1 Introduction

Let $(M, g_{_M})$ be a Riemannian manifold and let D denote the Levi-Civita connection of $(M, g_{_M})$ . For a smooth vector field X on M, the divergence of X denoted as $\text {div}(X)$ is defined as $trace(DX)$ . For a smooth function $f : M \rightarrow \mathbb {R}$ , the gradient of f at a point $p \in M$ denoted as $(\nabla f)(p)$ is defined by $\langle \nabla f (p), v \rangle _{g_{_M}} = df (p)(v)$ for all $v \in T_pM$ . Here, $T_pM$ is the tangent space to M at the point p. The Laplace–Beltrami operator $\Delta $ is defined by $\Delta f = \text {div}(\nabla f )$ . The Riemannian manifolds, we consider in this article, are going to be space forms, that is, the complete simply connected Riemannian manifolds with constant sectional curvature. It is well known that they are one of the following spaces: (i) the Euclidean space $\mathbb {E}^n$ , (ii) the unit sphere $S^n = \{(x_1, x_2, \ldots ,x_{n+1}) \in \mathbb {R}^{n+1} \, | \, \sum _{i=1}^{n+1} x_i^2 = 1\}$ with induced Riemannian metric $\langle \cdot , \cdot \rangle $ from the Euclidean space $\mathbb {R}^{n+1}$ , and (iii) the hyperbolic space ${\mathbb {H}^n := \{(x_1, x_2, \ldots ,x_{n+1}) \in \mathbb {R}^{n+1} \,| \, \sum _{i=1}^n x_i^2 - x_{n+1}^2 = -1 ~\mbox { and }~ x_{n+1}> 0\}}$ with the Riemannian metric induced from the bilinear form $(x, y) := \sum _{i=1}^n x_i\, y_i - x_{n+1}\, y_{n+1}$ . Here, $ x = (x_1, x_2,\ldots ,x_{n+1}), y = (y_1, y_2,\ldots ,y_{n+1}) \in \mathbb {R}^{n+1}$ . For details about the standard geometric terminologies used in this article, the reader may refer to [Reference do Carmo13].

Finding sharp bounds for Steklov–Dirichlet eigenvalues among a constrained family of doubly connected domains is an interesting problem which has caught attention of many authors [Reference Hong, Lim and Seo21, Reference Michetti25] owing to its physical significance. In this article, we consider a mixed Steklov–Dirichlet eigenvalue problem on domains with a spherical hole. Let $\Omega _{out}$ denote a bounded domain in the Riemannian manifold $ M, $ with smooth boundary denoted by $\partial \Omega _{\mathrm{out}} $ . Let $B_{r} \subset M $ be a ball of radius ${r}$ such that $\overline {B_{r}} \subset \Omega _{\mathrm{out}} $ . Consider the following Steklov–Dirichlet eigenvalue problem on $\Omega := \Omega _{\mathrm{out}} \setminus \overline {B_{r}}:$

(1.1) $$ \begin{align} \begin{cases} \Delta u=0 & \text{in} \,\, \Omega= \Omega_{\mathrm{out}} \setminus \overline{B_{r}},\\ u=0 & \text{on}\,\, \partial B_{r},\\ \frac{\partial u}{\partial \nu }=\sigma u & \text{on} \,\, \partial \Omega_{\mathrm{out}}. \end{cases} \end{align} $$

It is known (cf. [Reference Agranovich1]) that the boundary value problem (1.1) has a discrete spectrum $0 < \sigma _1 \leq \sigma _2 \leq \sigma _3 \leq \cdots \nearrow \infty $ , counted with multiplicity. For each $1 \leq k < \infty $ , the k-th eigenvalue of (1.1), i.e., $\sigma _{k}$ , admits the following variational characterization:

(1.2) $$ \begin{align} \sigma_{k}(\Omega)=\min_{E\in \mathcal{H}_{k, 0}(\Omega)} \, \max_{u(\neq 0) \in E} \frac{ \int_{\Omega} \| \nabla u \|^2 \,dv}{ \int_{\partial \Omega_{\mathrm{out}}} u^2 \,ds}, \end{align} $$

where ${\mathcal {H}_{k,0}(\Omega )}$ is the collection of all the k-dimensional subspaces of the Sobolev space $\tilde {H}_{0}^1 (\Omega ):=\{ u \in H^1(\Omega ) : u=0 \,\,\text {on} \,\, \partial B_{r}\}$ . The expression $\displaystyle R(u):=\frac {\int _{\Omega } \| \nabla u \|^2\, dv}{ \int _{\partial \Omega _{\mathrm{out}}} u^2 \,ds}$ is called a Rayleigh quotient. If v is an eigenfunction corresponding to a Steklov Dirichlet eigenvalue $\sigma $ , then $\displaystyle \sigma = R(v)= \frac { \int _{\Omega } \| \nabla v \|^2 \,dv}{\int _{\partial \Omega _{\mathrm{out}}} v^2 \,ds}$ .

