Electron. J. Differential Equations, Vol. 2018 (2018), No. 110, pp. 1-24.

Global regularity in Orlicz-Morrey spaces of solutions to nondivergence elliptic equations with VMO coefficients

Vagif S. Guliyev, Aysel A. Ahmadli, Mehriban N. Omarova, Lubomira Softova

Abstract:
We show continuity in generalized Orlicz-Morrey spaces $M_{\Phi,\varphi}(\mathbb{R}^n)$ of sublinear integral operators generated by Calderon-Zygmund operator and their commutators with BMO functions. The obtained estimates are used to study global regularity of the solution of the Dirichlet problem for linear uniformly elliptic operator $\mathcal{L}=\sum_{i,j=1}^n a^{ij}(x)D_{ij}$ with discontinuous coefficients. We show that $\mathcal{L} u\in M_{\Phi,\varphi}$ implies the second-order derivatives belong to $M_{\Phi,\varphi}$.

Submitted September 11, 2017. Published May 10, 2018
Math Subject Classifications: 35J25, 35B40, 42B20, 42B35, 46E30.
Key Words: Generalized Orlicz-Morrey spaces; Calderon-Zygmund integrals; commutators; VMO; elliptic equations; Dirichlet problem.

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Vagif S. Guliyev
Ahi Evran University
Department of Mathematics
40100 Kirsehir, Turkey
email: vagif@guliyev.com
Aysel A. Ahmadli
Dumlupinar University
Department of Mathematics
40100 Kytahya, Turkey
email: aysel.ahmadli@gmail.com
Mehriban N. Omarova
Baku State University
AZ1141 Baku, Azerbaijan
email: mehribanomarova@yahoo.com
Lubomira G. Softova
Department of Mathematics
University of Salerno
Fisciano, Italy
email: lsoftova@unisa.it

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