2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 2002, pp 1–6. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

A note on the moving hyperplane method ^{∗}

Céline Azizieh & Luc Lemaire

Abstract

We make precise the domain regularity needed for having the monotonicity and symmetry results recently proved by Damascelli and Pacella on p-Laplace equations. For this purpose, we study the continuity and semicontinuity of some parameters linked with the moving hyperplane method.

Résumé. Dans [1], Ph. Clément et le premier auteur ont établi par des méthodes de continuation des résultats d’existence pour des problèmes du type −Δ_{p}u = f(u) dans Ω, u = 0 sur ∂Ω, u > 0 sur Ω, où 1 < p ≤ 2, Ω ⊂ ℝ^{N} est un domaine borné convexe et f : ℝ → [0 + ∞) est continue. La preuve de ces théorèmes fait appel aux récents résultats de monotonie et de symétrie établis par Damascelli et Pacella dans [3], résultats dont la démonstration nécessitait la continuité ou semi-continuité de certains paramètres géométriques liés à la méthode des moving hyperplanes. Notre but est ici de préciser les hypoth‘eses de régularité et de convexité du domaine Ω qui sont nécessaires pour satisfaire les différentes conditions de continuité des paramètres en question.

1 Results

Let us consider the problem

−Δ_{p}u = f(u) in Ω,
u = 0 on ∂Ω,
u ∈ C^{1}(Ω), u > 0 in Ω

(1.1)

where 1 < p ≤ 2, Ω ⊂ ℝ^{N} is a bounded convex domain, Δ_{p} is the p-laplacian operator defined by Δ_{p}u = div(|∇u|^{p−2}∇u) and f : ℝ → [0, +∞) is continuous on ℝ, locally Lipschitz continuous on (0, +∞) and satisfies

∃C_{0}, C_{1} > 0 such that C_{0}|u|^{q} ≤ f(u) ≤ C_{1}|u|^{q} ∀u ∈ ℝ^{+}

∗Mathematics Subject Classifications: 35J60, 35B99.

Key words: p-Laplacian, symmetry of solutions, nonlinear elliptic equations.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published October 21, 2002.

The second author is supported by an Action de Recherche Concertée de la Communauté Française de Belgique

where q > p−1. In [1], Ph. Clément and the first author proved the existence of a nontrivial positive solution to (1.1) by using continuation methods and establishing a priori estimates for the solutions of some nonlinear eigenvalue problem associated with (1.1). The desired a priori estimates use a blow up argument as well as some monotonicity and symmetry results proved by Damascelli and Pacella in [4] and generalizing to the p-laplacian operator with 1 < p < 2 the well known results of Gidas–Ni–Nirenberg from [5] and Berestycki–Nirenberg in [2]. In their proof, Damascelli and Pacella use a new technique consisting in moving hyperplanes orthogonal to directions close to a fixed one. To be efficient, this procedure needs some continuity of some parameters linked with the moving plane method (see the functions λ_{1}(ν) and a(ν) defined below). Therefore they assume in their result that ∂Ω is smooth to insure this continuity (and only for that reason). However, such a smoothness hypothesis does not appear in the case p = 2 in the classical moving plane procedure (see [2]).

Our purpose here is to give more precision on the regularity of the domain Ω that is needed to have the continuity of the function a(ν) and the lower semicontinuity of λ_{1}(ν), and so to have the monotonicity and symmetry results of [4]. This question is also important concerning the existence result from [1]. Specifically, we ask that the domain be of class C^{1}, and we also discuss convexity conditions relating to the continuity of λ_{1}(ν).

Remark that some symmetry results for solutions of elliptic partial differential equations have also been obtained by Brock by using the continuous Steiner symmetrization (cf. [3]).

In this paper, Ω will denote an open bounded domain in ℝ^{N} with C^{1} boundary. We will say that Ω is strictly convex if for all x, y ∈ Ω and for all t ∈ (0, 1), (1 − t)x + ty ∈ Ω. For any direction ν ∈ ℝ^{N}, |ν| = 1, we define

a(ν) := inf_{x∈Ω} x.ν

and for all λ ≥ a(ν),

Ω_{λ}^{ν} := {x ∈ Ω | x.ν < λ},
T_{λ}^{ν} := {x ∈ Ω | x.ν = λ} (≠ ∅ for a(ν) < λ < −a(−ν)).

