Eigenvalue pinching on convex domains in space forms

E. Aubry^{⋆}, J. Bertrand, B. Colbois

Abstract. In this paper, we show that the convex domains of ℍ^{n} which are almost extremal for the Faber-Krahn or the Payne-Polya-Weinberger inequalities are close to geodesic balls. Our proof is also valid in other space forms and allows us to recover known results in ℝ^{n} and 𝕊^{n}.

1. Introduction

This paper aims to study some optimal inequalities involving the first eigenvalues of the Dirichlet spectrum of convex domains in space forms, and to ask how stable they are. The paper essentially deals with the most intricate case of the hyperbolic space.

The inequalities we are interested in are the Faber-Krahn inequality and the Payne-Polya-Weinberger inequality. The Faber-Krahn inequality asserts that among all bounded domains with the same volume in a given space form, the geodesic ball has the smallest first Dirichlet eigenvalue. Moreover, the geodesic ball is the unique minimizer (up to an isometry) among smooth domains. In this setting, such an inequality is stable if a bounded domain Ω whose λ_{1}(Ω) is close to λ_{1}(B) (B is a geodesic ball with the same volume as Ω), is close for the Hausdorff distance to B (up to an isometry). This general statement does not hold true, because it is possible to attach very long and thin tentacles to a ball without affecting significantly the volume and the spectrum. In fact, for Euclidean domains, weaker forms of stability have been established. One form is to prove that a domain whose first Dirichlet eigenvalue is close to the first eigenvalue of a suitable ball, ressembles a ball up to sets of small volume (see [15] for a precise statement). The other form is to consider only convex bodies, in this case the Faber-Krahn inequality is stable [14]. The stability of the Faber-Krahn inequality has also been established for convex domains in ℍ^{2} and 𝕊^{2} [3].

⋆ Partially supported by FNRS Swiss Grant n^{∘}20-101469.

The first result of this paper is to prove the stability of the Faber-Krahn inequality for convex domains in a space form of arbitrary dimension and arbitrary curvature. In the sequel, we will denote by X^{1} = (𝕊^{n}, can), X^{0} = (ℝ^{n}, can) and X^{−1} = (ℍ^{n}, can) the space forms of curvature 1, 0 and −1 respectively. Except when stated otherwise, the results in this paper hold true for δ ∈ {−1, 0, 1}.

Theorem 1.1. Let V_{0} > 0. Let λ_{1}^{∗}(V_{0}) be the first Dirichlet eigenvalue of a geodesic ball B of volume V_{0} in X^{δ}. For any ϵ > 0, there exists η > 0 such that, if Ω is a convex domain of volume V_{0} in X^{δ} and if λ_{1}(Ω) ≤ λ_{1}^{∗}(V_{0}) + η then, up to an isometry,

d_{H}(Ω, B) ≤ ϵ,

where d_{H} denotes the Hausdorff distance. In the case δ = 0, we have η = η′(ϵ)V_{0}^{−2/n}.

Remark 1.1. We do not assume that the convex domains are bounded.

The method developed is the same whatever the space form. Nevertheless, the case δ = −1 is considerably harder. The primary difficulty is that the hyperbolic space comprises unbounded convex domains with finite volume therefore, have a discrete Dirichlet spectrum. This is contrary to the case of ℝ^{n}, where an upper bound of the type DiamΩ ≤ C(Vol Ω, λ_{1}(Ω), n) holds. To deal with this difficulty, we need to prove the thus for unsolved Faber-Krahn inequality for unbounded convex domains.

Proposition 1.1 (Faber-Krahn Inequality). Let Ω be a convex set in X^{δ} of finite volume V_{0}. The first Dirichlet eigenvalue of Ω satisfies

λ_{1}(Ω) ≥ λ_{1}^{∗}(V_{0})

where λ_{1}^{∗}(V_{0}) denotes the first Dirichlet eigenvalue of a geodesic ball of volume V_{0}. Moreover, the equality λ_{1}(Ω) = λ_{1}^{∗}(V_{0}) implies that Ω is isometric to a geodesic ball.

Remark 1.2. The difficulty is in proving the case of equality.

The second result of this paper concerns the stability of the Payne-Polya-Weinberger inequality (PPW inequality for short). This famous conjecture has been proved by M.S. Ashbaugh and R.D. Benguria [1].

Theorem 1.2 ([1]). Let Ω be a smooth bounded domain in Euclidean space (resp. a smooth domain included in a hemisphere in 𝕊^{n}). Then, the following inequality holds

\frac{λ_{2}}{λ_{1}} (Ω) ≤ \frac{λ_{2}}{λ_{1}} (B),

where B is an arbitrary Euclidean ball (resp. a spherical ball such that Vol B = Vol Ω). Moreover the equality is achieved if and only if Ω is isometric to a geodesic ball.

Let us notice that the ratio \frac{λ_{2}(B)}{λ_{1}(B)} is scale-invariant in Euclidean space and that M. Ashbaugh and R. Benguria also showed in [2], that the ratio of the two first eigenvalues of a geodesic ball in 𝕊^{n} is an increasing function of the radius r (if r ≤ π/2). Consequently, the PPW inequality follows directly from the theorem below.

Theorem 1.3 ([1,2,4]). Let Ω be a smooth bounded domain in X^{δ} and such that Ω is included in an hemisphere if δ = 1. The second Dirichlet eigenvalue of Ω satisfies

λ_{2}(Ω) ≤ λ_{2}(B)

where B is a geodesic ball such that λ_{1}(B) = λ_{1}(Ω). Moreover, the equality holds if and only if Ω is isometric to B.

It is shown in [4] that λ_{2}/λ_{1} is a decreasing function of the radius of hyperbolic balls and that the PPW is false in ℍ^{n}. This theorem can be seen as a generalized PPW inequality on space forms.

We prove the following stability results.

Theorem 1.4. Let Ω be a convex domain of ℝ^{n} or 𝕊^{n}, whose volume is equal to V_{0}. For any ϵ > 0 there exists η > 0 such that for all Ω as above, the assumption \frac{λ_{2}(Ω)}{λ_{1}(Ω)} ≥ \frac{λ_{2}(B)}{λ_{1}(B)} − η implies

d_{H}(Ω, B) ≤ ε,

where B is a (well-centered) geodesic ball of volume V_{0}.

Remark 1.3. The previous result was already known in Euclidean space, it has been proved by A. Melas [14].

Theorem 1.5. Let Ω be a convex domain of X^{δ} with λ_{1}(Ω) = λ (λ > \frac{(n−1)^{2}}{4} if δ = −1). For any ε > 0, there exits η such that for all Ω as above, the assumption λ_{2}(Ω) ≥ λ_{2}^{∗}(λ) − η implies

d_{H}(Ω, B) ≤ ε,

where λ_{2}^{∗}(λ) is the second Dirichlet eigenvalue of a (well-centered) geodesic ball B of X^{δ} such that λ_{1}(B) = λ.

Remark 1.4. We make no hypothesis about the volume of the convex domains we consider, not even the finiteness. This represents the main difference between this latter theorem and Theorems 1.4 and 1.1.

As for the Faber-Krahn inequality, it is necessary to generalize the PPW inequality to a more general setting, above all the characterization of the case of equality, in order to prove Theorems 1.4 and 1.5 (see Theorems 4.1 and 4.2 for precise statements).

The method developed to solve these stability questions is a rather general method, and based on the following abstract stability lemma. The proof is straightforward, therefore omitted.

Let X be a topological space and f : X → ℝ be a function. We say that f is coercive if there exists a compact subset K of X such that inf_{X\K} f > inf_{X} f (we set inf_{∅} f = +∞).

Lemma 1.1. Let X be a topological space. If f : X → ℝ is coercive and lower semi-continuous then f is bounded below, reaches its minimal value and the set M_{f} = f^{−1}{inf f} of its minima satisfies the following stability property: for any neighborhood U of M_{f}, there exists η > 0 such that

f^{−1}(] − ∞, inf f + η]) ⊂ U.

This lemma is very close to the so-called lower semi-continuity and compacity method. This is typically used in calculus of variations to deal with the problem of minimisers existence (see [18, Chapter 1]). It can be applied to a wide variety of problems (as large as the lower semi-continuity and compacity method). It does not, however, give an explicit η.

Our proof also shows that the infimum of the functional λ_{1} (resp. \frac{λ_{1}}{λ_{2}} ) on unbounded convex domains of ℍ^{n} with a given volume (resp. with a given λ_{1}) is strictly larger than those on bounded domains. To our knowledge, this is also a new result.

The paper is organized as follows:

In section 2 we define a metric on the space C of convex, bounded domains in X^{δ}.

In section 3 we show that the eigenvalues and volume functions are continuous on C.

In section 4 we extend the classical Faber-Krahn and Payne-Polya-Weinberger (as its generalized version) inequalities to the set of convex unbounded domains. This level of generality is required in our proof even if this set is restricted to bounded convex domains for the proof of the coercivity.

Finally, we reduce the proof of the stability theorems to the proof of the coercivity of the functionals λ_{1} and λ_{1}/λ_{2} on the set of bounded convex domains of given volume (resp. given λ_{1}) and prove the coercivity of these functionals in sections 5 and 6. For that purpose, we prove several new qualitative results on the spectrum and the eigenfunctions of domains in space forms. For instance, we prove that a convex Euclidean domain with a spectral gap is bounded (hence has a discrete spectrum) and that its diameter is bounded from above by C(n)(\frac{1+λ_{1}}{λ_{2}−λ_{1}} )^{3/2} where C(n) is a universal (explicitable) constant.

