Symmetry and asymmetry of minimizers of a class of noncoercive functionals

Friedemann Brock; Gisella Croce; Olivier Guibé; Anna Mercaldo

doi:10.1515/acv-2017-0005

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Symmetry and asymmetry of minimizers of a class of noncoercive functionals

Friedemann Brock , Gisella Croce , Olivier Guibé und Anna Mercaldo

Veröffentlicht/Copyright: 9. September 2017

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Aus der Zeitschrift Advances in Calculus of Variations Band 13 Heft 1

Abstract

In this paper we prove symmetry results for minimizers of a noncoercive functional defined on the class of Sobolev functions with zero mean value. We prove that the minimizers are foliated Schwarz symmetric, i.e., they are axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis. In the two-dimensional case, we show a symmetry breaking.

Keywords: Foliated Schwarz symmetry; Euler equation; symmetry breaking

MSC 2010: 49J40; 49K20; 49K30

1 Introduction

Consider the functional

v∈H01(Ω)→12∫Ω|∇v|2

subjected to the constraint ∫Ωv2=1, where Ω is the unit ball in the plane. Its critical values are the eigenvalues of the classical fixed membrane problem

(1.1){-Δ⁢u=λ⁢uin ⁢Ω,u=0on ⁢∂⁡Ω.

It is known that the first eigenfunctions are positive and Schwarz symmetric, that is, radial and decreasing in the radial variable. By contrast, the second eigenfunctions are sign-changing; they are not radial, but they are symmetric with respect to the reflection at some line ℝ⁢e, and they are decreasing in the angle arccos⁡[x|x|⋅e]∈(0,π). These properties can be seen as a spherical version of the Schwarz symmetry along the foliation of the underlying ball Ω by circles. For this reason, this property is called foliated Schwarz symmetry in the literature.

In the last years much interest has been devoted to the shape of sign changing minimizers of integral functionals, see, for example, [8, 14, 23, 7, 20]. In [14] Girao and Weth studied the symmetry properties of the minimizers of the problem

(1.2)v→∥∇⁡v∥2∥v∥p,v∈H1⁢(Ω),∫Ωv=0

for 2≤p<2*. In view of the zero average constraint, (1.2) is similar to the problem of finding the second eigenfunctions of problem (1.1). They proved that the minimizers are foliated Schwarz symmetric.

In [14] Girao and Weth pointed out another interesting phenomenon related to the shape of the minimizers of (1.2). If p is close to 2, then any minimizer of the above functional is anti-symmetric with respect to the reflection at the hyperplane {x⋅e=0}. By contrast, the minimizers are not anti-symmetric when N=2 and p is sufficiently large. A similar break of symmetry was already observed in [12, 11, 16, 3, 10, 19] for the minimizers of the functional

v→∥v′∥Lp⁢(0,1)∥v∥Lq⁢(0,1),v∈W1,p⁢((0,1)),v⁢(0)=v⁢(1),∫01v=0.

Indeed, it has been shown that any minimizer is an anti-symmetric function if and only if q≤3⁢p.

In this paper we will prove similar symmetry results for the minimizers of a generalized version of the functional studied by Girao and Weth in [14]. We consider

(1.3)λθ,p(Ω)=inf{∫Ω|∇⁡v|2(1+|v|)2⁢θdx,v∈W1,q(Ω),v≠0,∫Ωvdx=0,∥v∥Lp⁢(Ω)=1},

where Ω is either a ball or an annulus centered in the origin in ℝN, N≥2, and θ and q satisfy

(1.4)0<2⁢θ<1,

(1.5)q=2⁢N⁢(1-θ)N-2⁢θif ⁢N≥3,

(1.6)2⁢(1-θ)≤q<2if ⁢N=2,

(1.7)1<p<q*if ⁢N≥3,

(1.8)1<p<+∞if ⁢N=2.

Observe that if one defines

Ψ⁢(ξ):=∫0ξ(1+|t|)-θ⁢𝑑t=sgn⁡ξ1-θ⁢[(1+|ξ|)1-θ-1],ξ∈ℝ,

then our functional is the integral of |∇⁡Ψ⁢(u)|2, that is, (1.3) is equivalent to

λθ,p(Ω)=inf{∫Ω|∇Ψ(v)|2dx,v∈W1,q(Ω),v≠0,∫Ωvdx=0,∥v∥Lp⁢(Ω)=1}.

The main feature of this functional is that it is not coercive on H01⁢(Ω), even if it is well defined on this Sobolev space. The lack of coercivity has unpleasant consequences for the minimizers of

v→∫Ω[|∇⁡v|2(1+|v|)2⁢θ-G⁢(x,v)]⁢𝑑x

for functions G having various growth assumptions. Indeed, it was shown in [6, 2, 18, 13, 21, 4] that the minimizers are less regular than the minimizers of coercive functionals on H1⁢(Ω).

After recalling the definition of foliated Schwarz symmetry and proving some new sufficient conditions for this symmetry in Section 3, we will prove the foliated Schwarz symmetry of the minimizers for N≥2. As already pointed out, the same result has been obtained by Girao and Weth in [14] in the ‘coercive’ case, that is, for θ=0. We observed that in their proof, Girao and Weth make use of a well-known regularity result of the solutions of the Euler equation. In our case, we have to prove the analogous regularity result for our noncoercive functional (see Section 4). Actually we are able to prove the foliated Schwarz symmetry of the minimizers of a more general functional, that is, we consider

(1.9)λθ,p(Ω)=inf{∫Ω|∇⁡v|2-F⁢(|x|,v)(1+|v|)2⁢θdx,v∈W1,q(Ω),v≠0,∫Ωvdx=0,∥v∥Lp⁢(Ω)=1},

where we assume that F:ℝ+×ℝ→ℝ is a measurable function in r=|x|∈[0,+∞) and continuously differentiable in t∈ℝ, which satisfies

(1.10)F⁢(r,0)=0

and the growth conditions

(1.11)|F⁢(r,t)|≤c⁢(1+|t|)p,c>0,

(1.12)|Ft⁢(r,t)|≤C1⁢(1+|t|)p-1,C1>0,

for any r∈[0,+∞), t∈ℝ. If p∈(1,2), we add the requirement

(1.13)t⁢(1+|t|)⁢Ft⁢(r,t)-2⁢θ⁢|t|⁢F⁢(r,t)≤0

for any r∈[0,+∞), t∈ℝ.

In the last two sections we will focus on the two-dimensional setting, in the case where Ω is a ball. We will prove that there exists a unique minimizer, which is anti-symmetric, for p=2 and sufficiently small θ. However, the minimizers are not anti-symmetric for p sufficiently large. This shows a symmetry breaking phenomenon, which generalizes the results proved by Girao and Weth in the case θ=0. Note that because of the difficulty given by the lack of coercivity of our functional, our technique is quite different from the one used in [14].

2 Existence of a minimizer

In this section we prove the existence of a minimizer for problem (1.9) by adapting the technique of [12]. We will also make use of an estimate proved in [6] (see also [1, 5]).

Theorem 2.1.

Under assumptions (1.4)–(1.8) and (1.10)–(1.13), there exists a minimizer u which realizes λθ,p⁢(Ω) as defined in (1.9).

Proof.

