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.
KeywordsRiemannian Manifold Lipschitz Continuity Exterior Domain Entire Solution Distribution Solution
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