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

. 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 R × M where M is a complete Riemannian manifold.


Introduction
Let (M n , g) be a compact Riemannian manifold with a possibly non-empty boundary ∂M .For p ∈ (1, +∞), we consider the so-called p-Laplacian defined by ∆ p u = − div( ∇u p−2 ∇u) for any C 2 function.For p = 2, ∆ 2 is nothing else than the Laplace-Beltrami operator of (M n , g).
Over the past years, this operator ∆ 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 where ∂u ∂ν is the derivative of the function u with respect to the outward unit normal ν to the boundary ∂M .Note that for p = 2, (S) is the usual Steklov problem (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 R N , there exists a sequence of positive eigenvalues λ 0 = 0 < λ 1 λ 2 • • • λ k • • • 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 mentionned 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-Schnirelman principle.Moreover, the eigenvalue 0 is simple with constant eigenfunctions and is isolated, that is there is no eigenvalue between 0 and λ 1 .Then, the first positive eigenvalue of the Steklov problem λ 1 satisfies the following variational characterization , 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 ∂M .
Note that all the other eigenvalues λ 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 Ω with smooth boundary, then λ 1 1 R p−1 (resp. The aim of the present note is also to obtain upper bounds for the first nonzero eigenvalue λ 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 R n+1 , then the first positive eigenvalue of the Laplacian on M satisfies 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 R n+1 .More generally, Reilly obtained the following inequalities for r ∈ {0, • • • , n} 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, Extrinsic upper bounds for the first eigenvalue of the p-Steklov problem 51 if equality holds in one these inequality, then M is immersed in a geodesic sphere of radius n λ1(∆) .Note that, always in [9], Reilly also obtained similar estimates for higher codimension submanifolds.Namely, if (M n , g) is isometrically immersed into R N , N > n + 1, then for any even r ∈ {0, • • • , n} with equality if and only if M is minimally immersed in a geodesic sphere of 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 λ 1 (∆ p ) of the p-Laplace operator on closed submanifolds of R N .Namely, they proved that 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 σ 1 of the Steklov problem on submanifolds of R N .Namely, they proved the following estimate where (M n , g) is a compact submanifold of R N with boundary ∂M and H denote the mean curvature of ∂M .We denote by X the isometric immersion.
The limitting case is also characterized.Namely, they proved that equality occurs if and only if M is minimally immersed into B N 1 λ1 so that X(∂M ) ⊂ ∂B N 1 λ1 minimally and orthogonally.In particular, if n = N , equality occurs if and only if p = 2 and X(M ) = B N 1 λ1 .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 ∂M and p ∈ (1, +∞).Assume that (M n , g) is isometrically immersed into the Euclidean space R N by X.Let λ 1 the first eigenvalue of the p-Steklov problem Moreover, equality occurs in both inequality if and only if p = 2 and X is a minimal immersion of M into B N 1 λ1 so that X(∂M ) ⊂ ∂B N 1 λ1 minimally and orthogonally.In particular, if n = N , equality occurs if and only if p = 2 and X(M ) = B N 1 λ1 .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 ×R.Recently, Xiong [14] obtained extrinsic estimates of Reilly type for closed hypersurfaces of product spaces (R × N, dt 2 ⊕ h), where (N n , h) is a complete Riemannian manifold.In particular, he proved that the first eigenvalue α 1 of the Laplace operator and the first eigenvalue σ 1 of the Steklov problem for mean-convex hypersurfaces (bounding a domain for the second one) satisfy respectively 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 × R. Namely, we prove the following result.
Theorem 1.2.Let p 2 and (M n , g) be a complete Riemannian manifold.Consider (Σ n , g) a closed oriented Riemannian manifold isometrically immersed into the Riemannian product (R × M, g = dt 2 ⊕ g).Moreover, assume that Σ is meanconvex and bounds a domain Then, λ 1 satisfies Extrinsic upper bounds for the first eigenvalue of the p-Steklov problem 53 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 λ 1 : First, give the following lemma.
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 2 and 1 < p 2.
The case p 2. Using the coordinates X i , 1 i N , as test functions and summing for i from 1 to N , we get First, since p 2, we have since we have Hence, we obtain Moreover, by the Hölder inequality, we have We mulitply by and use the integral Hölder inequality to get We recall the Hsiung-Minkowski formula (see [5] for hypersurfaces and [4] for its generallization to submanifold of higher codimension) Using this identity, we get Extrinsic upper bounds for the first eigenvalue of the p-Steklov problem 55 which gives the desired upper bound for λ 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 1 λ1 so that X(∂M ) ⊂ ∂B N 1 λ1 minimally and orthogonally.In The case 1 < p 2. First, since p 2, we have On the other hand, by the Hölder inequality, we have Hence, using the last two inequalities in the variational characterization of λ 1 , we obtain The end of the proof is the same that in the case p 2, we mulitply by 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 quanities 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 1 R , where R is also the radius appearing in the estimates of Verma.Indeed, if n 2, we have Similarly, when 1 < p < 2, we have

