EXTRINSIC UPPER BOUNDS THE FIRST
EIGENVALUE OF THE p-STEKLOV PROBLEM ON
SUBMANIFOLDS

Julien Roth

► To cite this version:

Julien Roth. EXTRINSIC UPPER BOUNDS THE FIRST EIGENVALUE OF THE p-STEKLOV PROBLEM ON SUBMANIFOLDS. Communications in Mathematics, In press, Volume 30 (2022), Issue 1, pp.49-61. 10.46298/cm.9282 . hal-02466652v2

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Submitted on 7 Apr 2022

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# Extrinsic upper bounds for the first eigenvalue of the p-Steklov problem on submanifolds

Julien Roth

**Abstract.** We prove Reilly-type upper bounds for the first non-zero eigenvalue of the Steklov problem associated with the p-Laplace operator on submanifolds with boundary of Euclidean spaces as well as for Riemannian products $\mathbb{R} \times M$ where $M$ is a complete Riemannian manifold.

## 1 Introduction

Let $(M^n, g)$ be a compact Riemannian manifold with a possibly non-empty boundary $\partial M$. For $p \in (1, +\infty)$, we consider the so-called p-Laplacian defined by

$$ \Delta_p u = - \operatorname{div}(\|\nabla u\|^{p-2} \nabla u) $$

for any $C^2$ function. For $p = 2$, $\Delta_2$ is nothing else than the Laplace-Beltrami operator of $(M^n, g)$.

Over the past years, this operator $\Delta_p$, and especially its spectrum, has been intensively studied, mainly for Euclidean domains with Dirichlet or Neumann boundary conditions (see for instance [7] and references therein) but also on Riemannian manifolds [2], [8].

In the present paper, we will consider the Steklov problem associated with the p-Laplacian on submanifolds with boundary of the Euclidean space. Namely, we consider the p-Steklov problem which is the following boundary value problem

$$ \left\{ \begin{array}{ll} \Delta_p u = 0 & \text{in } M, \\ \|\nabla u\|^{p-2} \frac{\partial u}{\partial \nu} = \lambda |u|^{p-2} u & \text{on } \partial M, \end{array} \right. \qquad (S) $$

where $\frac{\partial u}{\partial \nu}$ is the derivative of the function $u$ with respect to the outward unit normal $\nu$ to the boundary $\partial M$. Note that for $p = 2$, (S) is the usual Steklov problem

<footnote>MSC 2020: 53C42, 58C40, 58J32</footnote>

<footnote>Keywords: eigenvalue estimates, p-Steklov problem, submanifolds</footnote>

<footnote>Affiliation:</footnote>

<footnote>Laboratoire d'Analyse et de Mathématiques Appliquées, Université Gustave Eiffel, Université Paris-Est Créteil, CNRS, F-77447 Marne-la-Vallée, France</footnote>

<footnote>E-mail: julien.roth@univ-eiffel.fr</footnote>
(the reader can for instance refer to [3] for an overview of results about the spectral geometry of the Steklov problem). Little is known about the spectrum of this p-Steklov problem. If $M$ is a domain of $\mathbb{R}^N$, there exists a sequence of positive eigenvalues $\lambda_0 = 0 < \lambda_1 \le \lambda_2 \le \dots \le \lambda_k \le \dots$ consisting in the variational spectrum and obtained by the Ljusternik-Schnirelmann theory (see [7], [12] for instance). One can refer to [1] for details about the Ljusternik-Schnirelmann principle. Note that, as mentioned in [8, Remark 1.1], the arguments used in [7] can be extended to domains on Riemannian manifolds and we have that there exists a non-decreasing sequence of variational eigenvalues obtained by the Ljusternik-Schnirelmann principle. Moreover, the eigenvalue 0 is simple with constant eigenfunctions and is isolated, that is there is no eigenvalue between 0 and $\lambda_1$. Then, the first positive eigenvalue of the Steklov problem $\lambda_1$ satisfies the following variational characterization

$$ \lambda_1 = \inf \left\{ \left. \frac{\int_M \|\nabla u\|^p dv_g}{\int_{\partial M} |u|^p dv_h} \right| u \in W^{1,p}(M) \setminus \{0\}, \quad \int_{\partial M} |u|^{p-2} u dv_h = 0 \right\}, $$

where $dv_g$ and $dv_h$ are the Riemannian volume forms respectively associated with the metric $g$ on $M$ and the induced metric $h$ on $\partial M$.

Note that all the other eigenvalues $\lambda_k$ of this sequence also have a variational characterization but we don't know if all the spectrum is contained in this sequence.

In a very recent paper, V. Sheela [13] obtain upper bound for the first eigenvalue of the p-Steklov problem (S) for Euclidean domain. Namely she proves that for a bounded domain $\Omega$ with smooth boundary, then $\lambda_1 \le \frac{1}{R^{p-1}}$ (resp. $\frac{n^{p-2}}{R^{p-1}}$) if $1 < p < 2$ (resp. $p \ge 2$), where $R > 0$ satisfies $V(\Omega) = V(B(R))$ where $B(R)$ is a ball of radius $R$.