We first observe that the growth assumption (1.11) on F and the condition ∥u∥Lp⁢(Ω)=1 in the functional imply that λθ,p∈ℝ. For any fixed n∈ℕ, let us define

Hn⁢(v)=∫Ω|∇⁡v|2-F⁢(|x|,v)(1+|v|)2⁢θ⁢𝑑x-(λθ,p⁢(Ω)+1n),

H∞⁢(v)=∫Ω|∇⁡v|2-F⁢(|x|,v)(1+|v|)2⁢θ⁢𝑑x-λθ,p⁢(Ω)

for any v∈W1,q⁢(Ω) such that v≠0, ∥v∥Lp⁢(Ω)=1 and ∫Ωv=0. By the definition of infimum, for any fixed n∈ℕ, there exists un∈W1,q⁢(Ω), un≠0, such that

(2.1)∥un∥Lp⁢(Ω)=1,∫Ωun⁢𝑑x=0,Hn⁢(un)<0.

Now, by the growth assumption (1.11) on F, since the functions un have Lp-norm equal to 1, we have

(2.2)∫Ω|F⁢(|x|,un)|(1+|un|)2⁢θ⁢𝑑x≤C,

where C is a positive constant which does not depend on n. From now on we will denote by C a positive constant which depends on the data and which can vary from line to line.

Since Hn⁢(un)<0, estimates (2.1) and (2.2) imply that

(2.3)∫Ω|∇⁡un|2(1+|un|)2⁢θ⁢𝑑x≤C.

Now we prove that |∇⁡un| is bounded in Lq⁢(Ω), that is, for any n∈ℕ,

(2.4)∥∇⁡un∥Lq⁢(Ω)≤C.

We adapt the estimate used in [6, Theorem 2.1], and we distinguish the cases N≥3 and N=2.

Let N≥3 with q=2⁢N⁢(1-θ)N-2⁢θ. We begin by applying the Hölder inequality, since q<2. Then we use estimate (2.3) and, since the mean value of un is null, by the Sobolev inequality, we get

∫Ω|∇un|qdx≤(∫Ω|∇⁡un|2(1+|un|)2⁢θdx)q2(∫Ω(1+|un|)2⁢θ⁢q2-qdx)1-q2

≤C(∫Ω|∇⁡un|2(1+|un|)2⁢θdx)q2(1+∫Ω|un|q*dx)1-q2

≤C(1+∫Ω|∇un|qdx)q*q⁢(1-q2),

where we have used the equality 2⁢θ⁢q2-q=q*. Since N≥3, we deduce that q*q⁢(1-q2)<1 and (2.4) is proved.

Let N=2. Similarly to above, using the Hölder inequality, estimate (2.3), the inclusion L2⁢q2-q⁢(Ω)⊂L2⁢q⁢θ2-q⁢(Ω) and the Sobolev inequality, we get

∫Ω|∇un|qdx≤(∫Ω|∇⁡un|2(1+|un|)2⁢θdx)q2(∫Ω(1+|un|)2⁢θ⁢q2-qdx)1-q2

≤(∫Ω|∇⁡un|2(1+|un|)2⁢θdx)q2(1+∫Ω|un|2⁢q2-qdx)(1-q2)⁢θ

≤C(1+∫Ω|∇un|qdx)θ.

Since θ<1, inequality (2.4) follows again.

By the Poincaré–Wirtinger inequality, since the mean value of un is zero, we deduce, by (2.4), that

(2.5)un⁢ is bounded in W1,q⁢(Ω),

and therefore there exists a function u∈W1,q⁢(Ω) such that, as n→∞, up to a subsequence,

(2.6)un→uin W1,q⁢(Ω) weakly,

(2.7)un→uin Lr⁢(Ω), 1≤r<q*,

(2.8)un→ua.e. in Ω.

Let N≥3 with q=2⁢N⁢(1-θ)N-2⁢θ. Note that (2.7), since p<q*, implies that

∫Ωu⁢𝑑x=0,∥u∥Lp⁢(Ω)=1.

We claim that

(2.9)H∞⁢(u)≤0.

Let

(2.10)Ψ⁢(ξ):=∫0ξ(1+|t|)-θ⁢𝑑t=sgn⁡ξ1-θ⁢[(1+|ξ|)1-θ-1],ξ∈ℝ,

and observe, by (2.3), that

∫Ω|∇Ψ(un)|2dx=∫Ω|∇⁡un|2(1+|un|)2⁢θdx≤C.

Moreover, Ψ⁢(un) is bounded in L2⁢(Ω), since 2⁢(1-θ)≤q, and un is bounded in Lq⁢(Ω) by (2.5). We infer that Ψ⁢(un) is bounded in W1,2⁢(Ω) and, up to a subsequence, Ψ⁢(un) converges weakly in W1,2⁢(Ω) to a limit which is necessarily Ψ⁢(u), by (2.8). Therefore, by the weak semi-continuity of the norm and the inequality in (2.1), as n→∞, up to a subsequence,

∥Ψ(u)∥W1,22≤lim infn→+∞∥Ψ(un)∥W1,22≤limn→+∞∫ΩF⁢(|x|,un)(1+|un|)2⁢θdx+limn→+∞(λθ,p(Ω)+1n).

To pass to the limit in the first term of the right-hand side, one can use the Lebesgue theorem. Indeed, the pointwise convergence is given by (2.8). The growth assumptions (1.11) on F and (2.7), since p<q*, imply the existence of a function h∈Lp⁢(Ω) such that

|F⁢(|x|,un)|(1+|un|)2⁢θ≤h⁢(x) a.e. in ⁢Ω.

Finally, we get

∫Ω|∇⁡u|2(1+|u|)2⁢θ⁢𝑑x≤∫ΩF⁢(|x|,u)(1+|u|)2⁢θ⁢𝑑x+λθ,p⁢(Ω),

that is, (2.9) holds. By the definition of λθ,p⁢(Ω), necessarily we have H∞⁢(u)=0. We observe that Ψ⁢(u)≠0, since ∥u∥Lp⁢(Ω)=1.

It remains to conclude the proof for the case N=2. Indeed, when N=2, we have 1<p<+∞ (see (1.8)) and 2⁢(1-θ)≤q<2 (see (1.6)), so that the convergences (2.6)–(2.8) do not imply, in general, that ∥u∥Lp⁢(Ω)=1. However, in view of (2.3) and since 2⁢(1-θ)≤q, we obtain that Ψ⁢(un) is bounded in W1,2⁢(Ω). From the Sobolev embedding theorem, it follows that Ψ⁢(un) is bounded in Lr⁢(Ω) for any 1≤r<+∞. Since Ψ⁢(ξ) grows like |ξ|1-θ, with 1-θ>0, we conclude that un is bounded in Lr⁢(Ω) for any 1≤r<+∞. We obtain that ∥u∥Lp⁢(Ω)=1, and the arguments developed in the case N=3 allow us to conclude that H∞⁢(u)=0. ∎

3 Identification of symmetry

In this section we generalize some known symmetry criteria (cf. [9]). We first introduce some notations and definitions. Let Ω be a domain that is radially symmetric with respect to the origin. In other words, Ω is either an annulus, a ball, or the exterior of a ball in ℝN. If u:Ω→ℝ is a measurable function, we will, for convenience, always extend u onto ℝN by setting u⁢(x)=0 for x∈ℝN∖Ω.

Definition 3.1.

Let ℋ0 be the family of open half-spaces H in ℝN such that 0∈∂⁡H. For any H∈ℋ0, let σH denote the reflection in ∂⁡H. We write

σH⁢u⁢(x):=u⁢(σH⁢x),x∈ℝN.

The two-point rearrangement with respect to H is given by

uH⁢(x):={max⁡{u⁢(x),u⁢(σH⁢x)}if ⁢x∈H,min⁡{u⁢(x),u⁢(σH⁢x)}if ⁢x∉H.