2. A distance on convex domains

In the following, we set s_{1}(t) = sin t, s_{0}(t) = t, s_{−1}(t) = sh t and c_{δ} = s′_{δ}. Let x_{0} denote a fixed point in X^{δ}.

Definition 2.1. Let C be the set of convex, bounded and open subsets Ω strictly included in X^{δ}, which contain the point x_{0}.

Remark 2.1. The isometry group of X^{δ} acts transitively on X^{δ}.

Remark 2.2. Each proper, convex set of the sphere is included in a hemisphere. Hence, up to the sphere itself, all convex domains Ω in 𝕊^{n} satisfy vol(Ω) ≤ Vol 𝕊^{n}/2 and λ_{1}(Ω) ≥ n.

So, Theorems 1.1, 1.4 and 1.5 are obvious in the case Ω = 𝕊^{n} and we just have to prove them for domains Ω ∈ C.

In the remaining part of this section we define a (proper) metric on C. We chose to work with a metric which has a better behaviour than the usual Hausdorff metric with respect to the volume and the Dirichlet spectrum. To define our metric, we need some facts on support functions.

2.1. Support functions

For any Ω ∈ C, the following function will be called the support of Ω.

ρ_{Ω} : v ∈ S_{x_{0}} ↦ sup{t ∈ ℝ_{+}/ exp_{x_{0}}(sv) ∈ Ω for all s ∈ [0, t]} ∈ ℝ_{+}

where S_{x_{0}} and exp_{x_{0}} are the set of unit tangent vectors and the exponential map of X^{δ} at x_{0} respectively. Note that, even on 𝕊^{n}, we have ρ_{Ω} ≤ R as soon as Ω ⊂ B(x_{0}, R).

The properties of ρ_{Ω} needed subsequently are summarized in the following lemma.

Lemma 2.1. The function ρ_{Ω} is a Lipschitz function. Under the assumption B(x_{0}, r) ⊂ Ω ⊂ B(x_{0}, R), its Lipschitz constant is bounded

above by s_{δ}(R)\sqrt{(\frac{s_{δ}(R)}{s_{δ}(r)})^{2} − 1} if δ ≠ 1, and by cotg r otherwise.

Moreover, the following equalities hold.

\overline{Ω} = exp_{x_{0}}\{t.v/ v ∈ S_{x_{0}}, 0 ≤ t ≤ ρ_{Ω}(v)\},

Ω = exp_{x_{0}}{t.v/ v ∈ S_{x_{0}}, 0 ≤ t < ρ_{Ω}(v)},
∂Ω = exp_{x_{0}}{ρ_{Ω}(v).v/ v ∈ S_{x_{0}}}.

Proof. Fix y_{0} = exp_{x_{0}}(ρ_{Ω}(u_{0})u_{0}) ∈ ∂ Ω and consider the geodesic double cone centered at y_{0} and tangent to the ball B(x_{0}, r). We claim that for each v ∈ S_{x_{0}} \ {u_{0}} close enough to u_{0}, the geodesic γ_{v}(t) = exp_{x_{0}}(tv) meets the cone in exactly two points Z(v), Z′(v). We also have l(d_{S_{x_{0}}} (v, u_{0})) ≤ ρ_{Ω}(v) ≤ L(d_{S_{x_{0}}} (v, u_{0})), where

l(d_{S_{x_{0}}} (v, u_{0})) = min{d(x_{0}, Z(v)), d(x_{0}, Z′(v))}

and

L(d_{S_{x_{0}}} (v, u_{0})) = max{d(x_{0}, Z(v)), d(x_{0}, Z′(v))}.

From elementary trigonometric computations (see appendix A for more details), we get

lim inf_{v→u_{0}} \frac{ρ_{Ω}(v) − ρ_{Ω}(u_{0})}{d_{S_{x_{0}}} (v, u_{0})} ≥ lim inf_{v→u_{0}} \frac{l(d_{S_{x_{0}}} (v, u_{0})) − l(0)}{d_{S_{x_{0}}} (v, u_{0})}

= l′(0) = −s_{δ}(d(x_{0}, y_{0}))\sqrt{(\frac{s_{δ}(d(x_{0}, y_{0}))}{s_{δ}(r)})^{2} − 1}

and

lim sup_{v→u_{0}} \frac{ρ_{Ω}(v) − ρ_{Ω}(u_{0})}{d_{S_{x_{0}}} (v, u_{0})} ≤ lim sup_{v→u_{0}} \frac{L(d_{S_{x_{0}}} (v, u_{0})) − L(0)}{d_{S_{x_{0}}} (v, u_{0})}

= L′(0) = s_{δ}(d(x_{0}, y_{0}))\sqrt{(\frac{s_{δ}(d(x_{0}, y_{0}))}{s_{δ}(r)})^{2} − 1},

which imply that ρ_{Ω} is Lipschitzian and give an upper bound on the Lipschitz constant, thanks to monotony properties of s_{δ}.

The last three equalities of the statement follow easily from the continuity of ρ_{Ω} and standard properties of the exponential map.

2.2. A distance on convex bounded domains

Definition 2.2. We set d(Ω_{1}, Ω_{2}) the metric defined on C by

d(Ω_{1}, Ω_{2}) = ∥ ln(\frac{ρ_{Ω_{1}}}{ρ_{Ω_{2}}} )∥_{∞}.

Proposition 2.1. (C, d) is a proper metric space (i.e. every closed and bounded subset of X is a compact set).

Proof. Let (Ω_{i})_{i∈ℕ} be a bounded sequence in C. Since there exist r and R such that B(x_{0}, r) ⊂ Ω_{i} ⊂ B(x_{0}, R) for every i ∈ ℕ, the functions ρ_{Ω_{i}} : S_{x_{0}} → [r, R] are equicontinuous (thanks to Lemma 2.1). Consequently, the sequence (ρ_{Ω_{i}})_{i∈ℕ} converges uniformly on S_{x_{0}} to a function ρ_{∞}, up to an extraction. Since r ≤ ρ_{∞} ≤ R, we also

have that lim_{i→∞} ∥ ln(ρ_{Ω_{i}}/ρ_{∞})∥_{∞} = 0. We set Ω_{∞} = {exp_{x_{0}}(t.v)/ v ∈

S_{x_{0}}, t ∈ [0, ρ_{∞}(v)[}, which is a bounded, star-shaped domain. The equality ρ_{Ω_{∞}} = ρ_{∞} holds because ρ_{∞} is continuous and exp_{x_{0}} is a diffeomorphism of a neighbourhood of B(0, R) onto a neighbourhood of B(x_{0}, R). It remains to prove that Ω_{∞} is convex.

Let y_{1} and y_{2} be any pair of points in Ω_{∞}. There exists only one minimizing geodesic γ from y_{1} to y_{2} in X^{δ} (Ω_{∞} is an open set of a hemisphere in the case δ=1). Since y_{1} and y_{2} are in Ω_{j} for all j large enough, we easily infer that γ ⊂ \overline{Ω}_{∞} = \{exp_{x_{0}}(t.v)/ v ∈ S_{x_{0}}, t ∈ [0, ρ_{∞}(v)]}. So for any r > 0 small enough, the union of the minimizing geodesic from y_{1} (resp. from y_{2}) to a point of B(y_{2}, r) (resp. of B(y_{1}, r)) is contained in \overline{Ω}_{∞}. Since y_{1} (resp. y_{2}) is in the injectivity domain of y_{2} (resp. y_{1}), the union of this two sets is an open neighbourhood of γ contained in \overline{Ω}_{∞} and the result is proved.

Corollary 2.1. For any R ≥ r > 0, the set of convex sets Ω in C such that B(x_{0}, r) ⊂ Ω ⊂ B(x_{0}, R) is a compact set.

3. Continuity of the volume and the eigenvalues

As proved in [9], any weak solution in H_{0}^{1}(Ω) of Δu = λu on a convex (in fact Lipschitzian) domain Ω belongs to C^{∞}(Ω)∩C^{0}(\overline{Ω}) and is equal to 0 on ∂Ω. Moreover, the Dirichlet spectrum of any open subset Ω of finite volume in X^{δ} is discrete [17, Corollary 10.10]. In this case, all the eigenvalues (λ_{k}(Ω)_{k∈ℕ^{∗}} satisfy the min-max principle below.

λ_{k}(Ω) = inf{m(E)/ E subspace of C_{c}^{∞}(Ω), dim E = k}

where m(E) = sup_{f∈E} \frac{∫_{Ω} |∇f|^{2}}{∫_{Ω} f^{2}} .

We will say that an arbitrary open set Ω has a spectral gap if λ_{1}(Ω) < λ_{2}(Ω) (where λ_{1}(Ω) and λ_{2}(Ω) are defined by the min-max principle). This implies that λ_{1}(Ω) is an eigenvalue of the Dirichlet problem and always occurs when the volume is finite.

Proposition 3.1. For any k ≥ 1, the following inequalities hold

|ln(\frac{λ_{k}(Ω_{1})}{λ_{k}(Ω_{2})})| ≤ Λ_{δ}[d(Ω_{1}, Ω_{2}), R]

and

|ln(\frac{Vol Ω_{1}}{Vol Ω_{2}} )| ≤ Λ′_{δ}[d(Ω_{1}, Ω_{2}), R],

where

Ω_{1} ∪ Ω_{2} ⊂ B(x_{0}, R), Λ_{δ}(s, t) = ln[e^{2s}(\frac{e^{2s}s_{δ}(te^{−2s})}{s_{δ}(t)})^{δ(n−1)}],

Λ′_{1}(s, t) = Λ′_{0}(s, t) = ns and Λ′_{−1} = ln[e^{ns}(\frac{e^{−s} sinh(t)}{sinh(e^{−s}t)})^{n−1}].

