Prescribing Gaussian and Geodesic Curvature on Disks

1 Introduction

The problem of prescribing the Gaussian curvature on a compact surface Σ under a conformal change of the metric is a classical one, and dates back to [4, 21, 22]. Let us denote by g the original metric, by g~ the new one and by eu the conformal factor (that is, g~=eu⁢g). This problem reduces to solving the problem

where Kg and Kg~ denote the curvatures with respect to g and g~, respectively. The solvability of this equation has been studied for a long time, and it is not possible to give here a comprehensive list of references.

If Σ has a boundary, then boundary conditions are in order. Homogeneous Dirichlet and Neumann boundary conditions have already been considered in the literature. In this paper, our aim is to prescribe not only the Gaussian curvature in Σ, but also the geodesic curvature on ∂⁡Σ. In this case, we are led with the boundary value problem

where hg and hg~ are the geodesic curvatures of ∂⁡Σ relative to g and g~, respectively.

Some versions of this problem have been studied in the literature. The case hg~=0 has been treated by Chang and Yang in [7]. Moreover, the case Kg~=0 has been treated in [6, 23, 25]. There is also some progress in the blow-up analysis, see [3, 10], although a complete description of the phenomenon is still missing.

The case of constants Kg~, hg~ has also been considered. For instance, Brendle [5] uses a parabolic flow to show that this problem admits always a solution for some constant curvatures. By using complex analysis techniques, explicit expressions for the solutions and the exact values of the constants are determined if Σ is a disk or an annulus; see [19, 20]. The case of the half-plane has also been studied; see [24, 15, 33]. However, the case in which both curvatures are not constant has not been much considered. In [9], some partial existence results are given, but they include a Lagrange multiplier which is out of control. Moreover, a Kazdan–Warner type of obstruction to existence has been found in [16]. In a forthcoming work, the case of K<0 in domains different from the disk is treated, and also a blow-up analysis is performed; see [26]. At present, as far as we know, those are the only works considering non-constant curvatures.

The higher-dimensional analogue of this question (that is, prescribing scalar curvature of a manifold and mean curvature of the boundary) has been studied more. The case of zero scalar curvature and constant mean curvature is known as the Escobar problem, in strong analogy with the Yamabe problem. In this regard, see [1, 11, 12, 13, 14, 17, 18, 28] and the references therein.

By integrating (1.1) and applying the Gauss–Bonnet theorem, one obtains

In this paper, we shall consider the case in which χ⁢(Σ)=1. By the Uniformization Theorem, we can pass via a conformal map to a disk, obtaining Kg~=0, hg~=1. Taking this into account, we can consider the problem

where now K, h are the curvatures to be prescribed.

Generally speaking, the case of a disk is especially challenging because of the non-compact action of the group of conformal maps of the disk, as happens in the Nirenberg problem for Σ=𝕊2. This issue has been only treated in [6] for K=0 (see also [10]). A blow-up analysis in this case for non-constant K, h is yet to be done, and will be the target of further research. In this paper, as a first step in the understanding of the problem, we shall impose symmetry conditions on K, h in order to rule out this phenomenon. This idea goes back to Moser [30] for the Nirenberg problem.

Being more specific, we fix a symmetry group as follows:

1. We denote by G one of the following groups of symmetries of the disk:

   1. The dihedric group 𝔻k with k≥3.

   2. The group of rotations with minimal angle 2⁢πk, k≥2.

   3. The whole group of symmetries O⁢(2).

Notice that none of the groups above has fixed points on 𝕊1, that is, for each x∈𝕊1 there exists g∈G such that g⁢(x)≠x. We say that a function f is G-symmetric if f⁢(x)=f⁢(g⁢(x)) for all g∈G and for all x in the domain of f.

Our main results are the following theorems.

We can also deal with changing sign curvatures K, h as long as their negative part is small.

One of the main goals of this paper is to find an original variational setting to this problem, which we think is natural and could be of use in future research on the topic. Let us be more specific. We define the parameter

In order to fix the ideas, let us assume that both K, h are nonnegative functions; by (1.2), 0<ρ<2⁢π.

We shall show that (1.3) is equivalent to

Observe that problem (1.4) is now invariant under addition of constants to u, and ρ is unknown here. This formulation may seem rather artificial but it has the advantage of being related to the critical points of the energy functional

We highlight the fact that the functional above depends on the couple (u,ρ), where u∈H1⁢(𝔻2) and ρ∈(0,2⁢π). The form of this energy functional seems to be completely new in the related literature.