The notion of two-point rearrangement was introduced more than fifty years ago as a set transformation in [24], and was applied to variational problems for the first time by Brock and Solynin in [9].

Note that one has u=uH if and only if u⁢(x)≥u⁢(σH⁢x) for all x∈H. Similarly, σH⁢u=uH if and only if u⁢(x)≤u⁢(σH⁢x) for all x∈H.

We will make use of the following properties of the two-point rearrangement (see [9]).

Lemma 3.2.

Let H∈H0.

If A∈C⁢([0,+∞),ℝ), u:Ω→ℝ is measurable and A⁢(|x|,u)∈L1⁢(Ω), then A⁢(|x|,uH)∈L1⁢(Ω) and
(3.1)∫ΩA⁢(|x|,u)⁢𝑑x=∫ΩA⁢(|x|,uH)⁢𝑑x.
If B∈L∞⁢(ℝ) and u∈W1,p⁢(Ω) for some p∈[1,+∞), then
(3.2)∫ΩB⁢(u)⁢|∇⁡u|p⁢𝑑x=∫ΩB⁢(uH)⁢|∇⁡uH|p⁢𝑑x.

Proof.

Since |σH⁢x|=|x|, for a.e. x∈H∩Ω, we have

A⁢(|x|,u⁢(x))+A⁢(|σH⁢x|,u⁢(σH⁢x))=A⁢(|x|,uH⁢(x))+A⁢(|σH⁢x|,uH⁢(σH⁢x))

and

B⁢(u⁢(x))⁢|∇⁡u⁢(x)|p+B⁢(u⁢(σH⁢x))⁢|∇⁡u⁢(σH⁢x)|p=B⁢(uH⁢(x))⁢|∇⁡uH⁢(x)|p+B⁢(uH⁢(σH⁢x))⁢|∇⁡uH⁢(σH⁢x)|p.

Now (3.1) and (3.2) follow from this by integration over H∩Ω. ∎

In order to study the symmetry of minimizers of (1.9), we introduce the notion of foliated Schwarz symmetrization of a function, that is, a function which is axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis.

Definition 3.3.

If u:Ω→ℝ is measurable, the foliated Schwarz symmetrizationu* of u is defined as the (unique) function satisfying the following properties:

There exists a function w:[0,+∞)×[0,π)→ℝ, w=w⁢(r,θ), which is nonincreasing in θ, such that
u*⁢(x)=w⁢(|x|,arccos⁡(x1|x|)),x∈Ω.
We have
ℒN-1⁢{x:a<u⁢(x)≤b,|x|=r}=ℒN-1⁢{x:a<u*⁢(x)≤b,|x|=r}
for all a,b∈ℝ, with a<b and r≥0.

Definition 3.4.

Let PN denote the point (1,0,…,0), the ‘north pole’ of the unit sphere 𝒮N-1. We say that u is foliated Schwarz symmetric with respect to PN if u=u*, that is, u depends solely on r and θ (the ‘geographical width’), and is nonincreasing in θ.

We also say that u is foliated Schwarz symmetric with respect to a point P∈𝒮N-1 if there exists a rotation about the origin ρ such that ρ⁢(PN)=P and u⁢(ρ⁢(⋅))=u*⁢(⋅).

In other words, a function u:Ω→ℝ is foliated Schwarz symmetric with respect to P if for every r>0 and c∈ℝ, the restricted superlevel set {x:|x|=r,u⁢(x)≥c} is equal to {x:|x|=r} or a geodesic ball in the sphere {x:|x|=r} centered at rP. In particular, u is axially symmetric with respect to the axis ℝ⁢P. Moreover, a measurable function u:Ω→ℝ is foliated Schwarz symmetric with respect to P∈𝒮N-1 if and only if u=uH for all H∈ℋ0, with P∈H.

The main result of this section is the following, which gives a tool to establish if a measurable function is foliated Schwarz symmetric with respect to some point P.

Theorem 3.5.

Let u∈Lp⁢(Ω) for some p∈[1,+∞), and assume that for every H∈H0, one has either u=uH or σH⁢u=uH. Then u is foliated Schwarz symmetric with respect to some point P∈SN-1.

Note that the above result has been shown for continuous functions in [23].

Theorem 3.6.

Let u∈C⁢(RN), and assume that for every H∈H0 one has either u=uH or σH⁢u=uH. Then u is foliated Schwarz symmetric with respect to some point P∈SN-1.

The idea in our proof is to use an approximation argument. Let φ∈C0∞⁢(ℝN), φ≥0, with ∫ℝNφ⁢(x)⁢𝑑x=1. Moreover, assume that φ is radial and radially nonincreasing, that is, there exists a nonincreasing function h:[0,+∞)→[0,+∞) such that φ⁢(x)=h⁢(|x|) for all x∈ℝN. For any function ε>0, define φε by φε⁢(x):=ε-N⁢φ⁢(ε-1⁢x), x∈ℝN. For any u∈Lloc1⁢(ℝN), let uε be the standard mollifier of u given by

uε⁢(x):=(u*φε)⁢(x)≡∫ℝNu⁢(y)⁢φε⁢(x-y)⁢𝑑y,x∈ℝN.

The following property is crucial. It allows a reduction to C∞-functions.

Lemma 3.7.

Let u∈Lp⁢(RN) for some p∈[1,+∞), and let H∈H0 be such that u=uH. Then uε=(uε)H for every ε>0.

Proof.

It is easy to see that

|x-y|=|σH⁢x-σH⁢y|≤|σH⁢x-y|=|x-σH⁢y|

whenever x,y∈H. Since u⁢(y)≥u⁢(σH⁢y) and since φε is radial and radially nonincreasing, for every x∈H, we have

uε⁢(x)-uε⁢(σH⁢x)=∫ℝNu⁢(y)⁢[φε⁢(x-y)-φε⁢(σH⁢x-y)]⁢𝑑x

=∫H{u⁢(y)⁢[φε⁢(x-y)-φε⁢(σH⁢x-y)]+u⁢(σH⁢y)⁢[φε⁢(x-σH⁢y)-φε⁢(σH⁢x-σH⁢y)]}⁢𝑑x

=∫H(u⁢(y)-u⁢(σH⁢y))⁢[φε⁢(x-y)-φε⁢(σH⁢x-y)]⁢𝑑x≥0.

The lemma is proved. ∎

Corollary 3.8.

Let u∈Lp⁢(RN) for some p∈[1,+∞), and let H∈H0 be such that σH⁢u=uH. Then we have σH⁢(uε)=(uε)H for every ε>0.

We are now able to prove Theorem 3.5.

Proof of Theorem 3.5.

Since for every H∈ℋ0 one has either u=uH or σH⁢u=uH, Lemma 3.7 and Corollary 3.8 apply. Then either uε=(uε)H or σH⁢uε=(uε)H for every ε>0. Since uε∈C∞⁢(ℝN), Theorem 3.6 tells us that uε is foliated Schwarz symmetric with respect to some point Pε∈𝒮N-1, for every ε>0. Since 𝒮N-1 is compact, there exist a sequence of positive numbers {εn} and a point P∈𝒮N-1 such that uεn is foliated Schwarz symmetric with respect to a point Pn∈𝒮N-1, and εn→0, Pn→P as n→+∞. Let ρn and ρ be rotations such that ρn⁢(N)=Pn, n∈ℕ, and ρ⁢(N)=P. Writing un:=uεn, we have that

(3.3)un⁢(ρn⁢(⋅))=(un)*⁢(⋅),n∈ℕ.

