The second Robin eigenvalue in non-compact rank-1 symmetric spaces

The Robin eigenvalue problem is to solve

where $\Omega$ is a bounded domain with Lipschitz boundary, $\Delta$ denotes the Laplace-Beltrami operator, $\nu$ denotes the outward unit normal to $\partial\Omega$, and $\alpha\in\mathbb{R}$ is the Robin parameter. The eigenvalues, denoted by $\lambda_{k,\alpha}(\Omega)$ for $k=1,2,\cdots$, are increasing and continuous in $\alpha$, and for each $\alpha$ satisfy

where each eigenvalue is repeated according to its multiplicity. The first eigenvalue is simple if $\Omega$ is connected; the first eigenfunction is positive. The Robin eigenvalue problem generates a global picture of the spectrum of the Laplace operator. Indeed, the Neumann $(\alpha=0)$, the Steklov ($\lambda=0$ and $-\alpha$ equals the Steklov eigenvalue) and the Dirichlet ($\alpha\to+\infty)$ eigenvalue problems are all special cases of the Robin eigenvalue problem. Hence, existing results on the Dirichlet, Neumann or Steklov eigenvalues naturally motivate the investigation on the Robin eigenvalues.

In Euclidean space, the classical Faber-Krahn inequality asserts that the ball uniquely minimizes the first Dirichlet eigenvalue among bounded domains with the same volume. When $\alpha>0$, the ball is also the unique minimizer of the first Robin eigenvalue $\lambda_{1,\alpha}(\Omega)$ among domains of the same volume in $\mathbb{R}^{n}$, as was shown in dimension two by Bossel [5] in 1986 and extended to all dimensions $n\geq 2$ by Daners [12] in 2006. An alternative approach via the calculus of variations was found by Bucur and Giacomini [9, 10] later. For negative values of $\alpha$, it was conjectured by Bareket [3] in 1977 that the ball would be the maximizer among domains in $\mathbb{R}^{n}$ with the same volume. However, in 2015, Freitas and Krejčiřik [14] disproved Bareket’s conjecture by showing that the ball is not a maximizer for sufficiently negative values of $\alpha$. In the same paper, the authors showed that in dimension two, the disk uniquely maximizes $\lambda_{1,\alpha}(\Omega)$ for $\alpha<0$ with $|\alpha|$ sufficiently small, and conjectured that the maximizer still has radial symmetry whenever $\alpha<0$ and should switch from a ball to a shell at some critical value of $\alpha$.

Let us turn to the shape optimization problem for the second Robin eigenvalue $\lambda_{2,\alpha}(\Omega)$. Suppose for the moment that $\Omega\subset\mathbb{R}^{n}$. When $\alpha>0$, both the second Dirichlet eigenvalue and the second Robin eigenvalue are uniquely minimized by the disjoint union of two equal balls among bounded Lipschitz domains of the same volume. This was proved by Kennedy [18]. When $\alpha=0$, we have $\lambda_{2,0}(\Omega)=\mu_{1}(\Omega)$, the first nonzero Neumann eigenvalue, for which the classical Szegö-Weinberger inequality [20, 21] states that among domains with the same volume, the ball uniquely maximizes $\mu_{1}(\Omega)$, see [7] for stability version as well. When $\alpha<0$, it is expected, cf. [17, Problem 4.41], that $\lambda_{2,\alpha}(\Omega)$ should be maximal on the ball for a range of Robin parameters. This expectation has recently been confirmed by Freitas and Laugesen via Szegö way [15] and Weineberger way [16] respectively. Precisely the following theorem holds true.

Inequality (1.2) asserts that the ball uniquely maximizes $\lambda_{2,\alpha}(\Omega)$ among domains of the same volume provided that $\alpha$ lies in a regime connecting the first nonzero Neumann eigenvalue $\mu_{1}$ and the first nonzero Steklov eigenvalue $\sigma_{1}$, namely $\alpha\in[-\frac{n+1}{n}R^{-1},0]$. Taking $\alpha=0$ and $\alpha=-1/R$ recovers the Szegö-Weinberger inequality [22] for $\mu_{1}(\Omega)$ and the Brock-Weinstock inequality [8] for $\sigma_{1}(\Omega)$ respectively, and both classical inequalities assert that the ball is the unique maximizer among domains with the same volume in Euclidean space. We mention that the stability version of the Brock-Weinstock inequality has been proved in [6].

It is known that the Szegö-Weinberger inequality holds for domains in the hemisphere, in the hyperbolic space [2] and in rank-1 symmetric spaces (ROSS) [1], asserting that the geodesic ball maximises the first nonzero Neumann eigenvalues among domains of the same volume. Binoy and Santhanam proved [4] the Brock-Weinstock inequality in hyperbolic space and non-compact ROSS, and Castillon and Ruffini proved [11] a stability version of the Brock-Weinstock inequality in non-compact ROSS and that the Brock-Weinstock inequality does not hold on sphere in general. Recently, in their work [19], the authors have extended the result of Freitas and Laugesen [16] to bounded domains in real hyperbolic spaces, and proved that in complete simply connected nonpositively curved space forms, geodesic balls uniquely maximize the second Robin eigenvalue among domains with the same volume, with negative Robin parameter in the regime connecting the first nontrivial Neumann and Steklov eigenvalues.