The aim of the present note is also to obtain upper bounds for the first non-zero eigenvalue $\lambda_1$ of the p-Steklov problem, but depending on the geometry of the boundary in the spirit of the classical Reilly upper bounds for the Laplacian on closed hypersurfaces. Reilly [9] showed that if $(M^n, g)$ is a closed connected and oriented Riemannian manifold isometrically immersed into $\mathbb{R}^{n+1}$, then the first positive eigenvalue of the Laplacian on $M$ satisfies

$$ \lambda_1(\Delta) \le \frac{n}{V(M)} \int_M H^2 dv_g, $$

where $H$ is the mean curvature of the immersion. Note that $M$ is not supposed to be embedded and so does not necessarily bounds a domain of $\mathbb{R}^{n+1}$. More generally, Reilly obtained the following inequalities for $r \in \{0, \dots, n\}$

$$ \lambda_1(\Delta) \left( \int_M H_r dv_g \right)^2 \le V(M) \int_M H_{r+1}^2 dv_g, $$

where $H_r$ and $H_{r+1}$ stands for the higher order mean curvatures (that we will define in Section 3). For $r = 0$, we recover the first mentioned inequality. In addition,
if equality holds in one these inequality, then $M$ is immersed in a geodesic sphere of radius $\sqrt{\frac{n}{\lambda_1(\Delta)}}$. Note that, always in [9], Reilly also obtained similar estimates for higher codimension submanifolds. Namely, if $(M^n, g)$ is isometrically immersed into $\mathbb{R}^N$, $N > n + 1$, then

$$ \lambda_1(\Delta) \left( \int_M H_r dv_g \right)^2 \leq V(M) \int_M \|H_{r+1}\|^2 dv_g, $$

for any even $r \in \{0, \dots, n\}$ with equality if and only if $M$ is minimally immersed in a geodesic sphere of $\mathbb{R}^N$. Note that, in the case of codimension greater that 1, $H_r$ is a function and $H_{r+1}$ is a normal vector field, contrary to the hypersurface case where both are functions (see again Section 3 for details).

Recently, Du and Mao [2] proved Reilly type upper bounds for the first eigenvalue $\lambda_1(\Delta_p)$ of the p-Laplace operator on closed submanifolds of $\mathbb{R}^N$. Namely, they proved that

$$ \lambda_1(\Delta_p) \leq \frac{n^{p/2}}{V(M)^{p-1}} \left( \int_M \|H\|^{\frac{p}{p-1}} dv_g \right)^{p-1} \begin{cases} N^{\frac{p-2}{2}} & \text{if } p \geq 2 \\ N^{\frac{2-p}{2}} & \text{if } 1 < p \leq 2 \end{cases} $$

Moreover, equality occurs if and only if $p = 2$ and $M$ is minimally immersed into a geodesic hypersphere. In particular, if $N = n + 1$, $M$ is a geodesic hypersphere. In addition, the authors proved analogous estimates with higher order mean curvatures.

On the other hand, Ilias and Makhoul [6] proved Reilly-type inequalities for the first eigenvalue $\sigma_1$ of the Steklov problem on submanifolds of $\mathbb{R}^N$. Namely, they proved the following estimate

$$ \sigma_1 V(\partial M)^2 \leq n V(M) \int_{\partial M} \|H\|^2 dv_g, $$

where $(M^n, g)$ is a compact submanifold of $\mathbb{R}^N$ with boundary $\partial M$ and $H$ denote the mean curvature of $\partial M$. We denote by $X$ the isometric immersion.

The limiting case is also characterized. Namely, they proved that equality occurs if and only if $M$ is minimally immersed into $B^N(\frac{1}{\lambda_1})$ so that $X(\partial M) \subset \partial B^N(\frac{1}{\lambda_1})$ minimally and orthogonally. In particular, if $n = N$, equality occurs if and only if $p = 2$ and $X(M) = B^N(\frac{1}{\lambda_1})$. Here again, analogous estimates with higher order mean curvatures were proven.