Proof. In the case δ = 1, we denote by y_{0} the antipodal point of x_{0} in 𝕊^{n}. For λ ∈]0, 1], we define the map

H_{λ} : X^{δ} (resp. X^{1} \ {y_{0}}) → X^{δ}  (1)
exp_{x_{0}}(tv) ↦ exp_{x_{0}}(λtv)

Set d = d(Ω_{1}, Ω_{2}). Since H_{e^{−d}}(Ω_{1}) ⊂ Ω_{2} we just have to bound the quotient λ_{k}(H_{λ}(Ω_{1}))/λ_{k}(Ω_{1}) for λ = e^{−d}.

For that purpose, we define a linear injective map Φ_{λ} : C_{c}^{∞}(Ω) → C_{c}^{∞}(H_{λ}(Ω)) by Φ_{λ}(f) = f ∘H_{1/λ}. Easy computations involving Jacobi fields give

λ( inf_{t∈]0,R]} \frac{s_{δ}(λt)}{s_{δ}(t)})^{n−1}∥f∥_{1} ≤ ∥Φ_{λ}(f)∥_{1} ≤ λ( sup_{t∈]0,R]} \frac{s_{δ}(λt)}{s_{δ}(t)})^{n−1}∥f∥_{1}

\frac{|d(Φ_{λ}(f))|^{2}(x)}{Φ_{λ}(|df|^{2})(x)} ≤ max(\frac{1}{λ^{2}} , \frac{s_{δ}^{2}(d(x_{0}, x)/λ)}{s_{δ}^{2}(d(x_{0}, x))})

The first inequality applied to f ≡ 1 gives the volume estimate. The two inequalities imply

\frac{∥d(Φ_{λ}(f))∥_{2}^{2}}{∥Φ_{λ}(f)∥_{2}^{2}} ≤ e^{Λ_{δ}(d,R)} \frac{∥df∥_{2}^{2}}{∥f∥_{2}^{2}} .

Using the min-max principle, we obtain

λ_{k}(Ω_{2}) ≤ λ_{k}(H_{e^{−d}}(Ω_{1})) ≤ e^{Λ_{δ}(d,R)}λ_{k}(Ω_{1}).

4. Extremal convex domains

4.1. Schwarz symmetrization on noncompact domains

The aim of this paragraph is to recall some basic properties of the Schwarz symmetrization. However we will not assume as usual, that the domain to be symmetrized is bounded. To replace this property, some additional assumptions on the functions to be symmetrized are sometimes needed.

Definition 4.1 (Schwarz symmetrization). Let f be a nonnegative function defined on an open set Ω in the space form X^{δ}. Let μ_{f} be the distribution function defined for s ≥ 0, by μ_{f}(s) = vol({f > s}) and let V : r ↦ V ol(B(r)) (r ≥ 0). The nonincreasing Schwarz symmetrization of f is

f^{∗} = μ_{f}^{♯} ∘ V ∘ d_{x_{0}},

where d_{x_{0}}(x) = d(x_{0}, x) and .^{♯} refers to the right inverse function of a nonincreasing function (i.e. u^{#}(s) = inf{t ≥ 0/u(t) ≤ s}). If the volume of Ω is finite, the Schwarz nondecreasing symmetrization of f is defined by

f_{∗} = μ_{f}^{♯} ∘ H ∘ d_{x_{0}},

where H : r ↦ vol (Ω) − V (r).

These symmetrized functions satisfy

μ_{f^{∗}} = μ_{f_{∗}} = μ_{f}.  (2)

Remark 4.1. For more details on symmetrization, we refer to [8,12,5].

Proposition 4.1. Let Ω be an open set of finite volume in the space form X^{δ}.

If u is in L^{2}(Ω) then u^{∗} is in L^{2}(Ω^{∗}) and

||u||_{L^{2}(Ω)} = ||u^{∗}||_{L^{2}(Ω^{∗})}.  (3)

In addition, the following inequalities hold

∫_{Ω_{∗}} f_{∗}g^{∗} ≤ ∫_{Ω} fg ≤ ∫_{Ω_{∗}} f^{∗}g^{∗}  (4)

for every nonnegative measurable functions f, g on Ω.

If u is now in H_{0}^{1}(Ω) then u^{∗} ∈ H_{0}^{1}(Ω^{∗}) and

∫_{Ω_{∗}} |∇u^{∗}|^{2} ≤ ∫_{Ω} |∇u|^{2}.  (5)

Proof. The proof of the statement (3) is an immediate consequence of (2), the inequality (4) is easy to check for simple functions and the general case follows by density [12]. The proof of (5) also relies on an approximation argument, a suitable dense subset is introduced in the lemma below. The assumption on the volume is then used to conclude, using Rellich’s Theorem on the symmetrized ball and the following inequality which is a direct consequence of (3) and (4).

||u^{∗} − v^{∗}||_{L^{2}(Ω^{∗})} ≤ ||u − v||_{L^{2}(Ω)}.

Lemma 4.1. Let f be a smooth nonnegative function in H_{0}^{1}(Ω), which is zero on ∂Ω and in C^{0}(\overline{Ω}), where Ω is an open set of finite volume in X^{δ}. Suppose that the level sets of f are compact sets (except maybe {f = 0}) of measure zero. Under these assumptions, μ_{f}^{♯} is absolutely continous, the symmetrized function f^{∗} is in H_{0}^{1}(Ω^{∗}) and satisfies (5).

Moreover, in case of equality in (5), the open set {f > 0} is a ball.

Proof. Let Reg(f) be the set of regular points of f which are included in {x ∈ Ω; f > 0}. By assumption, Reg(f) is an open set of full measure in {x ∈ Ω; f > 0}. As a consequence, we deduce that f^{∗} is continuously differentiable on an open set of full measure of {f^{∗} > 0} and satisfies the inequality (5) thanks to the coarea formula and the isoperimetric inequality (we refer to [6] for more details). We conclude that {f > 0} is a ball using a decreasing sequence of regular values which goes to 0 and the case of equality in the isoperimetric inequality.

Remark 4.2. The set of functions which satisfy the assumptions of the lemma above contains the smooth functions with compact support and only nondegenerate critical points, therefore it is dense in H_{0}^{1}(Ω) (see [6] and references herein).

Remark 4.3. In the sequel, we will use the Schwarz symmetrization on convex domains of ℍ^{n} whose the volume is not assumed to be finite. A priori, the nondecreasing Schwarz symmetrization cannot be defined in this setting, however the inequality

∫_{Ω_{∗}} f_{∗}g^{∗} ≤ ∫_{Ω} fg

remains true for a function f = F ∘ d_{x_{0}}, where F is a nonnegative and nondecreasing bounded function such that F is constant outside

a compact set, if we define f_{∗} as f_{∗}(x) = \{ \begin{matrix}(f|_{Ω ∩B(x_{0},r)})_{∗} & if |x| < r \\ ||f||_{∞} & otherwise \end{matrix}

for r large enough.

4.2. Faber-Krahn Inequality

In this section, we extend the Faber-Krahn inequality from the setting of smooth bounded domains to the setting of convex sets of finite volume. The main interest of the result below is the characterization of the case of equality. Proposition 4.2 (Faber-Krahn Inequality). Let Ω be a convex set in X^{δ} of finite volume V_{0}. The first Dirichlet eigenvalue of Ω satisfies

λ_{1}(Ω) ≥ λ_{1}^{∗}(V_{0})

where λ_{1}^{∗}(V_{0}) denotes the first Dirichlet eigenvalue of a geodesic ball with volume V_{0}. Moreover, the equality λ_{1}(Ω) = λ_{1}^{∗}(V_{0}) implies that Ω is isometric to a geodesic ball. Remark 4.4. The inequality can be proved using smooth approximations of Ω. However, the characterization of the case of equality without assuming that Ω is bounded, is crucial in our proof of Theorem 1.1, when δ = −1. Even when the domain is bounded, some regularity on the boundary is needed to deduce the case of equality. Indeed each ball with closed sets of capacity zero removed, satisfies the case of equality. Proof. The proof of the inequality follows from Proposition 4.1 and does not rely on the convexity of Ω. As the volume of Ω is assumed to be finite, the Dirichlet spectrum of Ω is discrete [17, Corollary 10.10] and the eigenfunctions belong to C^{∞}(Ω)∩C^{0}(\overline{Ω}) [9, Corollary 8.11 and theorem 8.29]. To prove the case of equality, it is sufficient to prove that the first eigenfunction (denoted by f_{1}) satisfies the assumptions of Lemma 4.1, which is a consequence of the lemma below. Indeed, thanks to this lemma and Sard’s Theorem, the set of singular values of f_{1} is a closed set of measure zero. Then, thanks to the fact that the function Δf_{1} = λ_{1}f_{1} is positive on Ω, we deduce that each level set of the first eigenfunction is of measure zero. Remark 4.5. Let us remark that the assumption on the finiteness of the volume is used only to prove that the bottom of the spectrum is an eigenvalue. It is also true for the lemma below; we will use this fact in Paragraph 4.3. Lemma 4.2. Under the assumptions of Proposition 4.2, the first Dirichlet eigenfunction f_{1} on Ω can be assumed to be positive and proper: for all s > 0, the set f_{1}^{−1}([s, +∞[) is a compact set. Proof. By the maximum principle, we can suppose f_{1} to be positive. To prove the second assertion, set y_{0} be a fixed point of Ω, R ≥ 1 and x_{0} ∈ Ω \ B(y_{0}, 2R). Recall (see for instance [7]) that there exists a constant C(n) such that,

