3 - Symmetry for a Non-linear Poisson Equation in a Symmetric Set Ω

Summary

The simplest case

In this chapter we prove theorems about positive solutions u of Δu + ƒ(u) = 0 in a bounded symmetric set Ω. Theorem 1.2 (which asserted spherical symmetry and monotonicity of u in a ball) appears as Corollary 3.5, with slightly less smoothness required of ƒ now than was demanded before, and again as Corollary 3.9, (b), for a function ƒ that may be discontinuous. Our main concern is with sets Ω and solutions u that are symmetrical only under reflection in a single, fixed hyperplane; results for this situation imply those for balls, as Lemma 1.8 may have indicated already.

Two definitions set the stage as regards ƒ and Ω.

Definition 3.1 Let W ⊂ ℝ. A function ƒ : W → ℝ is locally Lipschitz continuous iff, for each compact set E ⊂ W, there is a constant A(E) such that

the number A(E) is called a Lipschitz constant of ƒ|_E.

The function ƒ is uniformly Lipschitz continuous iff a single Lipschitz constant A(W) serves for all s, t ∈ W.

For example, if W = (0, ∞) and ƒ(t) = t⁻¹ + |t – 1| – t², then for the compact subset [δ, n] of (0, ∞) we may use the Lipschitz constant

But no single Lipschitz constant will serve for this function and for all s and t in (0, ∞).

Definition 3.2 Recall from Definition 1.7 that x^{μ, κ} denotes the reflection of any point x ∈ ℝ^N in the hyperplane T_μ(κ).