The main result of this note is the following estimate for the first non-zero eigenvalue of the Steklov problem associated with the p-Laplacian. Namely, we prove

**Theorem 1.1.** Let $(M^n, g)$ be a compact connected and oriented Riemannian manifold with nonempty boundary $\partial M$ and $p \in (1, +\infty)$. Assume that $(M^n, g)$ is isometrically immersed into the Euclidean space $\mathbb{R}^N$ by $X$. Let $\lambda_1$ the first eigenvalue
of the p-Steklov problem

$$ \left\{ \begin{array}{ll} \Delta_p u = 0 & \text{in } M, \\ \|\nabla u\|^{p-2} \frac{\partial u}{\partial \nu} = \sigma |u|^{p-2} u & \text{on } \partial M \end{array} \right. \qquad (S) $$

If $p \ge 2$, then $\lambda_1$ satisfies

$$ \lambda_1 \le N^{\frac{p-2}{2}} n^{\frac{p}{2}} \left( \int_{\partial M} \|H\|^{\frac{p}{p-1}} dv_h \right)^{p-1} \frac{V(M)}{V(\partial M)^p}. $$

If $1 < p \le 2$, then $\lambda_1$ satisfies

$$ \lambda_1 \le N^{\frac{2-p}{2}} n^{\frac{p}{2}} \left( \int_{\partial M} \|H\|^{\frac{p}{p-1}} dv_h \right)^{p-1} \frac{V(M)}{V(\partial M)^p}. $$

Moreover, equality occurs in both inequality if and only if $p = 2$ and $X$ is a minimal immersion of $M$ into $B^N(\frac{1}{\lambda_1})$ so that $X(\partial M) \subset \partial B^N(\frac{1}{\lambda_1})$ minimally and orthogonally. In particular, if $n = N$, equality occurs if and only if $p = 2$ and $X(M) = B^N(\frac{1}{\lambda_1})$.

After giving the proof of Theorem 1.1 in Section 2, we obtain more general inequalities involving higher order mean curvatures (Theorem 3.1 and Corollary 3.2).

We finish with an estimate for domains of products manifolds of the type $M \times \mathbb{R}$. Recently, Xiong [14] obtained extrinsic estimates of Reilly type for closed hypersurfaces of product spaces $(\mathbb{R} \times N, dt^2 \oplus h)$, where $(N^n, h)$ is a complete Riemannian manifold. In particular, he proved that the first eigenvalue $\alpha_1$ of the Laplace operator and the first eigenvalue $\sigma_1$ of the Steklov problem for mean-convex hypersurfaces (bounding a domain for the second one) satisfy respectively

$$ \alpha_1 \le n\kappa_+(M)\|H\|_{\infty} \quad \text{and} \quad \sigma_1 \le \kappa_+(M) \frac{\|H\|_{\infty}}{\inf_M H}. $$

In [11], we proved analogous estimates for a larger class of second order differential operators, Paneitz-type operators and Steklov problems. In the present note, we continue this study by considering the p-Steklov problem for domains of products $M \times \mathbb{R}$. Namely, we prove the following result.

**Theorem 1.2.** Let $p \ge 2$ and $(M^n, \bar{g})$ be a complete Riemannian manifold. Consider $(\Sigma^n, g)$ a closed oriented Riemannian manifold isometrically immersed into the Riemannian product $(\mathbb{R} \times M, \tilde{g} = dt^2 \oplus \bar{g})$. Moreover, assume that $\Sigma$ is mean-convex and bounds a domain $\Omega$ in $\mathbb{R} \times M$. Let $\lambda_1$ be the first eigenvalue of the p-Steklov problem on $\Omega$

$$ \left\{ \begin{array}{ll} \Delta_p u = 0 & \text{in } \Omega, \\ \|\nabla u\|^{p-2} \frac{\partial u}{\partial \nu} = \sigma |u|^{p-2} u & \text{on } \partial \Omega = \Sigma. \end{array} \right. \qquad (S) $$

Then, $\lambda_1$ satisfies

$$ \lambda_1 \le \left( \frac{\kappa_+(\Sigma) \|H\|_{\infty}}{\inf_{\Sigma} H} \right)^{p/2} \left( \frac{V(\Omega)}{V(\Sigma)} \right)^{1-\frac{p}{2}}. $$
**Remark 1.3.** Note that for $p = 2$, we recover the result of Xiong [14].

The proof of this last theorem will be given in Section 4.

## 2 Proof of Theorem 1.1

First, we recall the variational characterization of $\lambda_1$:

$$ \lambda_1 = \inf \left\{ \left| \frac{\int_M |\nabla u|^p dv_g}{\int_{\partial M} |u|^p dv_h} \right| \mid u \in W^{1,p}(M) \setminus \{0\} \right\}, \quad \int_{\partial M} |u|^{p-2} u dv_h = 0. $$

First, give the following lemma.

**Lemma 2.1.** There exists a point $y = (y_1, \dots, y_N) \in \mathbb{R}^N$ so that

$$ \int_{\partial M} |(X^i - y_i)|^{p-2} (X^i - y_i) dv_h = 0 $$

for all $i \in \{1, \dots, N\}$.