Since un→u, it follows that (un)*→u* in Lp⁢(ℝN), and since Pn→P, we also have that un⁢(ρn⁢(⋅))→u⁢(ρ⁢(⋅)) in Lp⁢(ℝN) as n→∞. This, together with (3.3), implies that u⁢(ρ⁢(⋅))=u*⁢(⋅). The theorem is proved. ∎

4 Symmetry of minimizers

In this section we study the properties of symmetry of minimizers of (1.9). The main result is the following.

Theorem 4.1.

Assume (1.11)–(1.13) if p∈(1,2). Then every minimizer of (1.9) is foliated Schwarz symmetric with respect to some point P∈SN-1.

Remark 4.2.

Condition (1.13) is equivalent to

(4.1)t⁢∂∂⁡t⁢(F⁢(r,t)(1+|t|)2⁢θ)≤0 for all ⁢(r,t)∈[0,+∞)×ℝ.

It is satisfied, for instance, if F⁢(r,t)=F⁢(r,-t) and if

(1+t)⁢Ft⁢(r,t)-2⁢θ⁢F⁢(r,t)≤0 for t≥0.

An example is

F⁢(r,t)=-c0⁢|t|α,α≥2⁢θ,c0≥0.

Observe that F satisfies the growth condition (1.11) with suitable c0 and α such that 2⁢θ≤α≤p.

Proof.

We divide the proof into four steps. Step 1. Let H∈ℋ0, and let u be a minimizer of (1.9). The Euler equation satisfied by u is

(4.2)-∇⁡(∇⁡u(1+|u|)2⁢θ)-θ⁢|∇⁡u|2⁢sgn⁡u(1+|u|)2⁢θ+1+c+d⁢|u|p-2⁢u=g⁢(|x|,u)in Ω,

∂⁡u∂⁡ν=0on ∂⁡Ω,

where c,d∈ℝ,

g⁢(r,t):=∂∂⁡t⁢(F⁢(r,t)2⁢(1+|t|)2⁢θ) for all ⁢(r,t)∈[0,+∞)×ℝ,

and ν denotes the exterior unit normal to Ω. Setting

I⁢(v):=∫Ω|∇⁡v|2-F⁢(|x|,v)(1+|v|)2⁢θ⁢𝑑x,

by Lemma 3.2, we have

uH≠0,uH∈W1,q⁢(Ω),∫ΩuH⁢𝑑x=0,∥uH∥Lp=1,I⁢(u)=I⁢(uH).

Hence, uH is a minimizer, too, so that it satisfies

(4.3)-∇⁡(∇⁡uH(1+|uH|)2⁢θ)-θ⁢|∇⁡uH|2⁢sgn⁡uH(1+|uH|)2⁢θ+1+c′+d′⁢|uH|p-2⁢uH=g⁢(|x|,uH)in ⁢Ω,

∂⁡uH∂⁡ν=0on ⁢∂⁡Ω,

where c′,d′∈ℝ. Step 2. We claim that u,uH∈W1,q⁢(Ω)∩W2,2⁢(Ω)∩C1⁢(Ω¯). Set

Φ⁢(η):=Ψ-1⁢(η),

where Ψ has been defined in (2.10). Let U:=Ψ⁢(u). Note that u=Φ⁢(U), uH=Φ⁢(UH),

Φ⁢(η)=([1+(1-θ)⁢|η|]11-θ-1)⁢sgn⁡η,

and that Φ is locally Lipschitz continuous. Rewriting (4.2) and (4.3) in terms of U and UH, we find

(4.4)-Δ⁢U+d⁢M⁢(U)=N⁢(|x|,U),

(4.5)-Δ⁢UH+d⁢M⁢(UH)=N⁢(|x|,UH)

in Ω, where

M⁢(t):=|Φ⁢(t)|p-2⁢Φ⁢(t)⁢(1+|Φ⁢(t)|)θ,

N⁢(r,t):=(g⁢(r,t)-c)⁢(1+|Φ⁢(t)|)θ

for any r∈[0,+∞), t∈ℝ.

Observe that, by the growth conditions (1.11)–(1.12) and the definition of Φ⁢(t), we have

|g⁢(r,t)|≤cθ⁢(1+|t|)p-1-2⁢θ,

|M⁢(t)|≤cθ′⁢(1+|t|)p-1+θ1-θ,

|N⁢(r,t)|≤cθ′′⁢(1+|t|)p-1-2⁢θ+θ1-θ.

Now, the growths of M and N allow us to apply classical techniques for Neumann problems (see [17, p. 272] and [22, p. 271]) to state that U∈H1⁢(Ω) is in fact C1,β⁢(Ω¯), with β∈(0,1). Therefore, u has the same regularity. Step 3. Integrating (4.2) and (4.3) gives

(4.6)-θ∫Ω|∇⁡u|2⁢sgn⁡u(1+|u|)2⁢θdx+c∫Ωdx+d∫Ω|u|p-2udx=∫Ωg(|x|,u)dx,

(4.7)-θ∫Ω|∇⁡uH|2⁢sgn⁡uH(1+|uH|)2⁢θdx+c′∫Ωdx+d′∫Ω|uH|p-2uHdx=∫Ωg(|x|,uH)dx.

Further, multiplying (4.2) and (4.3) with u and uH, respectively, and then integrating and using the constraints, yields

(4.8)∫Ω|∇⁡u|2⁢[1+(1-θ)⁢|u|](1+|u|)2⁢θ+1⁢𝑑x+d=∫Ωu⁢g⁢(|x|,u)⁢𝑑x,

(4.9)∫Ω|∇⁡uH|2⁢[1+(1-θ)⁢|uH|](1+|uH|)2⁢θ+1⁢𝑑x+d′=∫ΩuH⁢g⁢(|x|,uH)⁢𝑑x.

Now (4.6)–(4.9), together with Lemma 3.2, show that necessarily c=c′ and d=d′. Moreover, if p∈(1,2), then (4.1) holds, and so (4.8) yields d≤0. Step 4. Note that t→M⁢(t) is nondecreasing. Set h:=UH-U, and note that h≥0 in Ω∩H. We subtract (4.5) from (4.4) and split into two cases.

Let p≥2. Then we find that
-Δ⁢h=L⁢(x)⁢h in Ω∩H,
where
L(x):={N⁢(|x|,UH⁢(x))-N⁢(|x|,U⁢(x))-d⁢[M⁢(UH⁢(x))-M⁢(U⁢(x))]h⁢(x)if h⁢(x)>0,0if h⁢(x)=0
is a bounded function.
Let p∈(1,2). Then d≤0, and so
-Δ⁢h≥P⁢(x)⁢h in Ω∩H,
where
P(x):={N⁢(|x|,UH⁢(x))-N⁢(|x|,U⁢(x))h⁢(x)if h⁢(x)>0,0if h⁢(x)=0
is a bounded function.

Thus, in both cases the strong maximum principle tells us that either h⁢(x)≡0 or h⁢(x)>0 throughout Ω∩H. This implies that we have either u=uH or σH⁢u=uH in Ω. By Theorem 3.5, we deduce that u is foliated Schwarz symmetric. ∎

5 Anti-symmetry for p=2 in dimension 2

In this section we study symmetry properties of the solutions to (1.9) in the case p=2, Ω=B, where B is a ball in ℝ2, and F≡0. We will show that for small parameter values θ, there exists a unique minimizer of

v→∫B|∇⁡v|2(1+|v|)2⁢θ⁢𝑑x,v∈W1,q⁢(B),v≠0,∫Bv⁢𝑑x=0,∥v∥L2⁢(B)=1,

which is anti-symmetric. Recall that θ satisfies (1.4) and q satisfies (1.6). With abuse of notations, we will denote the infimum of the above functional by λθ,2⁢(B).