∀x_{0} ∈ X^{δ}, ∀v ∈ H_{0}^{1}(B(x_{0}, 1)), ∥v∥2_{\frac{2n}{n−2}} ≤ C(n)∥dv∥_{2}^{2}. (∗)

Note that in dimension n = 2, this inequality has to be replaced by ∥v∥_{4}^{2} ≤ C∥dv∥_{2}^{2} in what follows. A standard Moser’s iteration gives

f_{1}^{2}(x_{0}) ≤ A(n)(1 + λ_{1})^{γ(n)} ∫_{B(x_{0},1)} f_{1}^{2}  (6)

(where A(n) and γ(n) are constants that depend only on the dimension n, see appendix B for a proof), and from which we infer that

sup_{Ω\B(y_{0},2R)} f_{1}^{2} ≤ A(n)(1 + λ_{1})^{γ(n)} ∫_{Ω\B(y_{0},R)} f_{1}^{2}.

This gives the compactness of the sets f_{1}^{−1}([s, +∞[) for all s > 0 since ∫_{Ω\B(y_{0},R)} f_{1}^{2} → 0 when R → ∞ and f_{1} is continuous on the convex set \overline{Ω} and is equal to 0 on ∂Ω.

4.3. Payne-Polya-Weinberger Inequality

M.S. Ashbaugh and R.D. Benguria proved the Payne-Polya-Weinber-ger conjecture for smooth bounded sets of Euclidean space and smooth sets included in a hemisphere of the sphere [1,2]. We need to extend this inequality to (possibly non smooth) convex sets in the space form X^{δ} (δ ∈ {0, 1}). Theorem 4.1 (Payne-Polya-Weinberger Inequality). Let Ω be a convex set of finite volume V_{0} in X^{δ} (δ ∈ {0, 1}). Under these assumptions, the following inequality is satisfied,

\frac{λ_{2}}{λ_{1}} (Ω) ≤ \frac{λ_{2}^{∗}}{λ_{1}^{∗}} (V_{0}).

Moreover, the equality is achieved if and only if Ω is isometric to a geodesic ball. Remark 4.6. Actually, as in [1,2], the monotony properties of the ratio λ_{1}(B)/λ_{2}(B) with respect to the radius of the geodesic ball B of X^{δ} make this theorem a direct corollary of the following result. Theorem 4.2. Let Ω be a convex set in X^{δ} such that Ω ≠ X^{δ}. Then the spectral gap of Ω is smaller or equal to λ_{2}(B)−λ_{1}(B) (where B is a geodesic ball such that λ_{1}(B) = λ_{1}(Ω)). If the spectral gap is equal to λ_{2}(B) − λ_{1}(B), Ω is isometric to a geodesic ball.

Let us remark that contrary to the cases δ ∈ {0, 1}, the assumptions in Theorem 4.2 do not imply an upper bound on the volume of Ω in ℍ^{n}.

We will prove Theorems 4.1 and 4.2 simultaneously. The scheme of the proof is the same as in [1,2,4], so we will mainly focuse on the extra arguments needed in our setting. The fisrt step of the proof is the following proposition.

Proposition 4.3. Let Ω be an open subset of X^{δ} (included in a hemisphere if δ = 1) with a spectral gap, u_{1} an eigenfunction of Ω for the first eigenvalue and g be a positive, piecewise C^{1} function on [0, ∞[ (and with lim inf_{+∞} g > 0 if Ω is not bounded). Then, there exists a point x_{m} ∈ X^{δ} such that

λ_{2}(Ω) − λ_{1}(Ω) ≤ \frac{∫_{Ω} b(d(x_{m}, y))u_{1}^{2}(y) dy}{∫_{Ω} g^{2}(d(x_{m}, y))u_{1}^{2}(y) dy}

where b = g^{′2} + \frac{n−1}{s_{δ}^{2}}g^{2}.

Note that for the proof of Theorem 4.2, we can suppose that the spectral gap is non zero.

Proof. The min-max principle implies that

λ_{2}(Ω) − λ_{1}(Ω) ≤ \frac{∫_{Ω} |∇P |^{2}u_{1}^{2}}{∫_{Ω} P^{2}u_{1}^{2}} ,

for every non-zero function P such that P u_{1} is in H_{0}^{1}(Ω) and ∫_{Ω} P u_{1}^{2} = 0.

The next step consists in choosing n suitable test functions. For that purpose, we need the following lemma which extends a result of [1,2,4] (the proof is postponed to appendix C).

Lemma 4.3. For any u ∈ L^{2}(X^{δ}) (with support in a hemisphere if δ = 1) and any g : ℝ^{+} → ℝ^{+} continuous (bounded and with lim inf_{+∞} g > 0 if u has not compact support), there is x ∈ X^{δ} such that

∫_{X^{δ}} g(d(x, y))\frac{exp_{x}^{−1}(y)}{d(x, y)}u^{2}(y) dy = 0_{T_{x}X^{δ}}

In order to construct test functions, we apply this lemma to u = u_{1}.1_{Ω} and g a nonnegative, increasing and bounded function (g will be specified later). The functions P_{i} = g(r)X_{i}, where (r, X_{i}) are the geodesic coordinates at the point x_{m} given by Lemma 4.3, satisfy ∫_{Ω} P_{i}u_{1}^{2} = 0 for every i. To conclude the proof of Proposition 4.3, we just have to sum the n inequations given by the min-max principle applied to the P_{i} ,and note that ∑_{i} P_{i}^{2} = g^{2} and ∑_{i} |∇P_{i}|^{2} = b.

Now, we choose g a radial function such that the equality below holds.

λ_{2}(B) − λ_{1}(B) = \frac{∫_{B} bz^{2}}{∫_{B} g^{2}z^{2}} ,

where z is a positive first eigenfunction of B. It is shown in [1,2,4] that we can choose g positive, nondecreasing and constant outside B and such that b is radial, positive and nonincreasing. We recall that B is such that λ_{1}(B) = λ_{1}(Ω). It remains to compare the spectral gaps. For that purpose, we first use properties of the Schwarz symmetrization (Proposition 4.1). We get

∫_{Ω} b u_{1}^{2} ≤ ∫_{Ω_{∗}} b^{∗}u_{1}^{∗2} ≤ ∫_{Ω_{∗}} b u_{1}^{∗2}

∫_{Ω} g u_{1}^{2} ≥ ∫_{Ω_{∗}} g_{∗}u_{1}^{∗2} ≥ ∫_{Ω_{∗}} g u_{1}^{∗2}.

The inequality involving the nonincreasing Schwarz symmetrization is valid without any assumption on the volume, thanks to remark 4.3. We conclude using the Chiti comparison result, which allows to compare z with u_{1}^{∗} on B. This comparison result is valid as soon as the first eigenfunction u_{1} satisfies the assumptions of Lemma 4.1 (this has been established in the proof of Proposition 4.2 and does not rely on any assumption on the volume), we refer to [1, pages 21-24] for more details. Using the Chiti comparison result, we get [1, page 607]

∫_{Ω_{∗}} g u_{1}^{∗2} ≥ ∫_{B} g z^{2} and ∫_{Ω_{∗}} b u_{1}^{∗2} ≤ ∫_{B} b z^{2}

and this concludes the proof of the inequality. The case of equality follows from the characterization of the equality in the Chiti comparison result.

5. Coercivity of the functional λ_{1}

We show that λ_{1} is coercive on appropriate subsets of C. We first need a control of the in-radii of elements of C.

5.1. In-radius estimate in C

For any bounded domain Ω in X^{δ}, let Inrad (Ω) be the maximum radius of a geodesic ball included in \overline{Ω} . Proposition 5.1. Let Ω be a bounded convex set in X^{δ}. Then

Inrad (Ω) ≥ \frac{π}{2\sqrt{λ_{1}(Ω) + (n − 1)}}.

This proposition has been proved by P. Li and S.T. Yau [13] for smooth domains of nonnegative mean curvature (see appendix D for a proof). It can be readily extended to any (non smooth) convex domains: indeed, for any ϵ > 0 small enough, there exists a smooth convex domain Ω_{ϵ} such that H_{1−ϵ}(Ω) ⊂ Ω_{ϵ} ⊂ H_{1+ϵ}(Ω), where H is the map defined by (1), p. 8 (see [10, Lemma 2.3.2] for the Euclidean case and use the Klein projective model of the hyperbolic space and the open hemisphere, to infer this property in ℍ^{n} and 𝕊^{n}). The continuity of λ_{1} on C allows to conclude.

5.2. Coercivity of λ_{1}

Subsequently, we denote by C_{V_{0}} the set of convex bounded domains Ω of X^{δ} with Vol Ω = V_{0} and B(x_{0}, Inrad (Ω)) ⊂ \overline{Ω} (note that C_{V_{0}} contains, up to isometry, all convex bounded domains of X^{δ} with volume V_{0}).

