citation: AIP Conf. Proc. 1389, 605 (2011); doi: 10.1063/1.3636803 View online: http://dx.doi.org/10.1063/1.3636803 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1389&Issue=1 Published by the American Institute of Physics.

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Positivity of Two-Dimensional Elliptic Differential Operators with Nonlocal Conditions Allaberen Ashyralyev∗,† , Sema Kaplan∗∗ and Yasar Sozen‡ ∗

Department of Mathematics Fatih University, 34500, Istanbul, Turkey, [email protected] † Department of Mathematics, ITTU, Ashgabat, Turkmenistan, [email protected] ∗∗ Department of Mathematics Fatih University, 34500, Istanbul, Turkey, [email protected] ‡ Department of Mathematics Fatih University, 34500, Istanbul, Turkey, [email protected] Abstract. In the present paper, the differential operator A defined by Au = −utt (t, x) − uxx (t, x) + u(t, x) with domain D(A) = {u(t, x) : utt , uxx , u ∈ C ([0, 1] × R) , u(0, x) = u(1, x), ut (0, x) = ut (1, x),t ∈ [0, 1], x ∈ R} is considered. The Green function for A is constructed. The estimates for the Green function are obtained and the positivity of A in the Banach space C ([0, 1] × R) is established. The structure of fractional spaces generated by A is investigated, the equivalence of the norm of these fractional spaces and Hölder spaces is established. The positivity of A in the Hölder space C2β ([0, 1] × R) , 0 < β < 1/2 is obtained. Keywords: Positive operator, Difference operators, Fractional spaces, Nonlocal boundary conditions, Green’s function PACS: 87.10.Ed

INTRODUCTION Various local and nonlocal boundary value problems for partial differential can be considered as an abstract boundary value problem for ordinary differential equation in a Banach space E with a densely defined unbounded operator A. As is well-known that the study of the various properties of partial differential equations is based on the positivity property of the differential in a Banach space [1]-[3]. Many researcher have been studied the positivity of wider class of differential operators [4]-[7]. In the study of positive operators, the important progress has been made from the viewpoint of the stability analysis of higher order accuracy difference schemes for partial differential equations. As is well known difference analogue of maximum principle and energy method are the most useful methods for stability analysis of difference schemes. When the maximum principle and energy method can not be used, the theory of positive difference operators allows us to investigate the stability and coercive stability of difference schemes in various norms for partial differential equations. In a number of works (see, e.g. [8]-[23], and the references therein) difference schemes were treated as operator equations in a Banach space and the investigation was based on the positivity property of the operator coefficient. The investigation of well-posedness of various types of parabolic and elliptic differential and difference equations is based on the positivity of elliptic differential and difference operators in various Banach spaces and on the structure of the fractional spaces generated by these positive operators. An excellent survey of works in the theory of fractional spaces generated by positive multidimensional difference operators in the space and its applications to partial differential equations was given in [8, 9, 14]. Let E be a Banach space and A : D(A) ⊂ E → E be a linear unbounded operator densely defined in E. We call A strongly positive in the Banach space E, if its spectrum σA lies in the interior of the sector of angle φ , 0 < 2φ < π , symmetric with respect to the real axis, and if on the edges of this sector, S1 (φ ) = {ρ eiφ : 0 ≤ ρ ≤ ∞ } and S2 (φ ) = {ρ e−iφ : 0 ≤ ρ ≤ ∞}, and outside of the sector the resolvent (λ − A)−1 is subject to the bound (A + λ )−1 ≤ E→E

M . 1 + |λ |

Numerical Analysis and Applied Mathematics ICNAAM 2011 AIP Conf. Proc. 1389, 605-608 (2011); doi: 10.1063/1.3636803 © 2011 American Institute of Physics 978-0-7354-0956-9/$30.00

605 Downloaded 29 Nov 2012 to 193.255.106.240. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions

(1)

The infimum of such angles is called spectral angle ϕ (A, E) of A. Throughout the present paper, M denotes positive constants, which may differ in time and thus is not a subject of precision. However, we will use M(α , β , · · · ) to stress the fact that the constant depends only on M(α , β , · · · ). For a positive operator A in the Banach space E, let us introduce the fractional spaces Eβ = Eβ (E, A) (0 < β < 1) consisting of those v ∈ E for which the norm vEβ = sup λ β A(λ + A)−1 vE + vE λ >0

is finite. In this paper, we consider the second order differential operator Au(t, x) = −utt − uxx + u

