On the Optimal Growth of Functions with Bounded Laplacian

Abstract

Using a compactness argument, we introduce a Phragmen Lindelof type theorem for functions with bounded Laplacian. The technique is very useful in studying unbounded free boundary problems near the infinity point and also in approximating integrable harmonic functions by those that decrease rapidly at infinity. The method is flexible in the sense that it can be applied to any operator which admits the standard elliptic estimate.