If we freeze the variable ρ, the form of this functional is adequate for the use of Moser–Trudinger-type inequalities (or Onofri-type inequalities) which are already available also for boundary terms. Indeed, by interpolating these inequalities we will show that I is bounded from below. We will gain coercivity in the u variable by imposing symmetry, as done first by Moser in [30]. Finally, we will need to exclude the possibility of obtaining minima at the endpoints ρ=0 or ρ=2⁢π. Those limit cases correspond to the problem in which K=0 or h=0, respectively, so some study of these cases is needed. By energy estimates we can assure that the minimum is attained at ρ∈(0,2⁢π), concluding the proof.

If either K or h changes sign, the above approach fails. We shall also give in Theorem 3.4 a more general result; as a corollary, and making use of a compactness result for minima of the functional I, we will obtain the perturbation result stated in Theorem 1.2.

The rest of the paper is organized as follows: In Section 2, we set the notation and the variational formulation of the problem. After that, an analysis of the properties of the energy functional is performed by means of Moser–Trudinger-type inequalities. Section 3 is devoted to the proof of Theorem 1.1, for which we first need to address the limiting cases ρ=0 and ρ=2⁢π. A more general version is also given. Finally, the proof of Theorem 1.2 is completed in Section 4.

2 Variational Setting

2.3 Moser–Trudinger Inequalities

The Moser–Trudinger inequality (see [7, 29, 30, 32]) and their variations are useful tools to deal with the non-linear terms of exponential type which appear in our functional. In particular, we are interested in weaker versions of these inequalities, also called Onofri-type inequalities.

Theorem 2.3.

Let Σ be a compact surface with C1 boundary. Then there exists a constant C∈R, depending only on Σ, such that

(2.1) log ⁢ ∫ Σ e u ≤ 1 16 ⁢ π ⁢ ∫ Σ | ∇ ⁡ u | 2 + C   for all  ⁢ u ∈ H 0 1 ⁢ ( Σ ) ,

and

(2.2) log ⁢ ∫ Σ e u ≤ 1 8 ⁢ π ⁢ ∫ Σ | ∇ ⁡ u | 2 + ⨍ Σ u + C   for all  ⁢ u ∈ H 1 ⁢ ( Σ ) .

The first inequality is classical, whereas the second is given in [7, Proposition 2.3 and the subsequent corollary]. In both cases the constant is optimal.

In order to address the non-linear boundary terms of the functional I, we will use an analogous version of Theorem 2.3 for the boundary of a compact surface that can be found for instance in [23].

Proposition 2.4.

Let Σ be a compact surface with C1 boundary. Then there exists a constant C>0, depending only on Σ, such that

log ⁢ ∫ ∂ ⁡ Σ e u ≤ 1 4 ⁢ π ⁢ ∫ Σ | ∇ ⁡ u | 2 + ⨍ ∂ ⁡ Σ u + C   for all  ⁢ u ∈ H 1 ⁢ ( Σ ) .

In the case of the disk, the above inequality is the so-called Lebedev–Milin inequality (with C=0, see for instance [31, (4’)]).

By interpolating the previous inequalities, we will obtain a lower bound for the functional I. First, we notice that inequality (2.2) can be manipulated so that the mean value of u in ∂⁡Σ replaces the mean in Σ.

Corollary 2.5.

Let Σ be a compact surface with C1 boundary. There exists a constant C∈R, depending only on Σ, such that

log ⁢ ∫ Σ e v ≤ 1 8 ⁢ π ⁢ ∫ Σ | ∇ ⁡ v | 2 + ⨍ ∂ ⁡ Σ v + C   for all  ⁢ v ∈ H 1 ⁢ ( Σ ) .

Proof.

We consider the problem

(2.3) { - Δ ⁢ w = - 4 ⁢ π | Σ | in  ⁢ Σ , ∂ ⁡ w ∂ ⁡ η = 4 ⁢ π | ∂ ⁡ Σ | on  ⁢ ∂ ⁡ Σ .

Let us point out that (2.3) is solvable in H1⁢(Σ) because

∫ ∂ ⁡ Σ 4 ⁢ π | ∂ ⁡ Σ | = - ∫ Σ 4 ⁢ π | Σ | = 4 ⁢ π .

