Some Liouville theorems and applications

We give exposition of a Liouville theorem established in [6] which is a novel extension of the classical Liouville theorem for harmonic functions. To illustrate some ideas of the proof of the Liouville theorem, we present a new proof of the classical Liouville theorem for harmonic functions. Applications of the Liouville theorem, as well as that of earlier ones in [5], can be found in [6, 7] and [9]. The Laplacian operator ∆ is invariant under rigid motions: For any function u on R and for any rigid motion T : R → R, ∆(u ◦ T ) = (∆u) ◦ T. Partially supported by NSF grant DMS-0401118. To appear in a volume in honor of Haim Brezis sixtieth birthday.