Strongly elliptic equations with periodic coefficients in two-dimensional space
Year: 2024
Type: article
Abstract: Regularity for strongly elliptic equations with highly oscillatory \varepsilon -periodic coefficients in two-dimensional space is addressed. In each \varepsilon -cell, the diffusion coefficients of the elliptic equations are \omega^{2}\in(1,\infty) in a small disk with radius \frac{\varepsilon\mu}{4}\,(<\frac{1}{4}) and 1 outside the disk of the cell. Two cases are considered. Case one is that \varepsilon , \mu , \omega are independent in the elliptic equations. So, the diffusion coefficients of the elliptic equations are \varepsilon -periodic and discontinuous. L^{p} -gradient estimate uniformly in \varepsilon , \mu , \omega for the elliptic solutions is derived. However, the integrability p\,(>2) of the solutions is not a large number. Case two is that \varepsilon , \mu\, (=\omega^{-1}) are independent in the elliptic equations. The diffusion coefficients of the elliptic equations are \varepsilon -periodic, discontinuous, and L^{1} -bounded. Lipschitz estimate uniformly in \varepsilon , \mu\,(=\omega^{-1}) for the elliptic solutions is obtained.
DOI: 10.4171/zaa/1785
Author: Li-Ming Yeh
Institution: National Yang Ming Chiao Tung University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
Funder: National Science Council
OpenAlex ID: W4405655477