We fix a solution w of (2.3) and apply (2.2) to v+w, obtaining

log ⁢ ∫ Σ e v ≤ 1 8 ⁢ π ⁢ ∫ Σ | ∇ ⁡ v | 2 + 1 4 ⁢ π ⁢ ∫ ∂ ⁡ Σ ∂ ⁡ w ∂ ⁡ η ⁢ v - 1 4 ⁢ π ⁢ ∫ Σ ( Δ ⁢ w ) ⁢ v + ⨍ Σ v + C .

Finally, we use that w solves (2.3):

log ⁢ ∫ Σ e v ≤ 1 8 ⁢ π ⁢ ∫ Σ | ∇ ⁡ v | 2 + ∫ ∂ ⁡ Σ v - ⨍ Σ v + ⨍ Σ v + C = 1 8 ⁢ π ⁢ ∫ Σ | ∇ ⁡ v | 2 + ∫ ∂ ⁡ Σ v + C . ∎

In a similar way, one can obtain a modified version of Proposition 2.4 in which the mean value of u on Σ substitutes the mean on ∂⁡Σ.

Corollary 2.6.

Let Σ be a compact surface with C1 boundary. There exists C∈R, depending only on Σ, such that

log ⁢ ∫ ∂ ⁡ Σ e u ≤ 1 4 ⁢ π ⁢ ∫ Σ | ∇ ⁡ u | 2 + ⨍ Σ u + C   for all  ⁢ u ∈ H 1 ⁢ ( Σ ) .

The combined use of inequality (2.3) and Corollary 2.5 allows us to prove that I is bounded from below in H1⁢(𝔻2).

Proposition 2.7.

There exists a constant C∈R such that Iρ⁢(u)≥C for every u∈X and every ρ∈[0,2⁢π].

Proof.

Let us define f:(0,2⁢π)→ℝ as the correction term in (1.5), that is,

f ⁢ ( ρ ) = 4 ⁢ ( 2 ⁢ π - ρ ) ⁢ log ⁡ ( 2 ⁢ π - ρ ) + 2 ⁢ ρ + 2 ⁢ ρ ⁢ log ⁡ ρ .

It is clear that

lim ρ → 0 ⁡ f ⁢ ( ρ ) = 8 ⁢ π ⁢ log ⁡ ( 2 ⁢ π ) , lim ρ → 2 ⁢ π ⁡ f ⁢ ( ρ ) = 4 ⁢ π + 4 ⁢ π ⁢ log ⁡ ( 2 ⁢ π ) .

Then f can be continuously extended to the compact interval [0,2⁢π]. Thus, there exists a constant M>0 such that |f⁢(ρ)|≤M for all ρ∈[0,2⁢π]. Moreover, since K and h are continuous, there exist M1,M2∈ℝ such that

log ⁢ ∫ 𝔻 2 K ⁢ e u ≤ log ⁢ ∫ 𝔻 2 e u + C , log ⁢ ∫ 𝕊 1 h ⁢ e u / 2 ≤ ∫ 𝕊 1 e u / 2 + C .

Then for every a,b∈ℝ,

I ρ ⁢ ( u ) ≥ 1 2 ⁢ ∫ 𝔻 2 | ∇ ⁡ u | 2 - 2 ⁢ ρ ⁢ log ⁢ ∫ 𝔻 2 e u - 4 ⁢ ( 2 ⁢ π - ρ ) ⁢ log ⁢ ∫ 𝕊 1 e u / 2 + 2 ⁢ ∫ 𝕊 1 u + C

= 8 ⁢ π - 2 ⁢ a - b 16 ⁢ π ⁢ ∫ 𝔻 2 | ∇ ⁡ u | 2 + a 8 ⁢ π ⁢ ∫ 𝔻 2 | ∇ ⁡ u | 2 + b 16 ⁢ π ⁢ ∫ 𝔻 2 | ∇ ⁡ u | 2 - 2 ⁢ ρ ⁢ log ⁢ ∫ 𝔻 2 e u - 4 ⁢ ( 2 ⁢ π - ρ ) ⁢ log ⁢ ∫ 𝕊 1 e u / 2 + 2 ⁢ ∫ 𝕊 1 u + C .