*Proof.* Proceeding as in [13, Theorem 1], we consider the function $f: \mathbb{R}^N \to \mathbb{R}$ defined by

$$ f(y_1, \dots, y_N) = \frac{1}{p} \int_{\partial M} \sum_{i=1}^{N} |X^i - y_i|^p dv_h, $$

where $(X^1, \dots, X^N)$ are the Euclidean coordinates centered at the origin. This function is nonnegative and we denote by $\alpha > 0$ its infimum. By compactness of $\partial M$, there exists $R > 0$ so that $\partial M$ is contained in the ball $B(0, R)$. Moreover,

we set $\rho = \left( \frac{2\alpha p}{V(\partial M)} \right)^{\frac{1}{p}}$. Now, let $(y_1, \dots, y_N) \in \mathbb{R}^N$, outside the hypercube $[-R - \rho, R + \rho]^N$ so that there exists $i \in \{1, \dots, N\}$ so that $|y_i| > R + \rho$. Hence, for any $(X^1, \dots, X^N) \in \partial M$, we have $\sum_{i=1}^N |X^i - y_i|^p \ge \rho^p$. From this, we deduce that

$$ f(y_1, \dots, y_N) \ge \frac{1}{p} \int_M \rho^p dv_h = \frac{1}{p} \rho^p V(\partial M) = 2\alpha. $$

By continuity of $f$ and compactness of $[-R - \rho, R + \rho]^N$, the infimum $\alpha$ of $f$ is necessarily attained inside $[-R - \rho, R + \rho]^N$. Let $y_0 = (y_1, \dots, y_N)$ a point where the minimum is attained. At this point, $(\nabla f)_{y_0} = 0$ and we deduce that for each $i \in \{1, \dots, N\}$, we have

$$ \langle \nabla f, \partial_i \rangle_{y_0} = \int_{\partial M} |(X^i - y_i)|^{p-2} (X^i - y_i) dv_h = 0, $$

where $\{\partial_1, \dots, \partial_N\}$ is the canonical basis of $\mathbb{R}^N$.

<figure><img src="image_2.png" /></figure>

With this lemma, up to translation, we can use the coordinate functions as test functions without loss of generality.

From this point, we will consider separately the cases $p \ge 2$ and $1 < p \le 2$.
**The case $p \ge 2$.** Using the coordinates $X^i$, $1 \le i \le N$, as test functions and summing for $i$ from 1 to $N$, we get

$$ \lambda_1 \int_{\partial M} \sum_{i=1}^{N} |X^i|^p \le \int_{M} \sum_{i=1}^{N} \|\nabla X^i\|^p. $$

First, since $p \ge 2$, we have

$$ \sum_{i=1}^{N} \|\nabla X^i\|^p \le \left( \sum_{i=1}^{N} \|\nabla X^i\|^2 \right)^{\frac{p}{2}} = n^{\frac{p}{2}}, \quad (1) $$

since we have $\sum_{i=1}^{N} \|\nabla X^i\|^2 = n$ (see [10, Lemma 2.1] for instance).

Hence, we obtain

$$ \lambda_1 \int_{\partial M} \sum_{i=1}^{N} |X^i|^p dv_h \le n^{\frac{p}{2}} V(M). $$

Moreover, by the Hölder inequality, we have

$$ \|X\|^2 \le \left( \sum_{i=1}^{N} |X^i|^p \right)^{\frac{2}{p}} N^{\frac{p-2}{p}}, \quad (2) $$

which gives

$$ \sum_{i=1}^{N} |X^i|^p \ge \frac{1}{N^{\frac{p-2}{2}}} \|X\|^p $$

and so

$$ \lambda_1 \int_{\partial M} \|X\|^p dv_h \le n^{\frac{p}{2}} N^{\frac{p-2}{2}} V(M). $$

We multiply by $\left( \int_{\partial M} \|H\|^{\frac{p}{p-1}} dv_h \right)^{p-1}$ and use the integral Hölder inequality to get

$$ \lambda_1 \left| \int_{\partial M} \langle X, H \rangle dv_h \right|^p \le n^{\frac{p}{2}} N^{\frac{p-2}{2}} \left( \int_{\partial M} \|H\|^{\frac{p}{p-1}} dv_h \right)^{p-1} V(M). \quad (3) $$

We recall the Hsiung-Minkowski formula (see [5] for hypersurfaces and [4] for its generalization to submanifold of higher codimension)

$$ \int_{\partial M} (\langle H, X \rangle + 1) dv_h = 0. \quad (4) $$

Using this identity, we get

$$ \lambda_1 V(\partial M)^p \le n^{\frac{p}{2}} N^{\frac{p-2}{2}} \left( \int_{\partial M} \|H\|^{\frac{p}{p-1}} dv_h \right)^{p-1} V(M). $$
which gives the desired upper bound for $\lambda_1$.

Moreover, if equality occurs, then equality holds in all the above inequalities and in particular in the inequality (1), which implies that $p = 2$. Therefore, the end of the proof is similar to the proof of Ilias and Makhoul for the classical Steklov problem and we have that $X$ is a minimal immersion of $M$ into $B^N(\frac{1}{\lambda_1})$ so that $X(\partial M) \subset \partial B^N(\frac{1}{\lambda_1})$ minimally and orthogonally. In particular, if $n = N$, then $X(M) = B^N(\frac{1}{\lambda_1})$.