In this paper, we continue our study of the optimization problem for the second Robin eigenvalue for some negative Robin parameter [19], and prove the following quantitative Freitas-Laugesen inequality in non-compact ROSS.

By taking $\alpha=0$, Theorem 1.2 yields that geodesic balls uniquely maximize the first nonzero Neumann eigenvalue among domains of the same volume in noncomapct ROSS, recovering Theorem 2 of [1] proved by Aithal and Santhanam. By taking $\alpha=-\sigma_{1}(B)$, Theorem 1.2 implies that geodesic balls uniquely maximize the first nonzero Steklov eigenvalue among domains of the same volume in noncomapct ROSS, which recovers the result of Binoy and Santhanam in [4], and the result of Castillon and Ruffini in [11]. Noticing from [11] that the Brock-Weinstock inequality does not hold on sphere in general, so we mention here that Theorem 1.2 does not hold in compact space forms or compact ROSS.

This paper is organized as follows. In Section 2, we collect some known facts on non-comapct ROSS. In Section 3, we prove some properties of second Robin eigenvalues and eigenfuctions for geodesic balls in noncomapct ROSS. In Section 4, we prove a monotonicity lemma, which is useful in the proof of Theorem 1.2. Section 5 is devoted to the proof of Theorem 1.2.

Let $\mathbb{K}$ denote one of the following: the field $\mathbf{R}$ of real numbers, the field $\mathbf{C}$ of complex numbers, and the algebra $\mathbf{H}$ of quaternions or or the algebra $\mathbf{Ca}$ of octonions. Let $M^{m}=\mathbb{K}H^{n}$, a non-compact ROSS of dimension $m=kn$, where $k=\operatorname{dim}_{\mathbb{R}}(\mathbb{K})$. It is known that $M$ carries $k-1$ orthogonal complex structures $J_{1},\cdots,J_{k-1}$, and for any unit vector $X,Y\in T_{x}M$ with $Y$ orthogonal to $X,J_{1}X,\cdots,J_{k-1}X$ we have

where $1\leq i\leq k-1$ and $\operatorname{Rm}$ is the Riemannian curvature operator.

Now we collect some known facts about geodesic polar coordinates, see for example [1, Sect. 3]. In geodesic polar coordinates centered at a point $p\in M$, the Riemannian density function $J(r)$ is given by

where $r$ is the distance function to $p$. The Laplace operator $\Delta_{M}$ is given by

where

the mean curvature of the distance sphere $S_{r}:=\{x\in M:\operatorname{dist}(p,x)=r\}$, and $\Delta_{S_{r}}$ denotes the Laplacian of $S_{r}$. Moreover, the first non-zero eigenvalue of $\Delta_{S_{r}}$ is

and the associated eigenfunctions are the linear coordinate functions restricted to $\mathbb{S}^{m-1}$, denoted by $\psi_{i}(\theta)$ ($1\leq i\leq m$), satisfying

see Lemma 4.11 of [11] for details.

The theory of self-adjoint operators yields variational characterizations of Laplacian eigenvalues. In particular, the first two Robin eigenvalues are characterized by

and

where $u_{1}$ is the first eigenfunction associated with $\lambda_{1,\alpha}(\Omega)$, and the first nonzero Steklov eigenvalue $\sigma_{1}(\Omega)$ is characterized variationally by

where $d\mu_{g}$ is the Riemnnian measure induced by the metric $g$ and $dA_{g}$ is the induced measure on $\partial\Omega$.

Let $M$ be a non-compact ROSS and $B_{R}\subset M$ be a geodesic ball of radius $R$, and we consider the following Robin eigenvalue problem

Since $\lambda_{1,\alpha}$ is simple and $B_{R}$ is rotational symmetric, then the first eigenfunction is radial, hence $\lambda_{1,\alpha}(B_{R})$ is the first eigenvalue of

with $f^{\prime}(0)=0$ and $f^{\prime}(R)+\alpha f(R)=0$. By using the separation of variables technique, we see that the second eigenvalue $\lambda_{2,\alpha}(B_{R})$ of Problem (3.4) is either the second eigenvalue of (3.5) or the first eigenvalue of

with $g(0)=0$ and $g^{\prime}(R)+\alpha g(R)=0$, where $\lambda_{1}(S_{r})$ is given by (2.2). Moreover it’s easily seen that the first eigenvalue of (3.6) is characterized variationally by

and the associated eigenfunction $g(r)$ can be so chosen that $g(r)>0$ for $r\in(0,R]$ and $g^{\prime}(0)=1$.

We prove the following properties of the second eigenvalue and eigenfunctions of (3.4) for negative Robin parameter.

Relabelling the solution to (3.8) as $g_{1}$, we define (with a slight abuse of notation) the function $g:[0,\infty)\to[0,\infty)$ by

By definition, $g$ is continuously differentiable and non-decreasing on $(0,\infty)$. The following monotonicity lemma plays a key role in the proof of Theorem 1.2.