Let us denote by R_{λ}^{ν} the symmetry with respect to the hyperplane T_{λ}^{ν} and

x_{λ}^{ν} := R_{λ}^{ν}(x) ∀x ∈ ℝ^{N},
(Ω_{λ}^{ν})′ := R_{λ}^{ν}(Ω_{λ}^{ν}),
Λ_{1}(ν) := {μ > a(ν) | ∀λ ∈ (a(ν), μ), we have (1.2) and (1.3)},
λ_{1}(ν) := sup Λ_{1}(ν)

where (1.2), (1.3) are the following conditions:

(Ω_{λ}^{ν})′ is not internally tangent to ∂Ω at some point p ∉ T_{λ}^{ν}  (1.2)
for all x ∈ ∂Ω ∩ T_{λ}^{ν}, ν(x).ν ≠ 0,  (1.3)

where ν(x) denotes the inward unit normal to ∂Ω at x. Notice that Λ_{1}(ν) ≠ ∅ and λ_{1}(ν) < ∞ since for λ > a(ν) close to a(ν), (1.2) and (1.3) are satisfied and Ω is bounded.

Figure 1: Illustration of the notations

Propositions 1 and 2 below give sufficient conditions on Ω to guarantee the continuity of the functions a(ν) and λ_{1}(ν), as well as the lower semicontinuity of λ_{1}(ν).

Proposition 1 Let Ω be a bounded domain with C^{1} boundary. Then the function a(ν) is continuous with respect to ν ∈ S^{N−1}.

Proposition 2 Let Ω ⊂ ℝ^{N} be a bounded domain with C^{1} boundary. Then the function λ_{1}(ν) is lower semicontinuous with respect to ν ∈ S^{N−1}. If moreover Ω is strictly convex, then λ_{1}(ν) is continuous.

As a consequence of these results, we can give more precision on the conditions to impose to Ω in the monotonicity result of [4]. This result becomes:

Theorem 1.1 in Damascelli-Pacella [4] Let Ω be a bounded domain in ℝ^{N} with C^{1} boundary, N ≥ 2 and g : ℝ → ℝ be a locally Lipschitz continuous function. Let u ∈ C^{1}(Ω̄) be a weak solution of

−Δ_{p}u = g(u) inΩ
u > 0 inΩ,
u = 0 on ∂Ω

where 1 < p < 2. Then, for any direction ν ∈ ℝ^{N} and for λ in the interval (a(ν), λ_{1}(ν)], we have u(x) ≤ u (x_{λ}^{ν}) for all x ∈ Ω_{λ}^{ν}. Moreover \frac{∂u}{∂ν}(x) > 0 for all x ∈ Ω_{λ_{1}(ν)}^{ν}\Z where Z = {x ∈ Ω | ∇u(x) = 0}.

Below we prove Propositions 1 and 2 and we give a counterexample of a C^{∞} convex but not strictly convex domain for which λ_{1}(ν) is not continuous everywhere.

Proof of Proposition 1: Let us fix a direction ν ∈ S^{N−1}. We shall prove that for all sequence ν_{n} → ν with |ν_{n}| = 1, there exists a subsequence still denoted by ν_{n} such that a(ν_{n}) → a(ν). Since Ω is bounded, (a(ν_{n})) is also bounded, so passing to an adequate subsequence, there exists ā ∈ ℝ such that a(ν_{n}) → ā. We will show that ā = a(ν). Suppose by contradiction that ā ≠ a(ν). Then either ā < a(ν) or ā > a(ν). Case 1: ā < a(ν): Since

a(ν) = inf_{x∈Ω} x.ν = min_{x∈Ω} x.ν = min_{x∈∂Ω} x.ν,

there exists x_{n} ∈ ∂Ω such that

x_{n}.ν_{n} = a(ν_{n}).  (1.4)

Passing again to a subsequence, there exists x ∈ ∂Ω such that x_{n} → x and taking the limit of (1.4), we get x.ν = ā < a(ν), a contradiction with the definition of a(ν). Case 2: ā > a(ν): There exists x ∈ ∂Ω with x.ν = a(ν). For n large, |x.ν_{n} − x.ν| = |x.ν_{n} − a(ν)| is small, and since a(ν_{n}) → ā > a(ν), for n large enough we have x.ν_{n} < a(ν_{n}), contradicting the definition of a(ν_{n}). □