**The case** $1 < p \le 2$. First, since $p \le 2$, we have

$$ \|X\|^p = \left( \sum_{i=1}^{N} |X^i|^2 \right)^{\frac{2}{p}} \le \sum_{i=1}^{N} |X^i|^p. \qquad (5) $$

On the other hand, by the Hölder inequality, we have

$$ \sum_{i=1}^{N} \|\nabla X^i\|^p \le N^{\frac{2-p}{p}} \left( \sum_{i=1}^{N} \|\nabla X^i\|^2 \right)^{\frac{2}{p}} = N^{\frac{2-p}{p}} n^{\frac{2}{p}}. $$

Hence, using the last two inequalities in the variational characterization of $\lambda_1$, we obtain

$$ \lambda_1 \int_{\partial M} \|X\|^p dv_h \le n^{\frac{2}{p}} N^{\frac{2-p}{p}} V(M). $$

The end of the proof is the same that in the case $p \le 2$, we multiply by $(\int_{\partial M} \|H\|^{\frac{p}{p-1}} dv_h)^{p-1}$, use the integral Hölder inequality and the Hsiung-Minkowski formula (4).

If equality holds, then equality occurs in (5). Thus, here again $p = 2$ and we conclude as previously.

**Remark 2.2.** When the hypersurface bounds a Euclidean domain, that is ($N = n$), the above upper bounds and the upper bounds of Verma are not express in term of the same quantities so that one cannot say if one is sharper than the other. However, we can remark that in the case where the domain is a ball of radius $R$, both upper bounds coincide since in that case, the mean curvature $||H||$ is constant and is equal to $\frac{1}{R}$, where $R$ is also the radius appearing in the estimates of Verma. Indeed, if $n \ge 2$, we have

$$ \lambda_1 \le N^{\frac{p-2}{2}} n^{\frac{p}{2}} \left( \int_{\partial M} \|H\|^{\frac{p}{p-1}} dv_h \right)^{p-1} \frac{V(M)}{V(\partial M)^p} = n^{p-1} \frac{1}{R^p} \frac{V(B(R))}{V(S(R))} = \frac{n^{p-2}}{R^{p-1}}, $$

since $\frac{V(B(R))}{V(S(R))} = \frac{R}{n}$. Similarly, when $1 < p < 2$, we have

$$ \lambda_1 \le N^{\frac{2-p}{2}} n^{\frac{p}{2}} \left( \int_{\partial M} \|H\|^{\frac{p}{p-1}} dv_h \right)^{p-1} \frac{V(M)}{V(\partial M)^p} = n \frac{1}{R^p} \frac{V(B(R))}{V(S(R))} = \frac{1}{R^{p-1}}. $$
### 3 Inequalities with higher order mean curvatures

In this section, we extend Theorem 1.1 to estimates with higher order mean curvatures. It will appear as a particular casde of a more general result. Before stating the result, we briefly give some recalls.

First of all, let $T$ be a divergence-free symmetric (1,1)-tensor. We associate with $T$ the second order differential operator $L_T$ defined by $L_T u := -\operatorname{div}(T \nabla u)$, for any $C^2$ function $u$ on $\partial M$. We also associate with $T$ the following normal vector field:

$$H_T = \sum_{i,j=1}^{n} \langle T e_i, e_j \rangle B(e_i, e_j), \qquad (6)$$

where $B$ is the second fundamental form of the immersion of $M$ into $\mathbb{R}^N$ and $\{e_1, \dots, e_n\}$ is a local orthonormal frame of $T\partial M$. Moreover, we recall the following generalized Hsiung-Minkowski formula (see [4], [10] for details and proofs)

$$\int_{\partial M} (\langle X, H_T \rangle + \operatorname{tr}(T)) dv_h = 0. \qquad (7)$$

Now, we can state the following

**Theorem 3.1.** Let $(M^n, g)$ be a compact connected and oriented Riemannian manifold with nonempty boundary $\partial M$ and $p \in (1, +\infty)$. Assume that $(M^n, g)$ is isometrically immersed into the Euclidean space $\mathbb{R}^N$ by $X$ and let $T$ be a symmetric and divergence-free (2,0)-tensor on $\partial M$. Let $\lambda_1$ the first eigenvalue of the $p$-Steklov problem

$$\left\{ \begin{array}{ll} \Delta_p u = 0 & \text{in } M, \\ \|\nabla u\|^{p-2} \frac{\partial u}{\partial \nu} = \sigma |u|^{p-2} u & \text{on } \partial M \end{array} \right. \qquad (S)$$