Combining Corollary 2.1 and Proposition 5.1, we get

Corollary 5.1. For any M > 0, the set of elements Ω of C (resp. C_{V_{0}}) with Ω ⊂ B(x_{0}, M) and λ_{1}(Ω) ≤ M is compact.

5.2.1. case δ=1. Corollary 5.1 shows the compactness of the set {Ω ∈ C_{V_{0}}/ λ_{1}(Ω) ≤ M}. This implies that λ_{1} is coercive. Actually, C_{V_{0}} itself is compact (see section 6).

5.2.2. Case δ=0. In this case, {Ω ∈ C_{V_{0}}/ λ_{1}(Ω) ≤ M} is also a compact set and so λ_{1} is coercive. Indeed by Proposition 5.1, a convex domain Ω in this set contains the ball B(x_{0}, \frac{π}{2\sqrt{M+n−1}}). Set y ∈ Ω such that d(x_{0}, y) = diam Ω/4. Since Ω is convex, it contains the convex hull of B(x_{0}, \frac{π}{2\sqrt{M}}) ∪ \{y\} whose volume must be bounded from above by V_{0}. We deduce that diam Ω is bounded from above by a function of M and V_{0}. We conclude by Corollary 5.1.

5.2.3. Case δ=−1. We cannot argue as easily as in the previous case because in ℍ^{n}, the volume of the convex hull of B(x_{0}, \frac{π}{2\sqrt{M+n−1}})∪\{y\} does not tend to ∞ with d(x_{0}, y). We will prove simultaneously the coercivity of λ_{1} and the property

inf_{C′} λ_{1}(Ω) > λ_{1}^{∗}(V_{0}),  (7)

where C′ = {Ω unbounded convex sets; vol(Ω) = V_{0}}.

These two facts prove Theorem 1.1. First, we need to establish some lemmata.

Lemma 5.1. Let Ω be a domain of a complete Riemannian manifold (M^{n}, g). Then for any R ≥ 1, α, γ ∈]0, 1[ and y_{0} ∈ M, we have

min[λ_{1}(Ω ∩ B(y_{0}, R)), λ_{1}(Ω \ B(y_{0}, γR))]

≤ \frac{1}{(1 − R^{−α})^{2}} [λ_{1}(Ω) + \frac{8}{(1 − γ)^{2}R^{2(1−α)}} ]

where λ_{1} stands for the bottom of the spectrum, Ω can be of infinite volume and we have set λ_{1}(∅) = ∞.

Proof. The proof relies on the variational characterization of the first eigenvalue.

We set N = E(R^{α}) + 1, B_{r} = B(y_{0}, r), A_{r,r′} = Ω ∩ (B_{r} \ B_{r′}) and r_{k} = γR + (1 − γ)R\frac{k}{N} for any integer 0 ≤ k ≤ N. Then, for any u ∈ H_{0}^{1}(Ω), we have

∫_{Ω} u^{2} ≥ ∑_{k=0}^{N−1} ∫_{A_{r_{k+1}, r_{k}}} u^{2} ≥ N ∫_{A_{r_{k_{0}+1}, r_{k_{0}}}} u^{2}

for at least one integer 0 ≤ k_{0} ≤ N − 1. Let ϕ and ψ be the two functions defined on ℝ^{+} by:

–ϕ is non-decreasing, ϕ = 0 on [0, \frac{r_{k_{0}}+r_{k_{0}+1}}{2} ], ϕ = 1 on [r_{k_{0}+1}, ∞[ and ∥∇ϕ∥_{∞} ≤ \frac{2N}{(1−γ)R},

–ψ is non-increasing, ψ = 1 on [0, r_{k_{0}}], ψ = 0 on [\frac{r_{k_{0}}+r_{k_{0}+1}}{2} , ∞[ and ∥∇ψ∥_{∞} ≤ \frac{2N}{(1−γ)R},

For g(x) = ψ(d(y_{0}, x))u(x) and h(x) = ϕ(d(y_{0}, x))u(x), we have

∫_{Ω∩B_{\frac{r_{k_{0}} +r_{k_{0}+1}}{2}}} g^{2} + ∫_{Ω\B_{\frac{r_{k_{0}} +r_{k_{0}+1}}{2}}} h^{2} = ∫_{Ω}(g + h)^{2}

≥ ∫_{Ω} u^{2} − ∫_{A_{r_{k_{0}+1}, r_{k_{0}}}} u^{2} ≥ \frac{N − 1}{N} ∫_{Ω} u^{2}

Since |dg + dh|^{2} = |(ψ + ϕ)du + ud(ψ + ϕ)|^{2}, we obtain

∫_{Ω∩B_{\frac{r_{k_{0}} +r_{k_{0}+1}}{2}}} |dg|^{2} + ∫_{Ω\B_{\frac{r_{k_{0}} +r_{k_{0}+1}}{2}}} |dh|^{2} = ∫_{Ω} |dg + dh|^{2}

≤ (1 + R^{−α}) ∫_{Ω}(ϕ + ψ)^{2}|du|^{2} + (1 + R^{α}) ∫_{Ω} u^{2}|dϕ + dψ|^{2}

≤ (1 + R^{−α}) ∫_{Ω} |du|^{2} + (1 + R^{α}) ∫_{A_{r_{k_{0}+1}, r_{k_{0}}}} u^{2} \frac{4N^{2}}{(1 − γ)^{2}R^{2}}

≤ (1 + R^{−α}) ∫_{Ω} |du|^{2} + (1 + R^{α}) ∫_{Ω} u^{2} \frac{4N}{(1 − γ)^{2}R^{2}} .

We infer

min[λ_{1}(Ω ∩ B(y_{0}, R)), λ_{1}(Ω \ B(y_{0}, γR))]

≤ min(\frac{∫_{Ω∩B_{\frac{r_{k_{0}} +r_{k_{0}+1}}{2}}} |dg|^{2}}{∫_{Ω∩B_{\frac{r_{k_{0}} +r_{k_{0}+1}}{2}}} g^{2}} , \frac{∫_{Ω\B_{\frac{r_{k_{0}} +r_{k_{0}+1}}{2}}} |dh|^{2}}{∫_{Ω\B_{\frac{r_{k_{0}} +r_{k_{0}+1}}{2}}} h^{2}} )

≤ \frac{∫_{Ω∩B_{\frac{r_{k_{0}} +r_{k_{0}+1}}{2}}} |dg|^{2} + ∫_{Ω\B_{\frac{r_{k_{0}} +r_{k_{0}+1}}{2}}} |dh|^{2}}{∫_{Ω∩B_{\frac{r_{k_{0}} +r_{k_{0}+1}}{2}}} g^{2} + ∫_{Ω\B_{\frac{r_{k_{0}} +r_{k_{0}+1}}{2}}} h^{2}}

≤ \frac{1}{(1 − R^{−α})^{2}} [\frac{∫_{Ω} |du|^{2}}{∫_{Ω} u^{2}} + \frac{8}{(1 − γ)^{2}R^{2(1−α)}} ]

This lemma implies the following result.

Lemma 5.2. For any V_{0} > 0 there exist C(V_{0}, n) > λ_{1}^{∗}(V_{0}) and R(V_{0}, n) > 0 such that, for any bounded convex set Ω which satisfies vol(Ω) ≤ V_{0} and λ_{1}(Ω) ∈ [λ_{1}^{∗}(V_{0}), C(V_{0}, n)], we have

λ_{1}(Ω) ≤ λ_{1}(Ω ∩ B(x_{0}, R)) ≤ \frac{1}{(1 − R^{−α})^{2}} [λ_{1}(Ω) + \frac{32}{R^{2(1−α)}} ]

for any α ∈]0, 1[, R ≥ R(V_{0}, n, α) and x_{0} such that B(x_{0}, Inrad(Ω))⊂ \overline{Ω}.

Proof. We set r(V_{0}, n) = _{\frac{π}{\sqrt{2λ_{1}^{∗}(V_{0})+n−1}}} and

C(V_{0}, n) = min[2λ_{1}^{∗}(V_{0}), \frac{λ_{1}^{∗}(V_{0}) + λ_{1}^{∗}(V_{0} − Vol B(x_{0}, r(V_{0}, n)/2))}{2}].

Then, by Proposition 5.1, we have B(x_{0}, r(V_{0}, n)/2) ⊂ Ω and so Vol (Ω\B(x_{0}, R/2)) ≤ V_{0}−Vol B(x_{0}, r(V_{0}, n)/2) for any R ≥ r(V_{0}, n). By the Faber-Krahn inequality, this implies that λ_{1}(Ω \ B(x_{0}, R/2)) is larger than λ_{1}^{∗}(V_{0} − Vol B(x_{0}, r(V_{0}, n)/2)). Now, we can choose

R(V_{0}, n) large enough in order to have \frac{1}{(1−R^{−α})^{2}} [C(V_{0}, n)+ \frac{32}{R^{2(1−α)}} ] ≤

λ_{1}(Ω \ B(x_{0}, R/2)) for any R ≥ R(V_{0}, n). Lemma 5.1 then applies.

Now, we prove (simultaneously) the coercivity of λ_{1} and (7). By definition of the bottom of the spectrum, it is sufficient to prove that every sequence of bounded convex domains (Ω_{i})_{i∈ℕ} such that vol(Ω_{i}) ≤ V_{0} and lim_{i} λ_{1}(Ω_{i}) = λ_{1}^{∗}(V_{0}), converges, up to isometries and extraction, to B(x_{0}, r_{0}) (vol(B(x_{0}, r_{0})) = V_{0}).

