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We study asymptotically regular elliptic equations with-logarithmic growth under the assumptions that the variable exponent satisfies strong log-Hölder continuity and the underlying domain is Reifenberg flat. To ensure a global Calderón–Zygmund estimate for such asymptotically regular problem, we make use of approximating the solutions for asymptotically regular problem by the solutions for regular problem while the gradients of the solutions close to infinity.

complex variables and elliptic equations

Calderón–Zygmund estimate for asymptotically regular elliptic equations with  L                                      p(x)                   -logarithmic growth