Strongly elliptic equations with periodic coefficients in
two-dimensional space

Regularity for strongly elliptic equations with *ϵ*-periodic highly
oscillatory coefficients in two-dimensional space is concerned. In each
*ϵ*-cell, the diffusion coefficients of the elliptic equations are
ω 2 ∈ ( 1 , ∞ ) in a small disk with radius ϵµ 4 ( < 1 4 ) and
1 outside the disk of the cell. Two cases are considered. Case one is
that *ϵ,µ,ω* are independent in the elliptic equations. So the
diffusion coefficients of the elliptic equations are *ϵ*-periodic
and discontinuous. L p -gradient estimate uniform in *ϵ,µ,ω* for
the elliptic solutions is derived. However, the integrability *p* (
*>*2) of the solutions is not a large number. Case two
is that ϵ , µ ( = ω − 1 ) are independent in the elliptic equations. The
diffusion coefficients of the elliptic equations are *ϵ*-periodic,
discontinuous, and L 1 -bounded. Lipschitz estimate uniform in ϵ , µ ( =
ω − 1 ) for the elliptic solutions is obtained.