As the functional I is invariant under the addition of constants, we can assume that ∫Σu=0 and apply Corollary 2.5 and Proposition 2.4, taking a=2⁢ρ and b=4⁢(2⁢π-ρ), and thus obtaining:

I ρ ⁢ ( u ) ≥ - 2 ⁢ ρ ⁢ ⨍ 𝕊 1 u - 2 ⁢ ( 2 ⁢ π - ρ ) ⁢ ⨍ 𝕊 1 u + 2 ⁢ ∫ 𝕊 1 u + C = C .

We highlight that the constant C does not depend on ρ. ∎

Proposition 2.7 states that the functional I is bounded from above, but we do not have coercivity. The reason for this is the non-compact action of the conformal group of the disk. This effect appears, for instance, also in the Nirenberg problem in the sphere and makes the problem rather difficult.

We will show now that we can gain coercivity by restricting ourselves to spaces of symmetric functions. In order to do that, we introduce local versions of the inequalities above. This idea dates back to [2], and these inequalities are known as Chen–Li-type inequalities (see [8] for more details).

Proposition 2.8.

Assume Σ to be a compact surface with C1 boundary, and let Σ1⊂Σ and δ>0 be such that (Σ1)δ∩∂⁡Σ=∅. Then for every ε>0 there exists a constant C∈R depending on ε and δ such that

16 ⁢ π ⁢ log ⁢ ∫ Σ 1 e u ≤ ∫ ( Σ 1 ) δ | ∇ ⁡ u | 2 + ε ⁢ ∫ Σ | ∇ ⁡ u | 2 + C   for all  ⁢ u ∈ H 1 ⁢ ( Σ ) ⁢ with  ⁢ ∫ Σ u = 0 .

The details of the proof of this precise statement can be found, for instance, in [27, Proposition 2.2], but the idea dates back to [8]. Roughly speaking, one applies (2.1) to the function u multiplied by a cut-off function in Σ1.

If the function u has mass in several separated regions satisfying the hypothesis of the propositions above, the obtained bounds improve by a factor of the number of such regions. This information is collected in the following corollary (see for instance [27, Lemma 2.4] for the case l=2; the case of general l is analogous).

Corollary 2.9.

Let Σ be a compact surface with C1 boundary, let l∈N and let Σ1,…,Σl⊂Σ for which there exists a δ>0 such that (Σi)δ∩(Σj)δ=∅ if i≠j. Assume that there exists γ∈(0,1l) such that

∫ Σ i e u ∫ Σ e u ≥ γ   for all  ⁢ i = 1 , … , l .

Then for every ε>0 there exists a constant C∈R depending on ε, δ and γ such that

8 ⁢ l ⁢ π ⁢ log ⁢ ∫ Σ e u ≤ ∫ Σ | ∇ ⁡ u | 2 + ε ⁢ ∫ Σ | ∇ ⁡ u | 2 + C   for all  ⁢ u ∈ H 1 ⁢ ( Σ ) ⁢ with  ⁢ ∫ Σ u = 0 .

Using the same techniques, we can give a localized version of Proposition 2.4.

Proposition 2.10.

Let Σ be a compact surface with C1 boundary, and let Γ1⊂∂⁡Σ. Then for every ε,δ>0 there exists a constant C∈R depending on ε and δ such that

4 ⁢ π ⁢ log ⁢ ∫ Γ 1 e u ≤ ∫ ( Γ 1 ) δ | ∇ ⁡ u | 2 + ε ⁢ ∫ Σ | ∇ ⁡ u | 2 + C   for all  ⁢ u ∈ H 1 ⁢ ( Σ ) ⁢ with  ⁢ ∫ Σ u = 0

Proof.

Following [8], we consider a cutoff function gδ:Σ→[0,1] satisfying

g δ = { 1 if  ⁢ x ∈ Γ 1 , 0 if  ⁢ x ∈ Σ ∖ ( Γ 1 ) δ / 2 .

We have gδ⁢u∈H1⁢(Σ), hence we can apply Corollary 2.6:

(2.4) 4 ⁢ π ⁢ log ⁢ ∫ Γ 1 e u = 4 ⁢ π ⁢ log ⁢ ∫ Γ 1 e g δ ⁢ u ≤ 4 ⁢ π ⁢ log ⁢ ∫ ∂ ⁡ Σ e g δ ⁢ u ≤ ∫ Σ | ∇ ⁡ ( g δ ⁢ u ) | 2 + 4 ⁢ π ⁢ ⨍ Σ g δ ⁢ u + C .