Various bounds for the first Steklov–Dirichlet eigenvalue relating the geometry of the underlying domains have been obtained for bounded domains in $\mathbb {R}^{2}$ . Using the theory of conformal mapping, Hersch and Payne [Reference Hersch and Payne19] obtained sharp upper bounds for the first Steklov–Dirichlet eigenvalue on planar doubly connected bounded domains. In [Reference Dittmar and Solynin12], using circular symmetrization and a distortion theorem for conformal mappings of an annulus, lower bounds for the first Steklov–Dirichlet eigenvalue were derived for doubly connected planar domains. For bounded domains in $\mathbb {R}^{n}$ , Banuelos et al. [Reference Bañuelos, Kulczycki, Polterovich and Siudeja6] obtained some classical inequalities comparing eigenvalues of the mixed Steklov–Dirichlet problem and mixed Steklov–Neumann problem.

Many isoperimetric bounds and monotonicity results for the first Steklov–Dirichlet eigenvalue have been derived for various domains in $\mathbb {R}^{n}$ [Reference Gavitone and Piscitelli16, Reference Hong, Lim and Seo20]. For instance, in [Reference Verma and Santhanam30], it was proved that, if $n \geq 3$ and $\Omega _{\mathrm{out}}$ is a ball of fixed radius, the first Steklov–Dirichlet eigenvalue $\sigma _{1}(\Omega )$ is maximum when $ \Omega _{\mathrm{out}}$ and $B_{r}$ are concentric. This result was later proved for planar annular domains and for domains contained in two-point homogeneous spaces [Reference Ftouhi14, Reference Seo27]. Generalizing this, Paoli et al. [Reference Paoli, Piscitelli and Sannipoli26] proved that concentric annular domain locally minimizes first Steklov–Dirichlet eigenvalue when $\Omega _{\mathrm{out}}$ varies over nearly spherical sets of fixed volume. Further, if $\Omega _{\mathrm{out}}$ is an open, bounded, and convex set contained in a suitable ball, Gavitone et al. [Reference Gavitone, Paoli, Piscitelli and Sannipoli15] proved that, under a volume constraint, $\sigma _{1}(\Omega )$ is maximum for concentric annular domain. In particular, they proved the following result.

Theorem 1.1 [Reference Gavitone, Paoli, Piscitelli and Sannipoli15, Theorem 1.1]

Fix $n \geq 2$ . Let $\Omega _{\mathrm{out}}$ be an open bounded convex set in $\mathbb {R}^{n}$ . Let $B_r$ be a ball in $\mathbb {R}^{n}$ of radius r such that $\overline {B_r} \subset \Omega _{\mathrm{out}}\subset B_{\bar {r}}$ , where $B_{\bar {r}} $ is the ball in $\mathbb {R}^{n}$ , centered at the origin, with radius $\bar {r}$ defined in terms of r and n, as

$$ \begin{align*} \bar{r} = \begin{cases} \begin{aligned} & r e^{\sqrt{2}} & \text{ if } n=2,\\ & r \left[ \frac{(n-1) + (n-2) \sqrt{2(n-1)}}{n-1} \right]^{\frac{1}{n-2}} & \text{ if } n \geq 3. \end{aligned} \end{cases} \end{align*} $$

Then, $\sigma _1 (\Omega _{\mathrm{out}}\backslash \overline {B_r} ) \leq \sigma _1 (B_{R,r})$ , where $B_{R,r}$ is the concentric annular domain with inner radius r and outer radius R such that Vol $(\Omega _{\mathrm{out}}\backslash \overline {B_r})$ = Vol $(B_{R,r})$ .

In [Reference Gavitone, Paoli, Piscitelli and Sannipoli15], the authors have also obtained sharp lower and upper bounds for $\sigma _{1}(\Omega )$ in terms of the minimal and maximal distances between the center of $\Omega _{\mathrm{out}}$ and its boundary $\partial \Omega _{\mathrm{out}}$ . Similar results have been proved for Steklov eigenvalues on star-shaped domains contained in unit hemisphere [Reference Verma29].

In this article, we generalize the above-mentioned results in three directions:

(1) First eigenvalue on star-shaped domains in non-Euclidean space forms: The aim is to find sharp lower and upper bounds for the first Steklov–Dirichlet eigenvalue of doubly connected star-shaped domains in non-Euclidean space forms (Theorem 2.1). These bounds are proved using a similar approach to that used in [Reference Gavitone, Paoli, Piscitelli and Sannipoli15], with appropriate changes due to the difference in the geometries of the non-Euclidean space forms.