In the following we will use the notations of the proof of Theorem 4.1. In particular, let uθ be a minimizer for λθ,2⁢(B), with corresponding constants c=cθ and d=dθ, see equation (4.6). By (4.8), we have

(5.1)dθ=-∫B|∇uθ|2[1+(1-θ)|uθ|)](1+|uθ|)2⁢θ+1⁢𝑑x.

We will also frequently work with the functions

Uθ:=Ψθ⁢(uθ),where ⁢Ψθ⁢(ξ)=sgn⁡(ξ)1-θ⁢[(1+|ξ|)1-θ-1]

(see (2.10)) and

Φθ⁢(η)=Ψθ-1⁢(η)=([1+(1-θ)⁢|η|]11-θ-1)⁢sgn⁡(η).

Our calculations will often contain a generic constant C that may vary from line to line, but will be independent of θ.

Furthermore, as a consequence of Theorem 4.1, we will assume that uθ is foliated Schwarz symmetric with respect to the positive x1-half axis, that is,

(5.2)uθ⁢(x1,x2)=uθ⁢(x1,-x2).

The anti-symmetry of uθ then reads as uθ⁢(x1,x2):=-uθ⁢(-x1,x2) if θ is small.

Lemma 5.1.

Under assumptions (1.4), (1.6), the function θ→λθ,2⁢(B) is decreasing. Moreover, λθ,2⁢(B)≤λ2⁢(B), where

λ2(B)=inf{∫B|∇u|2dx,u∈H1(B),∫Budx=0,∥u∥L2⁢(B)=1}.

Proof.

Let θ1<θ2, let uθ1 be a minimizer for λθ1,2⁢(B), and let 2⁢(1-θ1)≤q<2. Then we obtain

λθ1,2⁢(B)=∫B|∇⁡u1|2(1+|u1|)2⁢θ1⁢𝑑x≥∫B|∇⁡u1|2(1+|u1|)2⁢θ2⁢𝑑x≥λθ2,2⁢(B).

Next let u be a minimizer for λ2⁢(B). Then

λ2(B)≥∫B|∇u|2dx≥∫B|∇⁡u|2(1+|u|)2⁢θdx≥λθ,2(B).∎

Lemma 5.2.

Under assumptions (1.4), (1.6), let uθ be a minimizer for λθ,2⁢(B). Let also dθ be defined by (5.1). Then limθ→0⁡dθ=-λ2⁢(B).

Proof.

First we observe that

(5.3)-dθ=∫B|∇uθ|2(1+(1-θ)|uθ|))(1+|uθ|)2⁢θ+1⁢𝑑x≤∫B|∇⁡uθ|2(1+|uθ|)2⁢θ⁢𝑑x≤λθ,2⁢(B)≤λ2⁢(B),

by Lemma 5.1. On the other hand,

(5.4)-dθ≥∫B|∇⁡uθ|2⁢(1-θ)(1+|uθ|)2⁢θdx=(1-θ)∫B|∇Uθ|2dx.

Moreover, it is easy to see that ∥Uθ∥L2⁢(B) is uniformly bounded, since ∥uθ∥L2⁢(B)=1 and θ≤12. Therefore, ∥Uθ∥H1⁢(B) is also uniformly bounded. By compactness, as θ→0, Uθ converges weakly to some function V∈H1⁢(B) and strongly in L2⁢(B). By the lower semi-continuity of the norm, from (5.4), we get

(5.5)lim infθ→0⁡(-dθ)≥∥∇⁡V∥L2⁢(B)2.

Further, the a.e. limit of Uθ is the limit of uθ, say u. By the uniqueness of the limit, u=V a.e. in B. We recall that ∥uθ∥L2⁢(B)=1 and ∫Buθ⁢𝑑x=0. Since ∥Ψθ⁢(uθ)∥H1⁢(B), by the growth of Ψθ, we deduce that uθ→u in L2⁢(B) and that ∥V∥L2⁢(B)=1 and ∫BV⁢𝑑x=0. Together with (5.5), this implies that

(5.6)lim infθ→0⁡(-dθ)≥λ2⁢(B).

Now the lemma follows from inequalities (5.3) and (5.6). ∎

Proposition 5.3.

Let uθ be a minimizer for λθ,2⁢(B). Under assumptions (1.4), (1.6), the following hold:

We have ∥uθ∥W1,∞⁢(B)≤C, where C does not depend on θ.
Let u be the limit of uθ, as θ→0, in W1,r⁢(Ω), for every r∈(1,+∞). Then ∥u∥L2⁢(B)=1 and ∫Bu⁢𝑑x=0.

Proof.

The H1⁢(B)-norm of Uθ is uniformly bounded by Lemma 5.1. By multiplying equation (4.4) by Uθ and integrating on B, one has that the right-hand side of the equality is uniformly bounded, due to Lemma 5.1. The growth of M and Lemma 5.2 imply that ∥Uθ∥Lp-1+θ1-θ⁢(B)≤C. This allows us to use the bootstrap argument described in [22, p. 271]. ∎

Let vθ⁢(x1,x2):=-uθ⁢(-x1,x2). Then, from (4.6), we obtain

(5.7)cθ=θ|B|⁢∫B|∇⁡uθ|2⁢sgn⁡(uθ)(1+|uθ|)2⁢θ+1⁢𝑑x=-θ|B|⁢∫B|∇⁡vθ|2⁢sgn⁡(vθ)(1+|vθ|)2⁢θ+1⁢𝑑x.

Lemma 5.4.

Under assumptions (1.4), (1.6), let uθ be a minimizer for λθ,2⁢(B). Let also cθ be defined by (5.7). Then |cθ|≤C⁢θ⁢∥uθ-vθ∥L2⁢(B) for a positive constant C independent on θ.

Proof.

By multiplying equation (4.2) (with p=2 and g=0) by (1+|uθ|)θ, we have

(5.8)-∇⁡((1+|uθ|)-θ⁢∇⁡uθ)+cθ⁢(1+|uθ|)θ+dθ⁢uθ⁢(1+|uθ|)θ=0.

Integrating this gives

cθ⁢∫B(1+|uθ|)θ⁢𝑑x+dθ⁢∫Buθ⁢(1+|uθ|)θ⁢𝑑x=0,

since ∂⁡uθ∂⁡ν=0 on ∂⁡B. The first integral in this identity is bounded from below, and |dθ| is bounded by Lemma 5.2. If J denotes ∫Buθ⁢(1+|uθ|)θ⁢𝑑x, we deduce that

(5.9)|cθ|≤C⁢|J|

for a constant C independent on θ. A change of variables gives

J=12⁢∫B[uθ⁢(1+|uθ|)θ-vθ⁢(1+|vθ|)θ]⁢𝑑x.

Since ∫Buθ⁢𝑑x=∫Bvθ⁢𝑑x=0, we get

J=12⁢∫B(uθ-vθ)⁢[(1+|vθ|)θ-1]⁢𝑑x+12⁢∫Buθ⁢[(1+|uθ|)θ-(1+|vθ|)θ]⁢𝑑x.

Let J1 denote the first term and J2 the second one in this identity. A short computation shows that

|J1|≤θ2∫B|uθ-vθ||vθ|dx,|J2|≤θ2∫B|uθ-vθ||uθ|dx.