Let (Ω_{i})_{i∈ℕ} be such a sequence. Up to isometries, we can suppose that the fixed point x_{0} ∈ ℍ^{n} satisfies the condition B(x_{0}, Inrad(Ω_{i})) ⊂ Ω_{i} for every i. By the lemma above and Corollary 5.1, the sequence (Ω_{i} ∩ B_{R})_{i∈ℕ} is precompact in C for all R ≥ R(V_{0}, n). Up to a diagonal extraction, we can now suppose that for any n ∈ ℕ the sequence (Ω_{i} ∩ B_{n})_{i∈ℕ} converges to an element U_{n} of C. Using the continuity of λ_{1} and the volume on C, we get

λ_{1}^{∗}(V_{0}) ≤ λ_{1}(U_{n}) ≤ \frac{1}{(1 − n^{−1/2})^{2}} [λ_{1}^{∗}(V_{0}) + \frac{32}{n}]

and Vol (U_{n}) ≤ V_{0}

So λ_{1}(U_{n}) tends to λ_{1}^{∗}(V_{0}) and by the Faber-Krahn inequality, we must have Vol U_{n} → V_{0}. Moreover, (U_{n})_{n∈ℕ} is a nondecreasing sequence of convex sets for the inclusion. As a consequence, Ω = ⋃_{n} U_{n} is a convex domain of volume V_{0} and first eigenvalue λ_{1}(Ω) = λ_{1}^{∗}(V_{0}). By Proposition 4.2, Ω = B(x_{0}, r_{0}) and we infer that the sequence (Ω_{i})_{i∈ℕ} converges to B(x_{0}, r_{0}) in C.

6. Coercivity of the λ_{1}/λ_{2} functional

6.1. case δ = 0

We show that, on C_{V_{0}}, λ_{1}/λ_{2} tends to 1 when λ_{1} tends to ∞. By section 5.2.2, inf_{C_{V_{0}}} λ_{1}/λ_{2} < 1 and the fact that λ_{1}/λ_{2} is invariant under homothetie on the domains, this implies Theorem 1.4 in ℝ^{n}.

By a classical result due to Jones, for any Ω ∈ C_{V_{0}} there exists an ellipsoid E such that E ⊂ Ω ⊂ \sqrt{n}E. We easily infer that there is a n-rectangle R with edges of lengths L_{1} ≤ ⋯ ≤ L_{n}, such that R ⊂ Ω ⊂ nR. This gives

λ_{1}(Ω) ≤ λ_{1}(R) ≤ \frac{nπ^{2}}{L_{1}^{2}} and V_{0} ≤ L_{n}^{n−1}n^{n}L_{1},  (8)

and so L_{n} ≥ (\frac{V_{0}\sqrt{λ_{1}}}{πn^{n+ \frac{1}{2}}} )^{\frac{1}{n−1}} . Following [11], we can translate and rotate Ω so that R be centred in (0, …, 0) and the edge of R of length L_{n} be parallel to the last coordinate axis. We denote Ω(y) = {x ∈ ℝ^{n−1}/ (x, y) ∈ Ω} and λ(y) = λ_{1}(Ω(y)). Then, if f is an eigenfunction of Ω associated to the first eigenvalue, we have

∫_{Ω} f^{2} = ∫_{Ω} \frac{|∇f|^{2}}{λ_{1}(Ω)} ≥ ∫_{ℝ} ∫_{Ω(y)} \frac{|∇_{x}f(x, y)|^{2}}{λ_{1}(Ω)} dxdy

≥ ∫_{ℝ} \frac{λ_{1}(Ω(y))}{λ_{1}(Ω)} ∫_{Ω(y)} |f(x, y)|^{2}dxdy

Thus, there is y such that λ_{1}(Ω) ≥ λ_{1}(Ω(y)). By convexity of Ω we deduce that (1 − (\frac{L_{n}}{2} )^{− \frac{2}{3}} )Ω(y) × [y − (\frac{L_{n}}{2} )^{\frac{1}{3}} , y + (\frac{L_{n}}{2} )^{\frac{1}{3}} ] is contained in Ω and consequently,

λ_{1}(Ω) ≤ λ_{2}(Ω) ≤ λ_{2}((1 − (\frac{L_{n}}{2} )^{− \frac{2}{3}} )Ω(y) × [y − (\frac{L_{n}}{2} )^{\frac{1}{3}} , y + (\frac{L_{n}}{2} )^{\frac{1}{3}} ])

≤ \frac{λ_{1}(Ω(y))}{(1 − (\frac{L_{n}}{2} )^{− \frac{2}{3}} )^{2}} + \frac{2π^{2}}{(\frac{L_{n}}{2} )^{\frac{2}{3}}} ≤ \frac{λ_{1}(Ω)}{(1 − (\frac{L_{n}}{2} )^{− \frac{2}{3}} )^{2}} + \frac{2π^{2}}{(\frac{L_{n}}{2} )^{\frac{2}{3}}} . (9)

Since we have shown above that L_{n} → ∞ when λ_{1} → ∞, we obtain that λ_{1}/λ_{2} tend to 1 when λ_{1} tends to ∞.

Remark 6.1. The same method could be used to show that for any integers p ≤ q, λ_{p}/λ_{q} tends to 1 when λ_{1} tends to ∞ on C_{V_{0}}. We conlude that for any p ≤ q there exists a convex domain (to be determined) which minimizes the quotient λ_{p}/λ_{q}.

Remark 6.2. The inequations (9) imply that for any convex domain Ω of ℝ^{n}, x_{0} ∈ Ω and R > 0 such that B(x_{0}, R) does not contain Ω, we have

λ_{1}(Ω ∩ B(x_{0}, R)) ≤ λ_{2}(Ω ∩ B(x_{0}, R)) ≤ \frac{λ_{1}(Ω ∩ B(x_{0}, R))}{(1 − C(n)R^{− \frac{2}{3}} )^{2}} + \frac{C(n)}{R^{\frac{2}{3}}}

and so λ_{2}(Ω ∩ B(x_{0}, R)) tends to λ_{1}(Ω) when R tends to ∞. We conclude that a convex Euclidean domain with a spectral gap is bounded (hence has a discrete spectrum) and that its diameter is bounded from above by C(n)(\frac{1+λ_{1}}{λ_{2}−λ_{1}} )^{3/2}. This implies readily the coercivity of λ_{1}/λ_{2} on the set of convex Euclidean domains of fixed λ_{1}, from which we infer Theorem 1.5 in ℝ^{n}.

6.2. Case δ = 1

The coercivity λ_{1}/λ_{2} on the set of convex domains with λ_{1} = λ follows from Lemma 5.1. On C_{V_{0}}, this follows from the compactness of C_{V_{0}} which, by Corollary 5.1, is a consequence of the inequality Inrad (Ω) ≥ C(n)Vol (Ω). This inequality holds true for any convex domain of 𝕊^{n} as explained below.

First, using the inequality (8), based on the Jones ellipsoid, we get easily that for any convex domain contained in a geodesic ball of radius R in ℝ^{n}, we have Vol (Ω) ≤ n^{n}R^{n−1}Inrad (Ω). Now, since 𝕊^{n} can be covered by 2(n + 1) balls of radius R_{n} = arccos(\frac{1}{\sqrt{n+1}}) we infer that there is a point x_{0} in 𝕊^{n} such that Vol (Ω ∩ B(x_{0}, R_{n})) ≥ \frac{1}{2(n+1)}Vol (Ω). Using the canonical embedding of 𝕊^{n} in ℝ^{n+1}, we can project B(x_{0}, R_{n}) onto the tangent space T_{x_{0}}𝕊^{n} (using the origin of Euclidean space). This map P_{0} is a quasi-isometry from the ball B(x_{0}, R_{n}) in 𝕊^{n} to the geodesic ball B(x_{0}, \sqrt{n}) in Euclidean space, which preserves the convexity. Then, we have Inrad_{𝕊^{n}} (Ω) ≥ Inrad_{𝕊^{n}} (Ω ∩ B(x_{0}, R_{n}))

≥ C_{1}(n)Inrad_{T_{x_{0}}𝕊^{n}} (P_{0}(Ω ∩ B(x_{0}, R_{n})))

≥ C_{2}(n)Vol_{T_{x_{0}}𝕊^{n}}(P_{0}(Ω ∩ B(x_{0}, R_{n}))) ≥ C(n)Vol_{𝕊^{n}}(Ω)

6.3. Case δ = −1.

In this section, we prove simultaneously the coercivity of the functional λ_{1}/λ_{2} on bounded convex domains whose first eigenvalue is fixed and the property

sup_{C′} λ_{2}(Ω) < λ_{2}^{∗}(λ),  (10)

where C′ = {Ω unbounded convex sets; λ_{1}(Ω) = λ}. These two properties imply Theorem 1.5.

We need the following result whose proof follows easily from the min-max principle (see [16, theorem XIII.1]). Lemma 6.1. Let Ω be a convex domain in ℍ^{n} such that the bottom of the spectrum is an eigenvalue. Then for any fixed point x_{0} ∈ ℍ^{n}, we have

lim_{R→∞} λ_{i}(Ω ∩ B(x_{0}, R)) = λ_{i}(Ω), for i = 1, 2.

Thanks to this lemma, the coercivity property and the inequality (10) reduce to the fact below. Every sequence (Ω_{i})_{i∈ℕ} ∈ C such that lim_{i} λ_{1}(Ω_{i}) = λ and lim_{i} λ_{2}(Ω_{i}) = λ_{2}^{∗}(λ), converges (up to extraction) to a ball such that λ_{1}(B) = λ.