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 := − div(T ∇u), for any C 2 function u on ∂M .We also associate with T the following normal vector field: T e i , e j B(e i , e j ), (6) where B is the second fundamental form of the immersion of M into R N and {e 1 , . . ., e n } is a local orthonormal frame of T ∂M .Moreover, we recall the following generalized Hsiung-Minkowski formula (see [4], [10] for details and proofs) Now, we can state the following Theorem 3.1.Let (M n , g) be a compact connected and oriented Riemannian manifold with nonempty boundary ∂M and p ∈ (1, +∞).Assume that (M n , g) is isometrically immersed into the Euclidean space R N by X and let T be a symmetric and divergence-free (2, 0)-tensor on ∂M .Let λ 1 the first eigenvalue of the p-Steklov problem Then, the following holds Moreover, if H T does not vanish identically, then equality occurs in one of both inequalities if and only if p = 2 and 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 ∈ {1, . . ., n}, we set 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 H r .By convention, we set H 0 = 1.Moreover, always if r is even, we show easily that H Tr = c(r)H r+1 , where H Tr 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 ∂M X, H r+1 + H r dv h = 0 for any even r ∈ {0, . . ., n} if N > n + 1, and ∂M X, ν H r+1 + H r dv h = 0 for any r ∈ {0, . . ., n} if N = n + 1, where ν is the normal unit vector field on ∂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 ∂M and p ∈ (1, +∞).Assume that (M n , g) is isometrically immersed into the Euclidean space R N by X.Let λ 1 the first eigenvalue of the p-Steklov problem 1.If N > n + 1, and r ∈ {0, . . ., n − 1} is an even integer then we have (a) If p 2, then λ 1 satisifes 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 is a minimal immersion of M into B N 1 λ1 so that X(∂M ) ⊂ ∂B N 1 λ1 minimally and orthogonally.

If
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 1 λ1 .
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 R of the product R × M .First, obviously, up to a possible translation in the direction of R, we can assume that Σ tdv g = 0. Second, since Σ is mean-convex, we deduce that t does not vanish identically.Indeed, if t vanishes identically over Σ, then Σ is included in the slice {0} × M and thus is totally geodesic in the product R × M .This is a contradiction with the fact that Σ is mean-convex.Hence, t does not vanish identically and can be used as a test function.Thus, from the variational characterization of λ 1 , we have Hence, using ( 9) and ( 10), ( 8) becomes Hence, multiplying respectively by t and u, we get immediately t∆t = −nHut and u∆t = −nHu 2 which after integration over Σ gives Σ ∇t 2 dv g = Σ nHutdv g (12)
λ1 minimally and orthogonally and H T is proportional to X |∂M .(b) if N = n, M is a ball and tr(T ) is constant.Extrinsic upper bounds for the first eigenvalue of the p-Steklov problem 57