Since uθ and vθ are uniformly bounded by Proposition 5.3, this gives

|J|≤C⁢θ⁢∥uθ-vθ∥L2⁢(B).

The conclusion follows from estimate (5.9). ∎

Now we can prove the main result of the section.

Theorem 5.5.

There exists a number θ0>0, such that every minimizer uθ of λθ,2⁢(B) satisfying (5.2) is unique and anti-symmetric with respect to x1, that is,

uθ⁢(x1,x2):=-uθ⁢(-x1,x2)

for any 0<θ<θ0.

Proof.

We first prove that any minimizer is anti-symmetric. Let Uθ:=Ψθ⁢(uθ) and Vθ:=Ψθ⁢(vθ), where vθ⁢(x1,x2)=-uθ⁢(-x1,x2). Writing equation (5.8) in terms of Uθ, we have

-Δ⁢Uθ+cθ⁢(1+|uθ|)θ+dθ⁢uθ⁢(1+|uθ|)θ=0.

Similarly,

-Δ⁢Vθ-cθ⁢(1+|vθ|)θ+dθ⁢vθ⁢(1+|vθ|)θ=0.

Subtract both equations from each other. Assuming that Uθ-Vθ≠0 along a sequence θ→0, we multiply by Uθ-Vθ∥Uθ-Vθ∥L2⁢(B)2 and integrate. Then we obtain

∥∇⁡(Uθ-Vθ)∥L2⁢(B)2∥Uθ-Vθ∥L2⁢(B)2+cθ∥Uθ-Vθ∥L2⁢(B)⁢∫B[(1+|uθ|)θ+(1+|vθ|)θ]⁢Uθ-Vθ∥Uθ-Vθ∥L2⁢(B)⁢𝑑x

=-dθ∫B[uθ)(1+|uθ|)θ-vθ(1+|vθ)|)θ]Uθ-Vθ∥Uθ-Vθ∥L2⁢(B)2dx.

The second term of the left-hand side tends to zero, by Lemma 5.4 and since (1+|uθ|)θ+(1+|vθ|)θ is uniformly bounded by Proposition 5.3. To estimate the right-hand side, we first observe that -dθ→λ2⁢(B), by Lemma 5.2. Moreover, it is not difficult to prove the following estimate:

|uθ⁢(1+|uθ|)θ-vθ⁢(1+|vθ|)θ|≤(1+θ)⁢[1+(1-θ)⁢|ξθ|]2⁢θ1-θ⁢|Uθ-Vθ|,

where ξθ is between Uθ=Ψθ⁢(uθ) and Vθ=Ψθ⁢(vθ). By Proposition 5.3, we deduce that

(1+θ)⁢[1+(1-θ)⁢|ξθ|]2⁢θ1-θ→1 as θ→0,

uniformly in B.

Now, set

Wθ:=Uθ-Vθ∥Uθ-Vθ∥L2⁢(B).

By the above identity, the norms ∥∇⁡Wθ∥L2⁢(B) are uniformly bounded. Hence, there exists a function W~∈H1⁢(B) such that, along a subsequence, ∇⁡Wθ→∇⁡W~ weakly in L2⁢(B) and Wθ→W~ in L2⁢(B), and so ∥W~∥L2⁢(B)=1. This also implies

∥∇⁡W~∥L2⁢(B)2≤λ2⁢(B).

Next we claim that |∫BWθ⁢𝑑x|≤C⁢θ. Indeed, since ∫Buθ⁢𝑑x=∫Bvθ⁢𝑑x=0, we have

|∫B(Uθ-Vθ)⁢𝑑x|=∫B(Ψθ⁢(uθ)-uθ-Ψθ⁢(vθ)+vθ)⁢𝑑x

≤∫B|∫vθuθ(Ψθ′(t)-1)(uθ-vθ)dt|dx

≤θ∫B|uθ-vθ|dx.

Now, Φθ is locally Lipschitz continuous, uniformly in θ. By Proposition 5.3, we obtain

and the claim follows. This and the fact that Wθ→W~ in L2⁢(B) prove that ∫BW~⁢𝑑x=0. Then, by the definition of λ2⁢(B), we have that

∥∇⁡W~∥L2⁢(B)2≥λ2⁢(B).

Hence, W~ is a (nonzero) eigenfunction for the Neumann Laplacian in B. By the properties of Uθ and Vθ and by (5.2), one has Wθ⁢(x1,x2)=Wθ⁢(-x1,x2)=Wθ⁢(x1,-x2). Thus, the same symmetry properties hold for W~. But this is in contradiction with the shape of the eigenfunction in a ball, which is given by

Jn⁢(αn⁢k⁢|(x1,x2)|R)⋅{cos⁡(n⁢φ),l=1,sin⁡(n⁢φ),l=2(n≠0),

where we have used the polar coordinates, R is the radius of the ball, and αn⁢k are the positive roots of the derivative of the Bessel function Jn (see, for example, [15]). We thus have proved that any minimizer is anti-symmetric. Note that the anti-symmetry also implies that cθ=0, which can be seen by integrating (4.2).

It remains to prove that the minimizer is unique for small θ. Assume this is not the case. Then there is a sequence θ→0 along which there exist two distinct minimizers uθ and uθ′. Let the corresponding constants d of (4.2) be denoted by dθ and dθ′. Multiplying (4.2) by uθ and integrating by parts gives

dθ=-∫B|∇⁡uθ|2(1+|uθ|)2⁢θ⁢𝑑x+θ⁢∫B|∇⁡uθ|2⁢|uθ|(1+|uθ|)2⁢θ+1⁢𝑑x

(5.10)=-λ2,θ⁢(B)+θ⁢∫B|∇⁡uθ|2⁢|uθ|(1+|uθ|)2⁢θ+1⁢𝑑x.

Similarly,

(5.11)dθ′=-λ2,θ⁢(B)+θ⁢∫B|∇⁡uθ′|2⁢|uθ′|(1+|uθ′|)2⁢θ+1⁢𝑑x.

We define

gθ⁢(ξ):=|ξ|(1+|ξ|)2⁢θ+1.

Since the functions gθ are locally Lipschitz continuous, uniformly in θ, using Proposition 5.3, we can estimate

||∇⁡uθ|2⁢|uθ|(1+|uθ|)2⁢θ+1-|∇⁡uθ′|2⁢|uθ′|(1+|uθ′|)2⁢θ+1|=||uθ|(1+|uθ|)2⁢θ+1⁢(|∇⁡uθ|2-|∇⁡uθ′|2)+|∇⁡uθ′|2⁢(gθ⁢(uθ)-gθ⁢(uθ′))|

(5.12)≤C⁢(|∇⁡uθ-∇⁡uθ′|+|uθ-uθ′|).

Subtracting (5.11) from (5.10) and taking into account (5.12), we obtain

|dθ-dθ′|≤θ⁢∫B||∇⁡uθ|2⁢|uθ|(1+|uθ|)2⁢θ+1-|∇⁡uθ′|2⁢|uθ′|(1+|uθ′|)2⁢θ+1|⁢𝑑x

(5.13)≤C⁢θ⁢(∥uθ-uθ′∥L2⁢(B)+∥∇⁡(uθ-uθ′)∥L2⁢(B)).

Now we claim that

(5.14)|dθ-dθ′|≤C⁢θ⁢∥Uθ-Uθ′∥L2⁢(B).

As in the proof of Lemma 5.4, by multiplying equation (4.2) (with p=2 and g=0) by (1+|uθ|)θ, we have

-∇⁡((1+|uθ|)-θ⁢∇⁡uθ)+cθ⁢(1+|uθ|)θ+dθ⁢uθ⁢(1+|uθ|)θ=0.