Then, the following holds

1. If $p \ge 2$, then $\lambda_1$ satisfies

$$\lambda_1 \left| \int_{\partial M} \operatorname{tr}(T) \right|^p \le N^{\frac{p-2}{2}} n^{\frac{p}{2}} \left( \int_{\partial M} \|H_T\|^{\frac{p}{p-1}} \right)^{p-1} V(M).$$

2. If $1 < p \le 2$, then $\lambda_1$ satisfies

$$\lambda_1 \left| \int_{\partial M} \operatorname{tr}(T) \right|^p \le N^{\frac{2-p}{2}} n^{\frac{p}{2}} \left( \int_{\partial M} \|H_T\|^{\frac{p}{p-1}} \right)^{p-1} V(M).$$

Moreover, if $H_T$ does not vanish identically, then equality occurs in one of both inequalities if and only if $p = 2$ and

(a) if $N > n$, $X$ is a minimal immersion of $M$ into $B^N(\frac{1}{\lambda_1})$ so that $X(\partial M) \subset \partial B^N(\frac{1}{\lambda_1})$ minimally and orthogonally and $H_T$ is proportional to $X|_{\partial M}$.

(b) if $N = n$, $M$ is a ball and $\operatorname{tr}(T)$ is constant.
*Proof.* The proof is similar to the proof of Theorem 1.1 with the difference that we use the generalized Hsiung-Minkowski formula (7) instead of the classical one. In case of equality, we also get that $p = 2$ by equality in (1) or (5). Then, we conclude as in Theorem 1.1 by the argument of Ilias and Makhoul. ■

Now, let us consider higher order mean curvatures. For $r \in \{1, \dots, n\}$, we set

$$T_r = \frac{1}{r!} \sum_{\substack{i_1, i_1, \dots, i_r \\ j_1, j_1, \dots, j_r}} \epsilon \binom{i, i_1, \dots, i_r}{j, j_1, \dots, j_r} \langle B_{i_1 j_1} B_{i_2 j_2} \rangle \dots \langle B_{i_{r-1} j_{r-1}} B_{i_r j_r} \rangle e_i^* \otimes e_j^*,$$

if $r$ is even and

$$T_r = \frac{1}{r!} \sum_{\substack{i_1, i_1, \dots, i_r \\ j_1, j_1, \dots, j_r}} \epsilon \binom{i, i_1, \dots, i_r}{j, j_1, \dots, j_r} \langle B_{i_1 j_1} B_{i_2 j_2} \rangle \dots \langle B_{i_{r-1} j_{r-1}} B_{i_r j_r} \rangle B_{i_r, j_r} \otimes e_i^* \otimes e_j^*,$$

where the $B_{ij}$'s are the coefficients of the second fundamental form $B$ in a local orthonormal frame $\{e_1, \dots, e_n\}$ and $\epsilon$ is the standard signature for permutations. Here, $\{e_1^*, \dots, e_n^*\}$ is the dual coframe of $\{e_1, \dots, e_n\}$. By definition, the $r$-th mean curvature is $H_r = \frac{1}{c(r)} \text{tr}(T_r)$, where $c(r) = (n-r)\binom{r}{n}$. Note that $H_r$ is a real function if $r$ is even and a normal vector field if $r$ is odd, in this case, we will denote it by $\mathbf{H}_r$. By convention, we set $H_0 = 1$. Moreover, always if $r$ is even, we show easily that $H_{T_r} = c(r)\mathbf{H}_{r+1}$, where $H_{T_r}$ is given by the relation (6).

In the case of hypersurfaces, we can consider the higher order mean curvatures as scalar functions also for odd indices by taking $B$ as the real-valued second fundamental form.

By the symmetry of $B$, these tensors are clearly symmetric and it is also a classical fact that they are divergence-free (see [4] for instance). Hence, in this case, the Hsiung-Minkowski formula (7) becomes

$$\int_{\partial M} (\langle X, \mathbf{H}_{r+1} \rangle + H_r) dv_h = 0$$

for any even $r \in \{0, \dots, n\}$ if $N > n + 1$, and

$$\int_{\partial M} (\langle X, \nu \rangle H_{r+1} + H_r) dv_h = 0$$

for any $r \in \{0, \dots, n\}$ if $N = n + 1$, where $\nu$ is the normal unit vector field on $\partial M$ chosen to define the shape operator.