Then

∫ Σ | ∇ ⁡ ( g δ ⁢ u ) | 2 = ∫ Σ u 2 ⁢ | ∇ ⁡ g δ | 2 + 2 ⁢ ∫ Σ g δ ⁢ u ⁢ 〈 ∇ ⁡ u , ∇ ⁡ g δ 〉 + ∫ Σ ( g δ ) 2 ⁢ | ∇ ⁡ u | 2

(2.5) ≤ C δ ⁢ ∫ Σ u 2 + 2 ⁢ ∫ Σ g δ ⁢ u ⁢ | ∇ ⁡ u | ⁢ | ∇ ⁡ g δ | + ∫ ( Γ 1 ) δ | ∇ ⁡ u | 2 .

The central term can be bounded using Cauchy’s inequality, and thus obtaining

(2.6) ∫ Σ g δ ⁢ u ⁢ | ∇ ⁡ u | ⁢ | ∇ ⁡ g δ | ≤ C δ ⁢ ∫ Σ u ⁢ | ∇ ⁡ u | ≤ C ε , δ ⁢ ∫ Σ u 2 + ε ⁢ ∫ Σ | ∇ ⁡ u | 2 .

Combining (2.5) and (2.6), we obtain

(2.7) ∫ Σ | ∇ ⁡ ( g δ ⁢ u ) | 2 ≤ ∫ ( Γ 1 ) δ | ∇ ⁡ u | 2 + ε ⁢ ∫ Σ | ∇ ⁡ u | 2 + C ε , δ ⁢ ∫ Σ u 2 .

Also, we have the following bound for the mean value of gδ⁢u on ∂⁡Σ:

(2.8) ⨍ Σ g δ ⁢ u ≤ ⨍ Σ 1 2 ⁢ ( ( g δ ) 2 + u 2 ) ≤ 1 2 ⁢ ⨍ Σ ( g δ ) 2 + 1 2 ⁢ | Σ | ⁢ ∫ Σ u 2 ≤ C δ + C ⁢ ∫ Σ u 2 .

Now, apply both inequalities (2.7) and (2.8) to (2.4) to get

(2.9) 4 ⁢ π ⁢ log ⁢ ∫ Γ 1 e u ≤ ∫ ( Γ 1 ) δ | ∇ ⁡ u | 2 + ε ⁢ ∫ Σ | ∇ ⁡ u | 2 + C ε , δ ⁢ ∫ Σ u 2 + C .

Finally, we address the term ∫Σu2.

Let a∈ℝ, η=|{x∈Σ:u⁢(x)≥a}| and (u-a)+=max⁡{0,u-a}. Clearly, u≤(u-a)++a. We now apply formula (2.9) to the function (u-a)+:

4 ⁢ π ⁢ log ⁢ ∫ Γ 1 e u ≤ 4 ⁢ π ⁢ log ⁡ ( e a ⁢ ∫ Γ 1 e ( u - a ) + )

≤ 4 ⁢ π ⁢ a + log ⁢ ∫ Γ 1 e ( u - a ) +

≤ 4 π a + ∫ ( Γ 1 ) δ | ∇ ( u - a ) + | 2 + ε ∫ Σ | ∇ ( u - a ) + | 2 + C ε , δ ∫ Σ ( ( u - a ) + ) 2

(2.10) ≤ 4 ⁢ π ⁢ a + ∫ ( Γ 1 ) δ | ∇ ⁡ u | 2 + ε ⁢ ∫ Σ | ∇ ⁡ u | 2 + C ε , δ ⁢ ∫ Σ ( ( u - a ) + ) 2 .

By means of the Sobolev, Hölder and Poincaré–Wirtinger inequalities,

∫ Σ ( ( u - a ) + ) 2 = ∫ { x ∈ Σ : u ⁢ ( x ) ≥ a } ( ( u - a ) + ) 2

≤ η 1 / 2 ⁢ ( ∫ Σ ( ( u - a ) + ) 4 ) 1 / 2

≤ η 1 / 2 ⁢ ∥ ( u - a ) + ∥ H 1 ⁢ ( Σ ) 2

(2.11) ≤ C ⁢ η 1 / 2 ⁢ ∫ Σ | ∇ ⁡ u | 2 .