(2)

with domain D(A) = {u; utt , uxx , u ∈ C ([0, 1] × R) , u(0, x) = u(1, x), ut (0, x) = ut (1, x), x ∈ R}. Let us introduce the Banach space Cγ ([0, 1] × R) , γ ∈ (0, 1) of all continuous functions ϕ (t, x) defined on [0, 1] × R and satisfying a Hölder condition for which the following norm is finite ϕ Cγ ([0,1]×R) = ϕ C([0,1]×R) +

|ϕ (t1 , x1 ) − ϕ (t2 , x2 )| γ, (t1 ,x1 )=(t2 ,x2 ) (|t1 − t2 | + |x1 − x2 |) sup

where C ([0, 1] × R) is the Banach space of all continuous functions ϕ (t, x) defined on [0, 1] × R with the norm ϕ C([0,1]×R) = max sup |ϕ (t, x)|. t∈[0,1] x∈R

In the present article, the resolvent of the operator −A, i.e. Au + λ u = ϕ (t, x)

(3)

⎧ ⎨ −utt − uxx + (1 + λ )u = ϕ (t, x), 0 < t < 1, −∞ < x < ∞,

or

⎩ u(0, x) = u(1, x), u (0, x) = u (1, x), −∞ < x < ∞. t t

(4)

will be investigated. The Green function of A is constructed. The positivity of the operator A in the Banach space C ([0, 1] × R) is established. It is proved that for any β ∈ (0, 1/2) the norms in the spaces Eβ (E, A) and C2β ([0, 1] × R) are equivalent. The positivity of A in the Hölder spaces of C2β ([0, 1] × R) , β ∈ (0, 1/2) is proved.

THE GREEN FUNCTION OF A AND POSITIVITY OF A IN C Lemma 1. For λ > 0, equation (4) is uniquely solvable and the following formula holds: u(t, x) = (A + λ )−1 ϕ (t, x) =

where G(t, x) =

1 2π

R

eixσ

[0,1]×R

G(t − p, x − z)ϕ (p, z)d pdz,

√ √ 2 2 e−|t| σ +λ +1 + e−(1−|t|) σ +λ +1 dσ . √ √ 2 2 σ 2 + λ + 1 1 − e− σ +λ +1

(5)

(6)

Theorem 2. For λ > 0, we have the following estimate: (A + λ )−1 ≤ C([0,1]×R)→C([0,1]×R)

M . 1 + |λ |


(7)

THE STRUCTURE OF FRACTIONAL SPACES GENERATED BY A AND POSITIVITY OF A IN HÖLDER SPACES Clearly, the operator commutes A and its resolvent (A + λ )−1 . By the definition of the norm in the fractional space Eβ = Eβ (C([0, 1] × R), A), we get (A + λ )−1 Eβ →Eβ ≤ (A + λ )−1 C([0,1]×R)→C([0,1]×R) . Thus, from Theorem 2 it follows that A is a positive operator in the fractional spaces Eβ . Moreover, we have the following result Theorem 3. For β ∈ (0, 1/2), the norms of the spaces Eβ and the Hölder space C2β ([0, 1] × R) are equivalent. The proof of Theorem 3 is based on the formulas A(A + λ )−1 f (t, x) = f (t, x) =

1 f (t, x) + 1+λ ∞ 1 −∞ 0

∞ 1 −∞ 0

G(t − p, x − z) ( f (t, x) − f (p, z)) d pdz,

G(t − p, x − z, λ + t)A(λ + t + A)−1 d pdz.

for the positive operator A and on the estimates for the Green function of the resolvent equation (3). By the results of Theorem 2 and Theorem 3, we have the positivity of A in the Hölder space C2β ([0, 1] × R). Namely, Theorem 4. For λ > 0, we have the following estimate: (A + λ )−1

C2β ([0,1]×R)→C2β ([0,1]×R)

≤

M 1 . β (1 − 2β ) (1 + |λ |)

(8)

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