The Maximum Principle pp 181-221 | Cite as

# Applications

Chapter

## Abstract

A Cauchy-Liouville type theorem is a statement that under appropriate circumstances an entire solution (a solution defined over ℝ^{n}) 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 Δ*u* + *B*(*u, Du*) = 0, *x* ∈ ℝ^{n}, (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.

## Keywords

Riemannian Manifold Lipschitz Continuity Exterior Domain Entire Solution Distribution Solution
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## Copyright information

© Birkhäuser Verlag AG 2007