Proof of Proposition 2: We first prove the continuity of λ_{1}(ν) if Ω is strictly convex. Suppose by contradiction that there exists ν ∈ S^{N−1} such that λ_{1} is not continuous at ν. Then we can fix ϵ > 0 and a sequence (ν_{n}) ⊂ S^{N−1} such that ν_{n} → ν and |λ_{1}(ν) − λ_{1}(ν_{n})| > ϵ for all n ∈ ℕ. Passing to a subsequence still denoted by (ν_{n}), we can suppose that

either λ_{1}(ν) > λ_{1}(ν_{n}) + ϵ ∀n ∈ ℕ or λ_{1}(ν) < λ_{1}(ν_{n}) − ϵ ∀n ∈ ℕ.

Case 1: λ_{1}(ν) > λ_{1}(ν_{n}) + ϵ for all n ∈ ℕ. For any fixed n ∈ ℕ, we have the following alternative: either there exists x_{n} ∈ T_{λ_{1}(ν_{n})}^{ν_{n}} ∩ ∂Ω with ν(x_{n}).ν_{n} = 0,

or there exists x_{n} ∈ (∂Ω ∩ \overline{Ω_{λ_{1}(ν_{n})}^{ν_{n}}}) \ T_{λ_{1}(ν_{n})}^{ν_{n}} with (x_{n})_{λ_{1}(ν_{n})}^{ν_{n}} ∈ ∂Ω. Passing once again to subsequences, we can suppose that we are in one of the two situations above for all n ∈ ℕ. We treat below each situation and try to reach a contradiction. (1.a) For all n ∈ ℕ, there exists x_{n} ∈ T_{λ_{1}(ν_{n})}^{ν_{n}} ∩ ∂Ω with ν(x_{n}).ν_{n} = 0. Passing if necessary to a subsequence, there exist λ̄ ≤ λ_{1}(ν)−ϵ and x ∈ T ν ; λ̄ ∩∂Ω such that x_{n} → x and ν(x).ν = 0. This contradicts the definition of λ_{1}(ν). (1.b) For all n ∈ ℕ, there exists x_{n} ∈ (∂Ω ∩ \overline{Ω_{λ_{1}(ν_{n})}^{ν_{n}}}) \ T_{λ_{1}(ν_{n})}^{ν_{n}} with (x_{n})_{λ_{1}(ν_{n})}^{ν_{n}} ∈ ∂Ω. Passing if necessary to a subsequence, there exist λ̄ ≤ λ_{1}(ν)−ϵ and x ∈ ∂Ω∩\overline{Ων ; λ̄} such that x_{n} → x and xν ; λ̄ ∈ ∂Ω. If x ∉ T ν ; λ̄, we reach a contradiction with the definition of λ_{1}(ν). Suppose now that x ∈ T ν ; λ̄. Let us denote (x_{n})_{λ_{1}(ν_{n})}^{ν_{n}} by u_{n}. Since Ω is a C^{1} domain, it holds that ν(u_{n}).ν_{n} ≤ 0 for all n. By definition of λ_{1}(ν_{n}), ν(x_{n}).ν_{n} ≥ 0. If x ∈ T ν ; λ̄, x = lim x_{n} = lim u_{n} and so ν(x).ν = 0, which contradicts the definition of λ_{1}(ν). Observe that we do not use the convexity of the domain in Case 1. Case 2: λ_{1}(ν) < λ_{1}(ν_{n}) − ϵ for all n ∈ ℕ: As in the first case, either there exists x ∈ T_{λ_{1}(ν)}^{ν} ∩ ∂Ω with ν(x).ν = 0 or there exists x ∈ (∂Ω ∩ \overline{Ω_{λ_{1}(ν)}^{ν}}) \ T_{λ_{1}(ν)}^{ν} such that x_{λ_{1}(ν)}^{ν} ∈ ∂Ω. We treat the first situation in (2.a) and the second one in (2.b). (2.a) For ϵ small enough, T_{λ_{1}(ν)+ \frac{ϵ}{2}}^{ν} ∩ ∂Ω ≠ ∅. Since Ω is strictly convex, there exists x′ ∈ T_{λ_{1}(ν)+ \frac{ϵ}{2}}^{ν} ∩ ∂Ω such that

