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The objective of this paper is the connection between the problem of optimalregularity among solutions to elliptic divergence equations with measurablecoefficients with the Liouville property at infinity. Initially, we address thetwo-dimensional case by proving an Alt-Caffarelli-Friedman type monotonicityformula for non-negative subsolutions with disjoint supports, which allows toprove optimal regularity and the Liouville property for multiphase problems. Inthe higher-dimensional case we discuss the role of the monotonicity formula inthe characterization of the least growth at infinity and the exponent ofregularity as well. Finally, we establish this connection by combining blow-upand $G$-convergence argument.

Liouville theorems and optimal regularity in elliptic equations