We obtain directly from Theorem 3.1 the following corollary:

**Corollary 3.2.** Let $(M^n, g)$ be a compact connected and oriented Riemannian manifold with nonempty boundary $\partial M$ and $p \in (1, +\infty)$. Assume that $(M^n, g)$ is isometrically immersed into the Euclidean space $\mathbb{R}^N$ by $X$. Let $\lambda_1$ the first eigenvalue of the $p$-Steklov problem

$$\left\{ \begin{array}{ll} \Delta_p u = 0 & \text{in } M, \\ \|\nabla u\|^{p-2} \frac{\partial u}{\partial \nu} = \sigma |u|^{p-2} u & \text{on } \partial M. \end{array} \right.$$
1. If $N > n + 1$, and $r \in \{0, \dots, n - 1\}$ is an even integer then we have

(a) If $p \ge 2$, then $\lambda_1$ satisifes

$$ \lambda_1 \left| \int_{\partial M} H_r \right|^p \le N^{\frac{p-2}{2}} n^{\frac{p}{2}} \left( \int_{\partial M} \| \mathbf{H}_{r+1} \|^{\frac{p}{p-1}} \right)^{p-1} V(M). $$

(b) If $1 < p \le 2$, then $\lambda_1$ satisifes

$$ \lambda_1 \left| \int_{\partial M} H_r \right|^p \le N^{\frac{2-p}{2}} n^{\frac{p}{2}} \left( \int_{\partial M} \| \mathbf{H}_{r+1} \|^{\frac{p}{p-1}} \right)^{p-1} V(M). $$

Moreover, if $\mathbf{H}_{r+1}$ does not vanish identically, then equality occurs in one of both inequalities if and only if $p = 2$ and $X$ is a minimal immersion of $M$ into $B^N(\frac{1}{\lambda_1})$ so that $X(\partial M) \subset \partial B^N(\frac{1}{\lambda_1})$ minimally and orthogonally.

2. If $N = n + 1$ and $\in \{0, \dots, n - 1\}$ is any integer, then we have

(a) If $p \ge 2$, then $\lambda_1$ satisifes

$$ \lambda_1 \left| \int_{\partial M} H_r \right|^p \le N^{\frac{p-2}{2}} n^{\frac{p}{2}} \left( \int_{\partial M} |H_{r+1}|^{\frac{p}{p-1}} \right)^{p-1} V(M). $$

(b) If $1 < p \le 2$, then $\lambda_1$ satisifes

$$ \lambda_1 \left| \int_{\partial M} H_r \right|^p \le N^{\frac{2-p}{2}} n^{\frac{p}{2}} \left( \int_{\partial M} |H_{r+1}|^{\frac{p}{p-1}} \right)^{p-1} V(M). $$

Moreover, if $H_{r+1}$ does not vanish identically, then equality occurs in one of both inequalities if and only if $p = 2$ and $X(M) = B^N(\frac{1}{\lambda_1})$.

# 4 Proof of Theorem 1.2

In the spirit of the proofs of the results obtained in [14] and [11], we will use as test function the function $t$ which is the coordinate in the factor $\mathbb{R}$ of the product $\mathbb{R} \times M$. First, obviously, up to a possible translation in the direction of $\mathbb{R}$, we can assume that $\int_{\Sigma} t dv_g = 0$. Second, since $\Sigma$ is mean-convex, we deduce that $t$ does not vanish identically. Indeed, if $t$ vanishes identically over $\Sigma$, then $\Sigma$ is included in the slice $\{0\} \times M$ and thus is totally geodesic in the product $\mathbb{R} \times M$. This is a contradiction with the fact that $\Sigma$ is mean-convex. Hence, $t$ does not vanish identically and can be used as a test function. Thus, from the variational characterization of $\lambda_1$, we have

$$ \lambda_1 \int_{\Sigma} |t|^p dv_g \le \int_{\Omega} \|\tilde{\nabla} t\|^p dv_{\tilde{g}}. \qquad (8) $$
First, since $||\tilde{\nabla}t|| = 1$, we have

$$ \int_{\Omega} ||\tilde{\nabla}t||^p dv_{\tilde{g}} = V(\Omega) = \left( \int_{\Omega} ||\tilde{\nabla}t||^2 dv_{\tilde{g}} \right)^{\frac{p}{2}} V(\Omega)^{1-\frac{p}{2}}. \quad (9) $$

In addition, we have

$$ \int_{\Omega} ||\tilde{\nabla}t||^2 dv_{\tilde{g}} = - \int_{\Omega} t \tilde{\Delta} t dv_{\tilde{g}} + \int_{\Omega} \text{div}_{\tilde{g}}(t \tilde{\nabla}t) dv_{\tilde{g}}. $$

Since $\tilde{\Delta}t = 0$, using the Stokes theorem, we get

$$ \int_{\Omega} ||\tilde{\nabla}t||^2 dv_{\tilde{g}} = \int_{\Sigma} \langle t \tilde{\nabla}t, \nu \rangle dv_g = \int_{\Sigma} t u dv_g, $$

where $u$ is defined by $u = \langle \partial_t, \nu \rangle = \langle \tilde{\nabla}t, \nu \rangle$. Hence, by the Hölder inequality, we obtain

$$ \int_{\Omega} ||\tilde{\nabla}t||^2 dv_{\tilde{g}} \le \left( \int_{\Sigma} |t|^p dv_g \right)^{\frac{1}{p}} \left( \int_{\Sigma} |u|^{\frac{p}{p-1}} dv_g \right)^{\frac{p-1}{p}}. \quad (10) $$