ν(x′).ν < 0.  (1.5)

For ϵ > 0 small enough, there exists n_{0} ∈ ℕ such that for all n ≥ n_{0}, the sets T_{λ_{1}(ν)+ \frac{ϵ}{2}}^{ν_{n}} ∩ ∂Ω are non empty and since they are compact, we can choose a sequence (x_{n}) satisfying

x_{n} ∈ T_{λ_{1}(ν)+ \frac{ϵ}{2}}^{ν_{n}} ∩ ∂Ω, |x′ − x_{n}| = min \{|x′ − y| : y ∈ T_{λ_{1}(ν)+ \frac{ϵ}{2}}^{ν_{n}} ∩ ∂Ω\} .

Passing if necessary to a subsequence, x_{n} → y for some y ∈ T_{λ_{1}(ν)+ \frac{ϵ}{2}}^{ν} ∩ ∂Ω such that

|x′ − y| = lim_{n→∞} dist(x′, T_{λ_{1}(ν)+ \frac{ϵ}{2}}^{ν_{n}} ∩ ∂Ω),

but since this limit is equal to 0, we infer that x′ = y. Now, since λ_{1}(ν) < λ_{1}(ν_{n}) − ϵ for all n ∈ ℕ, ν(x_{n}).ν_{n} > 0 for all n and thus ν(x′).ν ≥ 0, a contradiction with (1.5). (2.b) The convexity of Ω implies that x_{λ_{1}(ν)+ \frac{ϵ}{2}}^{ν} ∉ Ω. Now, x_{λ_{1}(ν)+ \frac{ϵ}{2}}^{ν_{n}} → x_{λ_{1}(ν)+ \frac{ϵ}{2}}^{ν} , so that

x_{λ_{1}(ν)+ \frac{ϵ}{2}}^{ν_{n}} ∉ Ω  (1.6)

for n large enough. But since x.ν < λ_{1}(ν) by definition of x, we also have x.ν_{n} < λ_{1}(ν) < λ_{1}(ν) + \frac{ϵ}{2} for n sufficiently large, and so

x ∈ (∂Ω ∩ Ω_{λ_{1}(ν)+ \frac{ϵ}{2}}^{ν_{n}} ) \ T_{λ_{1}(ν)+ \frac{ϵ}{2}}^{ν_{n}}

for these values of n. This fact together with (1.6) contradicts the definition of λ_{1}(ν_{n}).

The proof of the lower semicontinuity follows from Case 1, which uses only the C^{1} regularity of the domain. □

A counterexample in ℝ^{2}

This is an example of a convex but not strictly convex domain in ℝ^{2}. It contradicts case (2.a) in the proof and indeed, case (2.a) is the only one using the strict convexity. The example can be made smooth. In fact all is required is a convex domain in ℝ^{2} whose boundary contains a piece of (straight) line, say of length L. Then for ν parallel to the line, there exists a sequence ν_{n} → ν such that λ_{1}(ν_{n}) ≥ λ_{1}(ν) + \frac{L}{2}.

A variation of this construction will produce similar examples in higher dimensions.

λ_{1}(ν_{n}) > λ_{1}(ν) + ϵ

Figure 2: Counterexample of a smooth convex but not strictly convex domain for which λ_{1}(ν) is not continuous everywhere.

References

[1] C. Azizieh and Ph. Clément, A priori estimates for positive solutions of p-Laplace equations, J. Differential Equations, 179, No 1, (2002), 213-245.

[2] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. 22 (1991), 1-37.

[3] F. Brock, Continuous rearrangement and symmetry of solutions of elliptic problems, Proc. Indian Acad. Sci. Math. Sci., 110, No 2, (2000), 157-204.

[4] L. Damascelli and F. Pacella, Monotonicity and symmetry of solutions of p−Laplace equations, 1 < p < 2, via the moving plane method, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., IV, Ser.26, (1998), 689-707.

[5] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and Related Properties via the Maximum principle, Comm. Math. Phys. 68, (1979), 209-243.

Céline Azizieh (e-mail: kazizieh@ulb.ac.be) Luc Lemaire (e-mail: llemaire@ulb.ac.be) Département de Mathématique Université Libre de Bruxelles CP 214 - Campus Plaine 1050 Bruxelles - Belgique