(2) Higher eigenvalues on symmetric Euclidean domains (Isoperimetric Inequality): The aim is to find optimal shapes for the higher Steklov–Dirichlet eigenvalues (Theorem 5.1) under a volume constraint and with certain symmetry assumptions on Euclidean domains. To prove this result, we plug suitable test functions into the variational characterization of the eigenvalue and in order to use this characterization, we develop a substantial list of auxiliary results concerning the behavior of a few relevant functions appearing therein. In Remark 5.2, we justify by giving suitable examples that the symmetry assumption in the statement of Theorem 5.1 is crucial.

(3) First eigenvalue on star-shaped Euclidean domains (Isoperimetric Inequality): We further remark that Theorem 1.1 of [Reference Gavitone, Paoli, Piscitelli and Sannipoli15] can easily be extended from convex Euclidean domains to star-shaped Euclidean domains (Theorem 2.2).

We bring to the readers’ notice that the eigenvalue optimization problems considered in [Reference Aithal and Chorwadwala2, Reference Chorwadwala and Ghosh7–Reference Chorwadwala and Vemuri11, Reference Harrell, Kröger and Kurata17, Reference Hersch18, Reference Kesavan22] are for the first Dirichlet eigenvalue of the Laplacian, the p-Laplacian, ( $1< p<\infty $ ), and are of certain general linear elliptic operators. In [Reference Kiwan and El Soufi23], the authors deal with the second Dirichlet eigenvalue of the Laplacian. In these problems, the authors considered the Dirichlet eigenvalue problem on doubly connected domains of the form $B \setminus P$ under a volume constraint, where B is a disk, while P is either a smaller disk or a domain having a dihedral symmetry with some monotonicity assumption on its boundary. In each of these articles, it was proved that the concentric configuration, among the admissible family of domains, maximizes the first Dirichlet eigenvalue of the operator under consideration.

We further mention here similar problems considered for the Neumann Eigenvalues (cf. [Reference Anoop, Bobkov and Drábek3–Reference Anoop and Verma5, Reference Verma and Santhanam30]).

This article is organized as follows: In Section 2, we discuss the geometry of star-shaped domains in non-Euclidean space forms. We then derive sharp bounds for the first Steklov–Dirichlet eigenvalue on doubly connected star-shaped domains with a spherical hole (Theorem 2.1). We also state a generalization of Theorem 1.1 from convex domains to star-shaped domains (Theorem 2.2). This theorem can be proved following the ideas in [Reference Gavitone, Paoli, Piscitelli and Sannipoli15]. In Section 3, we study the behavior of Steklov–Dirichlet eigenvalues and their corresponding eigenfunctions in annular domains. Certain integral inequalities on domains with given symmetry assumptions are proved in Section 4. In Section 5, we obtain isoperimetric bounds for higher eigenvalues (Theorem 5.1). The “order 4 symmetry assumption” on the domains in the statement of Theorem 5.1 is crucial. We illustrate this fact by providing example of a domain without any symmetry for which the theorem does not hold true. We also provide examples of symmetric domains having other symmetries but not having the symmetry of order 4 for which the theorem fails. These examples are explained in Remark 5.2.

We will be using the following notations in the rest of the article. For $s \in M$ and ${r> 0}$ , $B_{r}(s)$ would denote the open geodesic ball in the Riemannian manifold $(M,g)$ with center s and radius ${r}$ . When there is no confusion about the center s, we can denote $B_r(s)$ by $B_r$ . For $M=S^n$ , we take $r \in (0, \frac {\pi }{2}]$ . By $B_{R,r}$ we would mean the concentric annular domain in M with inner radius r and outer radius R. By $\tilde {\Omega }_{out}$ , we mean a star-shaped domain in a space form, centered at a point p in its interior. For us, $\tilde {\Omega } := \tilde {\Omega }_{\mathrm{out}} \setminus \overline {B_{r_1}(p)}$ when $\overline {B_{r_1}(p)} \subset \tilde {\Omega }_{\mathrm{out}}$ for $r_1>0$ . For $M=\mathbb {E}^n$ , without loss of generality, $B_r$ would mean $B_r(\underline {0})$ , the Euclidean ball with radius r centered at the origin. Let $0<R_1<R_2$ . Consider $B_{R_2} \setminus \overline {B_{R_1}}$ , the concentric annular domain in $\mathbb {R}^{n}$ centered at the origin with inner radius $R_1$ and outer radius $R_2$ . We set $\Omega _0:= B_{R_2} \setminus \overline {B_{R_1}}$ . For us, $\Omega _{\mathrm{out}}$ would mean a bounded domain in $\mathbb {R}^n$ having a symmetry of order $4$ with respect to the origin. For us, $\Omega := \Omega _{\mathrm{out}} \setminus \overline {B_{R_1}}$ when $\overline {B_{R_1}} \subset \Omega _{\mathrm{out}}$ .