Applications

Abstract

A Cauchy-Liouville type theorem is a statement that under appropriate circumstances an entire solution (a solution defined over ℝⁿ) of an elliptic equation must be constant.¹ For the Laplace equation in particular, it is enough that a solution u should be bounded, or even, at a minimum, that u(x) = o(|x|) as |x| → ∞. For quasilinear equations, and even for semilinear equations of the form Δu + B(u, Du) = 0, x ∈ ℝⁿ, (8.1.1) the same question is more delicate than might at first be expected, since a number of different kinds of behavior can be seen even for relatively simple examples.

Frequently called Liouville theorems in the literature. For a discussion of the relative contributions of Cauchy and Liouville, see reference [101].