Again by the Poincaré–Wirtinger inequality,

(2.12) a ⁢ η ≤ ∫ { x ∈ Σ : u ⁢ ( x ) ≥ a } u ≤ ∫ Σ | u | ≤ C ⁢ ( ∫ Σ | u | 2 ) 1 / 2 ≤ C ⁢ ( ∫ Σ | ∇ ⁡ u | 2 ) 1 / 2 .

From (2.12), using Cauchy’s inequality, we obtain

(2.13) a ≤ θ ⁢ ∫ Σ | ∇ ⁡ u | 2 + C 2 η 2 ⁢ θ   for all  ⁢ θ > 0 .

Mixing (2.10), (2.11) and (2.13), we obtain

4 ⁢ π ⁢ log ⁢ ∫ Γ 1 e u ≤ 4 ⁢ π ⁢ θ ⁢ ∫ Σ | ∇ ⁡ u | 2 + ∫ ( Γ 1 ) δ | ∇ ⁡ u | 2 + ε ⁢ ∫ Σ | ∇ ⁡ u | 2 + C ε , δ ⁢ η 1 / 2 ⁢ ∫ Σ | ∇ ⁡ u | 2 + C ,

and it is enough to take θ=14⁢π and η1/2≤εCε,δ to conclude the proof. ∎

Corollary 2.11.

Let Σ be a compact surface with C1 boundary, let l∈N and let Γ1,…,Γl⊂∂⁡Σ for which there exists a δ>0 such that (Γi)δ∩(Γj)δ=∅ if i≠j. Moreover, assume that there exists γ∈(0,1l) such that

(2.14) ∫ Γ i e u ∫ ∂ ⁡ Σ e u ≥ γ   for all  ⁢ i = 1 , … , l .

Then for every ε>0 there exists a constant C∈R depending on ε,δ and γ such that

4 ⁢ l ⁢ π ⁢ log ⁢ ∫ ∂ ⁡ Σ e u ≤ ∫ Σ | ∇ ⁡ u | 2 + ε ⁢ ∫ Σ | ∇ ⁡ u | 2 + C   for all  ⁢ u ∈ H 1 ⁢ ( Σ ) ⁢ with  ⁢ ∫ Σ u = 0 .

Proof.

First, we apply to each Γi the previous result, obtaining

4 ⁢ π ⁢ log ⁢ ∫ Γ i e u ≤ ∫ ( Γ i ) δ | ∇ ⁡ u | 2 + ε ⁢ ∫ Σ | ∇ ⁡ u | 2 + C .

Using (2.14), we get

4 ⁢ π ⁢ log ⁢ ∫ Γ i e u ≥ 4 ⁢ π ⁢ log ⁢ ∫ ∂ ⁡ Σ e u + C .

Then

4 ⁢ π ⁢ log ⁢ ∫ ∂ ⁡ Σ e u ≤ ∫ ( Γ i ) δ | ∇ ⁡ u | 2 + ε ⁢ ∫ Σ | ∇ ⁡ u | 2 + C .

Finally, summing over i∈{1,…,l}, we obtain

4 ⁢ l ⁢ π ⁢ log ⁢ ∫ ∂ ⁡ Σ e u ≤ ∫ ⊔ i ( Γ i ) δ | ∇ ⁡ u | 2 + ε ⁢ l ⁢ ∫ Σ | ∇ ⁡ u | 2 + C ≤ ∫ Σ | ∇ ⁡ u | 2 + ε ⁢ l ⁢ ∫ Σ | ∇ ⁡ u | 2 + C . ∎

We have just seen that the more regions the mass of a function is separated in, the better bounds we obtain using the local versions of the Moser–Trudinger inequalities. If an H1⁢(𝔻2) function is concentrated in an interior point of the disk, Proposition 2.8 gives us a lower bound which is sufficient to achieve coercivity, but that is not the case when a function concentrates around a boundary point. To avoid this we will restrict ourselves to consider functions satisfying a symmetry condition guaranteeing that a function cannot concentrate around a single point of the boundary. Hence we will obtain coercivity by interpolating Corollaries 2.9 and 2.11 with l=2.

Figure 1

If a symmetric function concentrates around x∈𝕊1,then it also concentrates around g⁢(x) for all g∈G.

We let G be a subgroup of the orthogonal transformation group of 𝔻2 such that the set of fixed points on 𝕊1 under the action of G is empty; in other words,

{ x ∈ 𝕊 1 : g ⁢ ( x ) = x ⁢  for all  ⁢ g ∈ G } = ∅ .