Moreover, the analog of this equality holds true for uθ′. In addition, by multiplying this equation by uθ-uθ′, we get

∫B(∇⁡uθ(1+|uθ|)θ-∇⁡uθ′(1+|uθ′|)θ)⁢∇⁡(uθ-uθ′)⁡d⁢x

+dθ⁢∫B[uθ⁢(1+|uθ|)θ-uθ′⁢(1+|uθ′|)θ]⁢(uθ-uθ′)⁢𝑑x+(dθ-dθ′)⁢∫Buθ′⁢(1+|uθ|)θ⁢(uθ-uθ′)⁢𝑑x=0.

Now we evaluate the three integrals on the left-hand side. For a small enough value of θ, we have

∫B(∇⁡uθ(1+|uθ|)θ-∇⁡uθ′(1+|uθ′|)θ)⁢∇⁡(uθ-uθ′)⁡d⁢x

=∫B1(1+|uθ|)θ⁢|∇⁡(uθ-uθ′)|2⁢𝑑x+∫B∇⁡uθ′⋅∇⁡(uθ-uθ′)⁡(1(1+|uθ|)θ-1(1+|uθ′|)θ)⁢𝑑x

(5.15)≥12⁢∥∇⁡(uθ-uθ′)∥L2⁢(B)2-C⁢∥∇⁡(uθ-uθ′)∥L2⁢(B)⁢∥uθ-uθ′∥L2⁢(B).

Moreover, as in the calculation of J in the previous arguments (see after (5.9)), we get

(5.16)|dθ⁢∫B[uθ⁢(1+|uθ|)θ-uθ′⁢(1+|uθ′|)θ]⁢(uθ-uθ′)⁢𝑑x|≤C⁢∥uθ-uθ′∥L2⁢(B)2,

∫B∇⁡uθ′⋅∇⁡(uθ-uθ′)⁡(1(1+|uθ|)θ-1(1+|uθ′|)θ)⁢𝑑x

(5.17)≤C⁢θ⁢∥uθ-uθ′∥L2⁢(B)⁢(∥uθ-uθ′∥L2⁢(B)+∥∇⁡(uθ-uθ′)∥L2⁢(B)).

Combining (5.15)–(5.17), from the Young inequality, we get

∥∇⁡(uθ-uθ′)∥L2⁢(B)2≤C⁢∥uθ-uθ′∥L2⁢(B)2.

Now as in the previous calculation, we get

∥uθ-uθ′∥L2⁢(B)≤∥Uθ-Uθ′∥L2⁢(B),

which gives (5.14).

Next we define

hθ⁢(ξ):=ξ⁢(1+|ξ|)θ-Ψθ⁢(ξ)=ξ⁢(1+|ξ|)θ-sgn⁡ξ1-θ⁢[(1+|ξ|)1-θ-1].

It is easy to see that hθ is locally Lipschitz continuous with

|hθ′⁢(ξ)|≤C⁢θ,|ξ|≤M,

where the constant C depends only on M (M>0). From this, using Proposition 5.3, we obtain

|hθ⁢(uθ)-hθ⁢(uθ′)|≤C⁢θ⁢|uθ-uθ′|.

Now let Uθ:=Ψθ⁢(uθ) and Uθ′:=Ψθ⁢(uθ′). Arguing as before, we first observe that

(5.18)|uθ-uθ|≤C⁢|Ψθ⁢(uθ)-Ψθ⁢(uθ′)|=C⁢|Uθ-Uθ′|.

By (4.2), we have

(5.19)-Δ⁢Uθ+dθ⁢uθ⁢(1+|uθ|)θ=0,

(5.20)-Δ⁢Uθ′+dθ′⁢uθ′⁢(1+|uθ′|)θ=0.

Subtracting (5.20) from (5.19), multiplying with (Uθ-Uθ′) and integrating by parts gives

0=∥∇⁡(Uθ-Uθ′)∥22+dθ⁢∫B(uθ⁢(1+|uθ|)θ-uθ′⁢(1+|uθ′|)θ)⁢(Uθ-Uθ′)⁢𝑑x

(5.21)+(dθ-dθ′)⁢∫Buθ′⁢(1+|uθ′|)θ⁢(Uθ-Uθ′)⁢𝑑x.

Now define Wθ:=(Uθ-Uθ′)⁢∥Uθ-Uθ′∥L2⁢(B)-1. Then, from (5.21), we obtain

0=∥∇⁡Wθ∥L2⁢(B)2+dθ⁢∫B(uθ⁢(1+|uθ|)θ-uθ′⁢(1+|uθ′|)θ)⁢Wθ⁢∥Uθ-Uθ′∥L2⁢(B)-1⁢𝑑x

+(dθ-dθ′)⁢∫Buθ′⁢(1+|uθ′|)θ⁢Wθ⁢∥Uθ-Uθ′∥L2⁢(B)-1⁢𝑑x.

=∥∇⁡Wθ∥L2⁢(B)2+dθ⁢∥Wθ∥L2⁢(B)2+dθ⁢∫B(hθ⁢(uθ)-hθ⁢(uθ′))⁢Wθ⁢∥Uθ-Uθ′∥L2⁢(B)-1⁢𝑑x

(5.22)+(dθ-dθ′)⁢∫Buθ′⁢(1+|uθ′|)θ⁢Wθ⁢∥Uθ-Uθ′∥L2⁢(B)-1⁢𝑑x.

Since ∥Wθ∥L2⁢(B)=1, and in view of estimates (5) and (5.18), we see that the last two terms in (5.22) tend to zero as θ→0. Hence, the functions Wθ are uniformly bounded in H1⁢(B). By passing to another subsequence if necessary, we find a function W¯∈H1⁢(B) such that Wθ→W¯ weakly in H1⁢(B) and Wθ→W¯ in L2⁢(B). Then, passing to the limit in (5.22), since lim infθ→0∥∇Wθ∥L2⁢(B)≥∥∇W¯∥L2⁢(B), we obtain

(5.23)∥∇⁡W¯∥L2⁢(B)2≤λ2⁢(B)⁢∥W¯∥L2⁢(B)2.

Since also ∫BW¯⁢𝑑x=0 and ∫BW¯2⁢𝑑x=1, we must have equality in (5.23), and W¯ is an anti-symmetric eigenfunction for the Neumann Laplacian in B, that is, W¯=u. In other words, we have

∫BWθ⁢Uθ⁢𝑑x→∫Bu2⁢𝑑x=1 as θ→0.

On the other hand, we calculate

∫BWθ⁢Uθ⁢𝑑x=∫BUθ2⁢𝑑x-∫BUθ′⁢Uθ⁢𝑑x∥Uθ-Uθ′∥L2⁢(B)

=1-∫BUθ′⁢Uθ⁢𝑑x2-2⁢∫BUθ′⁢Uθ⁢𝑑x

=12⁢1-∫BUθ′⁢Uθ⁢𝑑x→0 as ⁢θ→0,

which gives a contradiction. The proof of Theorem 5.5 is complete. ∎

6 Symmetry breaking in dimension 2

In this section we continue studying the two-dimensional case, assuming again that F≡0. We show that for p sufficiently large, the minimizers of λθ,p do not verify the properties of anti-symmetry described in the previous section; therefore a phenomenon of symmetry breaking occurs.

