A type of homogenization problem. (Q1865943)

The question studied in this paper is a homogenization problem, and it can be viewed as an analogous of the usual \(\Gamma\)-convergence type problem onto curved targets. Due to the constraint in the target, the proof techniques use the harmonic maps to construct comparison functions in proving the homogenization limit. Regularity result of minimizer is proved for fixed \(\varepsilon\). Similar estimates are proved on the size of the singular set as for minimizing harmonic maps. Moreover the homogenization limit theorem and uniform apriori estimates are proved independent of \(\varepsilon\). An application of uniform estimates is presented to obtain a uniform bound on the number of singularities in a special case.