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The Structure of Fractional Spaces Generated by a Two-dimensional Difference Operator in a Half Plane

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Ukrainian Mathematical Journal Aims and scope

We consider a difference-operator approximation \( {A}_h^x \) of the differential operator

$$ {A}^xu(x)=-{a}_{11}(x){u}_{x_1{x}_1}(x)-{a}_{22}(x){u}_{x_2{x}_2}(x)+\sigma u(x),\kern1em x=\left({x}_1,{x}_2\right), $$

defined in the region ℝ+ × ℝ with the boundary condition

$$ u\left(0,{x}_2\right)=0,\kern1em {x}_2\in \mathbb{R}. $$

Here, the coefficients aii(x), i = 1, 2, are continuously differentiable, satisfy the condition of uniform ellipticity \( {a}_{11}^2(x)+{a}_{22}^2(x)\ge \delta >0 \), and σ > 0. We study the structure of the fractional spaces generated by the analyzed difference operator. The theorems on well-posedness of difference elliptic problems in a Hölder space are obtained as applications.

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Authors and Affiliations

  1. Near East University, North Nicosia, Mersin, TRNC; Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

    A. Ashyralyev

  2. Yakuplu the Neighborhood Street Kubilay, Istanbul, Turkey

    S. Akturk

Authors
  1. A. Ashyralyev
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  2. S. Akturk
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 8, pp. 1019–1032, August, 2018.

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Ashyralyev, A., Akturk, S. The Structure of Fractional Spaces Generated by a Two-dimensional Difference Operator in a Half Plane. Ukr Math J 70, 1176–1191 (2019). https://doi.org/10.1007/s11253-018-1561-5

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  • DOI: https://doi.org/10.1007/s11253-018-1561-5

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