For instance, we can take G as the group of rotations generated by g⁢(z)=e(2⁢π⁢i)/k⁢z as well as the dihedral groups 𝔻k (k∈ℕ, k>1).

In the sequel, K and h will be assumed to be G-symmetric functions, and we set

H G 1 ⁢ ( 𝔻 2 ) = { u ∈ H 1 ⁢ ( 𝔻 2 ) : u ∘ g = u ⁢  for all  ⁢ u ∈ G }

and

𝕏 G = { u ∈ 𝕏 : u ∘ g = u ⁢  for all  ⁢ g ∈ G } .

As in Lemma 2.1, we observe that if K and h are G-symmetric functions somewhere positive, then 𝕏G is not empty.

Proposition 2.12.

Given ρ∈[0,2⁢π], the functional Iρ is coercive on XG, that is,

I ρ ( u ) → + ∞   ( ∥ u ∥ H 1 ⁢ ( 𝔻 2 ) → + ∞ , u ∈ 𝕏 G ) .

Proof.

Take a sequence (un) in 𝕏G. We know that Iρ is invariant under the addition of constants, so we can assume that ∫𝔻2un=0 for every n∈ℕ. We have

I ρ ⁢ ( u n ) ≥ 1 2 ⁢ ∫ 𝔻 2 | ∇ ⁡ u n | 2 - 2 ⁢ ρ ⁢ log ⁢ ∫ 𝔻 2 e u n - 4 ⁢ ( 2 ⁢ π - ρ ) ⁢ log ⁢ ∫ 𝕊 1 e u n / 2 + 2 ⁢ ∫ 𝕊 1 u n + C .

Then for any a,b∈ℝ one has

I ρ ⁢ ( u n ) ≥ 16 ⁢ π - 2 ⁢ a - b 32 ⁢ π ⁢ ∫ 𝔻 2 | ∇ ⁡ u n | 2 + a 16 ⁢ π ⁢ ∫ 𝔻 2 | ∇ ⁡ u n | 2 + b 32 ⁢ π ⁢ ∫ 𝔻 2 | ∇ ⁡ u n | 2 + 2 ⁢ ∫ 𝕊 1 u n

- 2 ⁢ ρ ⁢ log ⁢ ∫ 𝔻 2 e u n - 4 ⁢ ( 2 ⁢ π - ρ ) ⁢ log ⁢ ∫ 𝕊 1 e u n / 2 .

We can now apply Corollaries 2.9 and 2.11 with l=2 (see Figure 1):

I ρ ⁢ ( u n ) ≥ 16 ⁢ π - 2 ⁢ a - b 32 ⁢ π ⁢ ∫ 𝔻 2 | ∇ ⁡ u n | 2 + a ⁢ log ⁢ ∫ 𝔻 2 e u n - a ⁢ ε ⁢ ∫ 𝔻 2 | ∇ ⁡ u n | 2 + b ⁢ log ⁢ ∫ 𝕊 1 e u n / 2

- b ⁢ ε ⁢ ∫ 𝔻 2 | ∇ ⁡ u n | 2 - 2 ⁢ ρ ⁢ log ⁢ ∫ 𝔻 2 e u n - 4 ⁢ ( 2 ⁢ π - ρ ) ⁢ log ⁢ ∫ 𝕊 1 e u n / 2 + 2 ⁢ ∫ 𝕊 1 u n + C .

Choosing a=2⁢ρ and b=4⁢(2⁢π-ρ) and applying the trace inequality, we obtain

I ρ ⁢ ( u n ) ≥ ( 1 4 - ε ) ⁢ ∫ 𝔻 2 | ∇ ⁡ u n | 2 - 2 ⁢ C 2 ⁢ ∥ u n ∥ H 1 ⁢ ( 𝔻 2 ) + C , C 2 > 0 .

Finally, taking ε small enough and using the Poincaré–Wirtinger inequality, we obtain

I ρ ⁢ ( u n ) ≥ C 1 ⁢ ∥ u n ∥ H 1 ⁢ ( 𝔻 2 ) 2 - C 2 ⁢ ∥ u n ∥ H 1 ⁢ ( 𝔻 2 ) + C , C 1 , C 2 > 0 .

Again, we remark that the constant C1 is independent of ρ. ∎