Hence, using (9) and (10), (8) becomes

$$ \lambda_1 \le \frac{\left( \int_{\Sigma} |u|^{\frac{p}{p-1}} dv_{\tilde{g}} \right)^{\frac{p-1}{2}}}{\left( \int_{\Sigma} |t|^p dv_g \right)^{\frac{1}{2}}} V(\Omega)^{1-\frac{p}{2}}. \quad (11) $$

On the other hand, we have

$$ \begin{aligned} \Delta t &= - \text{div}_{\Sigma}(\tilde{\nabla}t) \\ &= - \sum_{i=1}^{n} \langle \nabla_{e_i}(\tilde{\nabla}t), e_i \rangle \\ &= - \sum_{i=1}^{n} \langle \tilde{\nabla}_{e_i}(\partial t - \langle \partial t, \nu \rangle \nu), e_i \rangle, \end{aligned} $$

where $\nu$ is a unit normal vector field. Moreover, since $\partial_t$ is parallel for $\tilde{\nabla}$ and $-\tilde{\nabla}_{(\cdot)}\nu$ is the shape operator $S$, we get

$$ \begin{aligned} \Delta t &= - \sum_{i=1}^{n} \langle \partial_t, \nu \rangle \langle Se_i, e_i \rangle \\ &= -nHu. \end{aligned} $$

Hence, multiplying respectively by $t$ and $u$, we get immediately $t\Delta t = -nHu$ and $u\Delta t = -nHu^2$ which after integration over $\Sigma$ gives

$$ \int_{\Sigma} ||\nabla t||^2 dv_g = \int_{\Sigma} nHu dv_g \quad (12) $$
$$ \int_{\Sigma} \langle S\nabla t, \nabla t \rangle dv_g = \int_{\Sigma} nHu^2. \qquad (13) $$

Note that for the second one, we have used the fact that $\nabla u = -S\nabla t$. Indeed, we have

$$ \nabla u = \sum_{i=1}^{n} e_i(u)e_i = \sum_{i=1}^{n} e_i(\langle \nu, \partial t \rangle)e_i = -\sum_{i=1}^{n} \langle Se_i, \partial_t \rangle e_i = -S(\nabla t). $$

Moreover, we have, using (13),

$$ \begin{aligned} n \inf_{\Sigma} (H) \int_M u^2 dv_g &\le \int_M nHu^2 dv_g \\ &\le \int_{\Sigma} \langle S\nabla t, \nabla t \rangle dv_g \\ &\le \kappa_+(\Sigma) \int_{\Sigma} \|\nabla t\|^2 dv_g, \end{aligned} $$

where $\kappa_+(\Sigma) = \max\{\kappa_+(x)|x \in M\}$ with $\kappa_+(x)$ the biggest principal curvature of $\Sigma$ at the point $x$. Now, we use (12) and the Hölder inequality to get

$$ \begin{aligned} n \inf_{\Sigma} (H) \int_{\Sigma} u^2 dv_g &\le \kappa_+(\Sigma) \int_{\Sigma} nHutdv_g \\ &\le n\kappa_+(\Sigma) \|H\|_{\infty} \int_{\Sigma} utdv_g \\ &\le n\kappa_+(\Sigma) \|H\|_{\infty} \left( \int_{\Sigma} |t|^p dv_g \right)^{\frac{1}{p}} \left( \int_{\Sigma} |u|^{\frac{p}{p-1}} dv_g \right)^{\frac{p-1}{p}}. \end{aligned} $$

Finally, using the Hölder inequality a last time, we have

$$ \begin{aligned} \inf_{\Sigma} (H) \left( \int_{\Sigma} |u|^{\frac{p}{p-1}} dv_g \right)^{\frac{2(p-1)}{p}} V(\Sigma)^{\frac{2-p}{p}} &\le \kappa_+(\Sigma) \|H\|_{\infty} \left( \int_M |t|^p dv_g \right)^{\frac{1}{p}} \left( \int_M |u|^{\frac{p}{p-1}} dv_g \right)^{\frac{p-1}{p}}. \end{aligned} $$

and so

$$ \frac{\left( \int_{\Sigma} |u|^{\frac{p}{p-1}} dv_g \right)^{\frac{p-1}{p}}}{\left( \int_M |t|^p dv_g \right)^{\frac{1}{p}}} \le \frac{\kappa_+ \Sigma \|H\|_{\infty}}{\inf_{\Sigma} (H)} V(\Sigma)^{\frac{p-2}{p}}. $$

Reporting this in (11), we get the desired inequality:

$$ \lambda_1 \le \left( \frac{\kappa_+(\Sigma) \|H\|_{\infty}}{\inf_{\Sigma} H} \right)^{p/2} \left( \frac{V(\Omega)}{V(\Sigma)} \right)^{1-\frac{p}{2}}. $$
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