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 xμ, κ denotes the reflection of any point x ∈ ℝN in the hyperplane Tμ(κ).

A first divergence theorem

Our task in this appendix is to extend the fundamental theorem of the calculus to functions defined on subsets of ℝN. If the result is to have a modest generality and to be of some use, then this task cannot be short and easy, for several reasons. First, it is not obvious how to pass from the merely local description of ∂Ω in Chapter 0, (viii), to the evaluation of integrals over ∂Ω as a whole; Definition D.I and Lemma D.2 are preparations for this step. Second, we must attend to the smoothness both of the function being integrated and of the boundary ∂Ω. Third, conditions that allow a straightforward proof of the divergence theorem (such as those in Theorem D.3) are often too restrictive for applications. Although we take only two primitive steps towards relaxing the conditions in Theorem D.3, those steps require a certain length if they are to be elementary and transparent.

Notation Throughout this appendix, ℝN is to have dimension N ≥ 2. The notation of Cartesian products will be taken beyond the definition of A × B in Chapter 0, (i); for example, (–β, β)N–1 denotes the cube Q′(0,β) described rather fully in Definition D.I, and {a} × [O,a]N–1 denotes a face of the cube (0, a)N, more precisely, the intersection of the hyperplane {x ∈ ℝN | x1 = a} and the closed cube [0,a]N.

During the academic year 1987–8 a group of young mathematicians at the University of Bath prepared (for the first time) a pamphlet, Master of Science in nonlinear mathematics, that contained the following entry.

PG14 Symmetry and the Maximum Principle

Naturally, the pamphlet did not state how this goal was to be reached in twenty lectures to students who could not be assumed to have any experience whatever of partial differential equations. Nor were detailed suggestions issued to me when, in the autumn of 1988, I joined the University of Bath and was ordered to give these lectures. What the authors of the pamphlet did do, however, was to attend the lectures themselves, to ask awkward questions, to imbue the course PG14 with their own youthful verve, and to appeal to my vanity by suggesting that I prepare something like the present book.

This explanation should indicate that the word Introduction in the title of the book is no gloss. I offer genuine apologies to B. Gidas, W.-M. Ni and L. Nirenberg for the extent to which I have used their paper Symmetry and related properties via the maximum principle (1979), to H. Berestycki and L. Nirenberg for my use of the easiest part of On the method of moving planes and the sliding method (1991), and to D. Gilbarg and N.S.