First, we show that a lower bound on the spectral gap implies some estimates on the first eigenfunction. Lemma 6.2. Let Ω be a bounded domain of ℍ^{n}. If u ∈ H_{0}^{1}(Ω) satisfies △ u = λ_{1}(Ω)u then there is a point x_{m} ∈ ℍ^{n} such that

(λ_{2}(Ω) − λ_{1}(Ω) − \frac{n−1}{sinh^{2}(R)}) ∫_{Ω\B(x_{m},R)} u^{2} ≤ \frac{n}{R^{2}} ∫_{Ω∩B(x_{m},R)} u^{2}

for any R > 0. This implies, for any R ≥ 2\sqrt{\frac{n−1}{λ_{2}(Ω)−λ_{1}(Ω)}},

λ_{1}(Ω) ≤ λ_{1}(Ω ∩ B(x_{m}, R))

≤ \frac{(1 + \frac{1}{R^{2}} )}{1 − \frac{4n}{(λ_{2}(Ω)−λ_{1}(Ω))R^{2}+4})}(λ_{1}(Ω) + \frac{4n}{(λ_{2}(Ω) − λ_{1}(Ω))R^{2} + 4}).

Proof. Proposition 4.3 applied to g(s) = s/R on [0, R] and g(s) = 1 on [R, ∞[ gives a point x_{m} ∈ ℍ^{n} such that

λ_{2}(Ω)−λ_{1}(Ω) ≤ \frac{\frac{n}{R^{2}} ∫_{Ω∩B(x_{m},R)} u^{2}(x)dx + \frac{n−1}{sinh^{2} R} ∫_{Ω\B(x_{m},R)} u^{2}(x)dx}{∫_{Ω\B(x_{m},R)} u^{2}(x)dx}

which gives the first estimate.

For the second estimate, we set ψ the non-increasing Lipschitzian function defined on ℝ^{+} by ψ = 1 on [0, R/2], ψ = 0 on [R, ∞[ and ∥∇ψ∥∞ = \frac{2}{R}. Then, we have

|d(ψu)|^{2} = ψ^{2}|du|^{2} + 2uψ(dψ, du) + u^{2}|dψ|^{2}
≤ (1 + \frac{1}{R^{2}} )ψ^{2}|du|^{2} + (1 + R^{2})|dψ|^{2}u^{2}

So, we infer

λ_{1}(Ω ∩ B(x_{m}, R)) ≤ \frac{∫_{Ω∩B(x_{m},R)} |d(ψu)|^{2}}{∫_{Ω∩B(x_{m},R)}(ψu)^{2}}

≤ (1 + \frac{1}{R^{2}} )\frac{∫_{Ω} ψ^{2}|du|^{2}}{∫_{Ω∩B(x_{m},R/2)} u^{2}} + (1 + R^{2})\frac{∫_{Ω} |dψ|^{2}u^{2}}{∫_{Ω∩B(x_{m},R/2)} u^{2}}

≤ (1+ \frac{1}{R^{2}} )\frac{∫_{Ω} |du|^{2}}{∫_{Ω} u^{2}} (1+ \frac{∫_{Ω\B(x_{m},R/2)} u^{2}}{∫_{Ω∩B(x_{m},R/2)} u^{2}} )+4(1+ \frac{1}{R^{2}} )\frac{∫_{Ω\B(x_{m},R/2)} u^{2}}{∫_{Ω∩B(x_{m},R/2)} u^{2}} .

By the first estimate, we have

\frac{∫_{Ω\B(x_{m},R/2)} u^{2}}{∫_{Ω∩B(x_{m},R/2)} u^{2}} ≤ \frac{4n}{(λ_{2}(Ω) − λ_{1}(Ω))R^{2} − 4(n − 1)}

from which we infer

λ_{1}(Ω ∩ B(x_{m}, R)) ≤

\frac{(1 + \frac{1}{R^{2}} )((λ_{2}(Ω) − λ_{1}(Ω))R^{2} + 4)}{(λ_{2}(Ω) − λ_{1}(Ω))R^{2} − 4(n − 1)}(λ_{1}(Ω)+\frac{4n}{(λ_{2}(Ω) − λ_{1}(Ω))R^{2} + 4})

Let (Ω_{i})_{i∈ℕ} ∈ C such that lim_{i} λ_{1}(Ω_{i}) = λ and lim_{i} λ_{2}(Ω_{i}) = λ_{2}^{∗}(λ). We can assume that λ_{2}(Ω_{i}) − λ_{1}(Ω_{i}) > \frac{λ_{2}^{∗}(λ)−λ}{2} > 0. Note that by the preceeding lemma, we infer that for any R ≥ 4\sqrt{\frac{n−1}{λ_{2}^{∗}(λ)−λ}} we

have λ_{1}(Ω_{i} ∩ B(x_{m}^{i}, R)) ≤ C(λ, n) (where C(λ, n) is a universal funtion and x_{m}^{i} is the center of mass of Ω_{i}). This implies, by Proposition 5.1, that we can suppose (up to isometry) x_{m}^{i} ∈ B(x_{0}, 4\sqrt{\frac{n−1}{λ_{2}^{∗}(λ)−λ}}) and B(x_{0}, r(λ, n)) ⊂ Ω_{i} for all i. Then, the sequence (Ω_{i} ∩ B(x_{0}, R)) is included in a compact set of C (see Corollary 5.1). By diagonal extraction, we can suppose that for any k ∈ ℕ the sequence (Ω_{i} ∩ B(x_{0}, k))_{i∈ℕ} converges to an element U_{k} of C. By continuity of λ_{1} on C, we have

λ ≤ λ_{1}(U_{k}) = lim_{i} λ_{1}(Ω_{i} ∩ B(x_{0}, k))

≤ lim_{i} λ_{1}(Ω_{i} ∩ B(x_{m}^{i}, k − 4\sqrt{\frac{n − 1}{λ_{2}^{∗}(λ) − λ}})) ≤ f(k, λ, n)

where f(k, λ, n) is a universal function given by the preceding lemma and that converge to λ when k tends to ∞. So λ_{1}(U_{k}) tends to λ. As in the subsection 5.2.3, we set Ω = ∪_{k}U_{k}. Then Ω is a convex domain with λ_{1}(Ω) = lim_{k} λ_{1}(U_{k}) = λ (since U_{k} = Ω ∩B(x_{0}, k)) and

λ_{2}(Ω) = lim_{k} λ_{2}(U_{k}) = lim_{k} lim_{i} λ_{2}(Ω_{i}∩B(x_{0}, k)) ≥ lim_{i} λ_{2}(Ω_{i}) = λ_{2}^{∗}(λ).

Then, we conclude by Theorem 4.2.

Appendices

A. Trigonometric computations

In this appendix, we perform the computations of l′(0) and L′(0) used in the proof of Lemma 2.1. We denote by β the half angle at y_{0} of the geodesic double cone tangent to the ball B(x_{0}, r). By the law of

sines, we have sin β = \frac{s_{δ}(r)}{s_{δ}(d(x_{0},y_{0}))} and ^{\frac{s_{δ}(l(t))}{sin β}} = ^{\frac{s_{δ}(l_{1}(t))}{sin t}}, where we

have set l_{1}(d(u_{0}, v)) = d(x_{0}, Z(v)). By letting t tend to 0, we get

l′_{1}(0) = ^{\frac{s_{δ}^{2}(d(x_{0},y_{0}))}{s_{δ}(r)}}. On the other hand, the cosine law gives us the equation c_{δ}(l) = c_{δ}(l_{1})c_{δ}(d(x_{0}, y_{0})) + δs_{δ}(l_{1})s_{δ}(d(x_{0}, y_{0})) cos β (resp. l^{2} = l_{1}^{2} + (d(x_{0}, y_{0}))^{2} − 2l_{1}d(x_{0}, y_{0}) cos β if δ = 0), whose derivative at t = 0 gives the relation l′(0) = −l′_{1}(0) cos β. We easily deduce

the relation l′(0) = −s_{δ}(d(x_{0}, y_{0}))\sqrt{(^{\frac{s_{δ}(d(x_{0},y_{0}))}{s_{δ}(r)}})^{2} − 1}. Note that for

L′(0) we just have to replace β by π − β in what preceed.

B. Moser’s iteration

In this section, we prove the inequality (6) used in the proof of lemma 4.2.

Set 0 ≤ η ≤ 1 a C^{1} function such that η ≡ 1 on B(x_{0}, αr) (for α ∈]0, 1[ and 1 ≥ r > 0), η ≡ 0 on X^{δ} \B(x_{0}, r) and |dη| ≤ 2/(1−α)r.

We fix m > 0 and β ≥ 0 and set h = inf(m, f_{1}), u = f_{1}h^{\frac{β}{2}} and ϕ = η^{2}h^{β}f_{1} ∈ H_{0}^{1}(Ω). Then we have

λ_{1} ∫_{Ω} η^{2}u^{2} = λ_{1} ∫_{Ω} ϕf_{1} ≥ ∫_{Ω}(df_{1}, dϕ)

≥ β ∫_{Ω} η^{2}h^{β}|dh|^{2} + \frac{1}{2} ∫_{Ω} η^{2}h^{β}|df_{1}|^{2} − 2 ∫_{Ω} |dη|^{2}h^{β}f_{1}^{2},

where we used 2ηf_{1}(df_{1}, dη) ≥ −\frac{1}{2}η^{2}|df_{1}|^{2} − 2f_{1}^{2}|dh|^{2}. This inequality, combined with the inequalities |d(ηu)|^{2} ≤ 2u^{2}|dη|^{2} + 2η^{2}|du|^{2} and |du|^{2} ≤ (1 + β)h^{β}(2β|dh|^{2} + |df_{1}|^{2}), gives

∫_{Ω} |d(uη)|^{2} ≤ (10 + 4λ_{1})(1 + β) ∫_{Ω} u^{2}(η^{2} + |dη|^{2}).