Let us denote by Was1,q⁢(B) the subset of the Sobolev space W1,q⁢(B) of the functions which are anti-symmetric with respect to the plane P≡{x∈ℝN+1:xN=0}, that is,

Was1,q⁢(B):={v∈W1,q⁢(B):u⁢(x′,-xN)=-u⁢(x′,xN)}.

Let

ℱ⁢(v)=∫B|∇⁡v|2(1+|v|)2⁢θ⁢𝑑x,v∈W1,q⁢(B),v≠0,∫Bv⁢𝑑x=0,∥v∥L2⁢(B)=1.

Recall that θ satisfies (1.4) and q satisfies (1.6). Let

λθ,p(B):=inf{ℱ(v),v∈W1,q(B),v≠0,∫Bvdx=0,∥v∥Lp⁢(B)=1}

and

λasθ,p(B):=inf{ℱ(v),v∈Was1,q(B),v≠0,∫Bvdx=0,∥v∥Lp⁢(B)=1}.

Observe that the existence of a function realizing λasθ,p⁢(B) can be proved as in Theorem 2.1. Let us also recall a well-known result. For any bounded smooth domain Ω⊂ℝ2, let

Λasp(Ω)=inf{∥∇v∥L2⁢(Ω)2,v∈Was1,2(Ω),v≠0,∫Ωvdx=0,∥v∥Lp⁢(Ω)=1}.

In [14] the behavior of Λasp⁢(Ω) is studied and it is proved that

(6.1)Λasp⁢(Ω)→0 as ⁢p→∞.

It is easy to prove the same result for λasθ,p⁢(B).

Proposition 6.1.

We have λasθ,p⁢(B)→0 as p→∞.

Proof.

Since ∫B|∇v|2dx≥ℱ(v), one has

Λasp(B)≥inf{ℱ(v),v∈Was1,2(B),v≠0,∫Bvdx=0,∥v∥Lp⁢(B)=1}

≥inf{ℱ(v),v∈Was1,q(B),v≠0,∫Bvdx=0,∥v∥Lp⁢(B)=1}

=λasθ,p⁢(B).

By (6.1), the conclusion follows. ∎

Now we can prove the main result of the section.

Theorem 6.2.

For p sufficiently large, λθ,p⁢(B)<λasθ,p⁢(B). Therefore, the minimizers of F are not anti-symmetric for p sufficiently large.

Proof.

Let vp be an eigenfunction for λasθ,p⁢(B). Then ∥vp∥Lp⁢(B)=1. Let B+={(x1,x2)∈B:x2>0}, and let u¯p be defined by

u¯p⁢(x)={vp⁢(x),x∈B+,0,x∈B∖B+.

Then

(6.2)∫B|∇⁡u¯p|2(1+|u¯p|)2⁢θ⁢𝑑x=λasθ,p⁢(B)2,∥u¯p∥Lp⁢(B)p=12.

We claim that

(6.3)∫Bu¯p⁢𝑑x→0 as p→∞.

By Proposition 6.1, we deduce that

λasθ,p⁢(Ω)=2⁢∥∇⁡Ψ⁢(u¯p)∥L2⁢(B)→0 as ⁢p→∞,

where Ψ has been defined in (2.10). Since u¯p=0 in B∖B+, we can use the Poincaré–Wirtinger inequality, which implies

∥Ψ⁢(u¯p)∥L2⁢(B)→0 as ⁢p→∞.

Therefore, up to subsequence, Ψ⁢(u¯p)→0 and up→0 a.e. in B. On the other hand, there exists a function h∈L2⁢(B) such that |Ψ⁢(u¯p)|≤h a.e. in B. By the definition of Ψ⁢(t), we deduce the existence of a function k∈L2⁢(1-θ)⁢(B) such that |u¯p|≤k a.e. in B. Hence, Lebesgue’s theorem applies and we get ∫B|u¯p|dx→0. This proves (6.3).

Next we define

u~p:=u¯p-1|B|⁢∫Bu¯p⁢𝑑x∥u¯p-1|B|⁢∫Bu¯p⁢𝑑x∥Lp⁢(B).

Then

λθ,p⁢(B)≤∫B|∇⁡u~p|2(1+|u~p|)2⁢θ⁢𝑑x=1∥u¯p-1|B|⁢∫Bu¯p⁢𝑑x∥Lp⁢(B)2⁢∫B|∇⁡u¯p|2(1+|u¯p-1|B|⁢∫Bu¯p⁢𝑑x|∥u¯p-1|B|⁢∫Bu¯p⁢𝑑x∥Lp⁢(B))2⁢θ⁢𝑑x.

Let ε>0 be sufficiently small. For a suitable p⁢(ε)>0 and for any p>p⁢(ε), by (6.2), one has

(6.4)λθ,p⁢(B)≤1[(1/2)1p-ε]2⁢∫B|∇⁡u¯p|2[1+|u¯p-1|B|⁢∫Bu¯p⁢𝑑x|(1/2)1p+ε]2⁢θ⁢𝑑x.

Let us set Mε=1+ε1-ε. We claim that

(6.5)G⁢(u¯p)≡1+|u¯p|1+|u¯p-1|B|⁢∫Bu¯p⁢𝑑x|(1/2)1p+ε≤Mε.

First of all, it is easy to verifies that for any p>p⁢(ε),

(6.6)1+|u¯p-1|B|⁢∫Bu¯p⁢𝑑x|≥1+||u¯p⁢|-1|B||⁢∫Bu¯p⁢𝑑x||≥1+||u¯p|-ε|≥(1-ε)⁢(1+|u¯p|).

Now we distinguish two cases.

If (1/2)1p+ε≤1, then
G⁢(u¯p)≤1+|u¯p|1+|u¯p-1|B|⁢∫Bu¯p⁢𝑑x|.
By (6.6), one has G⁢(u¯p)≤11-ε.
If (1/2)1p+ε>1, then, by (6.6),
1+|u¯p-1|B|⁢∫Bu¯p⁢𝑑x|(1/2)1p+ε≥1+|u¯p-1|B|⁢∫Bu¯p⁢𝑑x|(1/2)1p+ε≥1-ε(1/2)1p+ε⁢(1+|u¯p|).

Therefore, (6.5) is proved, that is,

(6.7)11+|u¯p-1|B|⁢∫Bu¯p⁢𝑑x|(1/2)1p+ε≤Mε1+|u¯p|.

Combining estimates (6.4) and (6.7), we get

λθ,p⁢(B)≤Mε2⁢θ[(1/2)1p-ε]2⁢∫B|∇⁡u¯p|2(1+|u¯p|)2⁢θ⁢𝑑x.

It is clear that

Mε2⁢θ[(1/2)1p-ε]2=(1+ε1-ε)2⁢θ⁢1[(1/2)1p-ε]2<2

for p>p⁢(ε). Therefore, for p sufficiently large, one has

λθ,p⁢(B)<2⁢∫B|∇⁡u¯p|2(1+|u¯p|)2⁢θ⁢𝑑x=λasθ,p⁢(B),

by (6.2). ∎

Communicated by Bernard Dacorogna

Funding statement: This work started during a visit of A. Mercaldo to Université de Rouen which was financed by the Fédération Normandie Mathématiques. The last author is a member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) which supported visitings of F. Brock to Università degli Studi di Napoli Federico II. All these institutions are gratefully acknowledged.

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Received: 2017-02-01

Accepted: 2017-08-14

Published Online: 2017-09-09

Published in Print: 2020-01-01

Artikel in diesem Heft

https://doi.org/10.1515/acv-2017-0005

Schlagwörter für diesen Artikel

Foliated Schwarz symmetry; Euler equation; symmetry breaking