Hence the Sobolev inequality (∗) applied to uη implies

(∫_{B(x_{0},αr)} h^{\frac{(2+β)n}{n−2}} )^{\frac{n−2}{n}} ≤ \frac{5C(n)(10 + 4λ_{1})(1 + β)}{(1 − α)^{2}r^{2}} ∫_{B(x_{0},r)} f_{1}^{(2+β)}

Then, we let m tend to ∞ and set r_{k} = \frac{1}{2^{\sqrt{k}}}, α_{k} = 2^{\sqrt{k}−\sqrt{k+1}} and

β_{k} = 2(\frac{n}{n−2})^{k} − 2. By multiplying the (2 + β_{k})-th square root of the inequalities obtained for 1 ≤ k ≤ K − 1 we infer

(∫_{B(x_{0},r_{K})} h^{2( \frac{n}{n−2} )^{K}})^{\frac{1}{2(n/n−2)^{K}}} ≤ A(n, K)(1 + λ_{1})^{γ(K)} ∫_{B(x_{0},r)} f_{1}.

By definition of r_{K}, we have [∫_{B(x_{0},r_{K})} f^{2( \frac{n}{n−2} )^{K}}]^{\frac{1}{2(n/n−2)^{K}}} tends to f(x_{0}) when K tends to +∞, meanwhile A(n, K) and γ(K) converge, which gives (6).

C. Proof of lemma 4.3

This lemma is essentially proven in [1,2,4] for u with compact support (which includes the case δ = 1) but we need to apply it to an eigenfunction u of a convex (unbounded) domain Ω ,and so we have to extend it in the case δ = 0, −1.

In the sequel of the proof, X denotes ℝ^{n} or ℍ^{n}. We fix x_{0} ∈ X and define

F : T_{x_{0}}X → T_{x_{0}}X
v ↦ d(exp_{x_{0}}^{−1})(∫_{X} g(d(v̄, y))\frac{exp−1 ; v̄(y)}{d(v̄, y)}u^{2}(y) dy)

where we have set v̄ = exp_{x_{0}}(v). We set m = lim inf_{+∞} g. Let R_{1} > 0 such that ∫_{X\B(x_{0},R_{1})} u^{2} ≤ min(\frac{m}{32∥g∥_{∞}} , \frac{1}{2}). Then for any v ∈ T_{x_{0}}X with |v| ≥ R_{1} we easily have

|F (v) − d(exp_{x_{0}}^{−1})(∫_{B(x_{0},R_{1})} g(d(v̄, y))\frac{exp−1 ; v̄(y)}{d(v̄, y)}u^{2}(y) dy)| ≤ \frac{m}{32}.

Note that d(exp_{x_{0}}^{−1}) ∘ exp−1 ; v̄(x_{0}) = −v and so we infer that for any v ∈ T_{x_{0}}X with |v| ≥ R_{1} we have

|F (v) + λ(v)\frac{v}{|v|}| ≤ ∥g∥_{∞} ∫_{B(x_{0},R_{1})}|\frac{exp−1 ; v̄(x_{0})}{d(v̄, x_{0})} − \frac{exp−1 ; v̄(y)}{d(v̄, y)}| dy + \frac{m}{32}

where we have set λ(v) = ∫_{B(x_{0},R_{1})} g(d(v̄, y))u^{2}(y) dy and used the fact that d(exp_{x_{0}}^{−1}) is a contraction. Then we have λ(v) ≥ \frac{m}{4} > 0 for any v with |v| ≥ R_{2} ≥ R_{1}. Note also that the integrand above measures the difference between the unit tangent vectors at v̄ to the minimizing geodesic from v̄ to x_{0} and y ∈ B(x_{0}, R_{1}). By the law of cosines, we can easily show that this quantity uniformly tends to zero on B(x_{0}, R_{1}) when |v| tends to +∞. Hence, there exists R_{3} > 0 such that for any v ∈ T_{x_{0}}X which satisfies |v| ≥ R_{3}, we have

|F (v) + λ(v)\frac{v}{|v|}| ≤ \frac{m}{16} and λ(v) ≥ \frac{m}{4}

We have to show that F is zero somewhere. If not, the following map (with R > R_{3}) is well-defined.

G : B(0, 1) ⊂ T_{x_{0}}X → S(0, 1) ⊂ T_{x_{0}}X
v ↦ \frac{F (−Rv)}{|F (−Rv)|}.

Moreover, the map G is continuous and satisfies

|G(v) − v| ≤ ^{\frac{2|F (−Rv)+λ(−Rv)|}{|F (−Rv)|}} ≤ 4/3 for any v ∈ S(0, 1). So, we could then easily construct a retraction from B(0, 2) to S(0, 2).

D. A result of Li and Yau

Lemma D.1 (Li-Yau). Let Ω be a bounded and smooth domain with positive mean curvature (for the exterior normal). If f an eigenfunction associated to the first eigenvalue of the Dirichlet problem on Ω, then we have,

|∇f|^{2} ≤ λ_{1}(∥f∥_{∞}^{2} − f^{2})

(resp.

|∇f|^{2} ≤ (λ_{1} + n − 1)(∥f∥_{∞}^{2} − f^{2})

if δ = −1).

Proof. Let F = \frac{|∇f|^{2}}{β−f^{2}} where β = (1 + ϵ)∥f∥_{∞}^{2}. Then we have

dF (v) = \frac{2|∇f|^{2}}{β − f^{2}} (\frac{Hess f(\frac{∇f}{|∇f|}, v)}{|∇f|} + \frac{fdf(v)}{β − f^{2}}).

If x_{0} is a point of ∂Ω then ν = ∇f(x_{0})/|∇f(x_{0})| is well-defined (by the strong maximum principle applied to f) and is the interior normal to Ω at x_{0}. We then have

dF (ν) = \frac{2|∇f|^{2}}{β − f^{2}} (\frac{Hess f(ν, ν)}{|∇f|} + \frac{f|∇f|}{β − f^{2}} ) ≥ 0,

since \frac{Hess f(ν,ν)}{|∇f|} = −\frac{△ f}{|∇f|} +μ(x_{0}), where μ(x_{0}) is the mean curvature of ∂Ω at x_{0}. We infer by the strong maximum principle that at a point x_{0} where F reaches its maximum on Ω̄ we must have

dF (x_{0}) = 0  and  △ F (x_{0}) ≥ 0

The first equation and our computation of dF imply that ∇f/|∇f| is an eigenvector of Hess f(x_{0}) with respect to g(x_{0}), associated to the eigenvalue −\frac{f|∇f|^{2}}{β−f^{2}} (we have ∇f(x_{0}) ≠ 0 since F ≠ 0). So we have |Hess f(x_{0})|^{2} ≥ f^{2}F^{2}.

From the Bochner formula \frac{1}{2}△ |∇f|^{2} = λ_{1}|∇f|^{2} − |Hess f|^{2} − Ric (∇f, ∇f) (where Ric denote the Ricci curvature tensor of X^{δ}) we infer that, at x_{0}, we have

|∇f|^{2}F − f△ fF + \frac{(β − f^{2})}{2}△ F = \frac{1}{2}△ ((β − f^{2})F )

≤ λ_{1}|∇f|^{2} − f^{2}F^{2} − δ(n − 1)|∇f|^{2}.

Since △ F (x_{0}) ≥ 0 and |∇f|^{2} = F (β − f^{2}) we readily obtain the estimate F (x_{0}) ≤ λ_{1} (resp. F (x_{0}) ≤ λ_{1} + n − 1 if δ = −1). Then, we just have to let ϵ tend to 0.

To get Proposition 5.1 in the case of a smooth convex domain, let f denote a positive eigenfunction associated to λ_{1} and z_{0} ∈ Ω a point where f(z_{0}) = ∥f∥_{∞}. Set γ a normal geodesic from z_{0} to a point y ∈ ∂Ω. By lemma D, we obtain

(arcsin(f ∘ γ/∥f∥_{∞}))′ ≥ −\sqrt{λ_{1} + δ(n − 1)}

and so that f ∘ γ(t) ≥ ∥f∥_{∞} cos(\sqrt{λ_{1} + δ(n − 1)}t). Since f(y) = 0, the geodesic ball B(z_{0}, \frac{π}{2\sqrt{λ_{1}+δ(n−1)}}) is included in Ω.

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Erwann AUBRY Université de Nice Sophia-Antipolis Laboratoire J.-A. Dieudonné UMR6621 (UNSA-CNRS) Parc Valrose F-06108 Nice Cedex (France) eaubry@math.unice.fr

Jérôme BERTRAND Scuola Normale Superiore Piazza dei Cavalieri, 7 I-56100 Italia j.bertrand@sns.it

Bruno COLBOIS Institut de mathématiques Université de Neuchâtel Rue Émile Argand, 11 Case postale 158 CH-2009 Neuchâtel bruno.colbois@unine.ch