Geometric bounds for Steklov eigenvalues

Geometric bounds for Steklov eigenvalues Luigi Provenzano École Polytechnique Fédérale de Lausanne, Switzerland Joint work with Joachim Stubbe June 20, 2017 luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 1 / 27

The Steklov problem We consider the Steklov eigenvalue problem on Ω R N { u = 0, in Ω, u ν = σu, on Ω. W. Stekloff, Sur les problèmes fondamentaux de la physique mathématique. Ann. Sci. École Norm. Sup., (3) 19 (1902), 191 259/455 490. If Ω is a bounded connected open set with Lipschitz boundary, then 0 = σ 0 < σ 1 σ 2 σ j +. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 2 / 27

The Steklov problem We consider the Steklov eigenvalue problem on Ω R N { u = 0, in Ω, u ν = σu, on Ω. W. Stekloff, Sur les problèmes fondamentaux de la physique mathématique. Ann. Sci. École Norm. Sup., (3) 19 (1902), 191 259/455 490. If Ω is a bounded connected open set with Lipschitz boundary, then 0 = σ 0 < σ 1 σ 2 σ j +. A. Girouard, I. Polterovich, Spectral geometry of the Steklov problem. Journal of Spectral Theory, 7 (2017), 321 359. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 2 / 27

Basic properties Variational characterization of Steklov eigenvalues: Ω σ j = u 2 dx Ω u2 dσ min max V H 1 (Ω), 0 u V dimv =j+1 luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 3 / 27

Basic properties Variational characterization of Steklov eigenvalues: Ω σ j = u 2 dx Ω u2 dσ min max V H 1 (Ω), 0 u V dimv =j+1 Weyl s asymptotic law (if Ω is piecewise C 1 ): ( ) σ j 2πω 1 1 N 1 j N 1 N 1 Ω as j +, where ω N 1 is the volume of the unit ball in R N 1. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 3 / 27

Geometric inequalities We have considered the issue of finding upper bounds for the Steklov eigenvalues. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 4 / 27

Geometric inequalities We have considered the issue of finding upper bounds for the Steklov eigenvalues. An open question is whether there exist bounds of the form ( ) 1 j N 1 σ j C N, Ω if N 3, where the constant C N depends only on the dimension. Remark: analogous inequalities hold for Dirichlet eigenvalues (Li-Yau, lower bounds), Neumann eigenvalues (Kröger, upper bounds) luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 4 / 27

Geometric inequalities We have considered the issue of finding upper bounds for the Steklov eigenvalues. An open question is whether there exist bounds of the form ( ) 1 j N 1 σ j C N, Ω if N 3, where the constant C N depends only on the dimension. Remark: analogous inequalities hold for Dirichlet eigenvalues (Li-Yau, lower bounds), Neumann eigenvalues (Kröger, upper bounds) and eigenvalues of the Laplacian on Riemannian manifolds (Buser, Cheng-Yang, Colbois-Maerten). luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 4 / 27

Geometric inequalities The problem is completely solved for simply connected Ω R 2 : σ 1 2π Ω R. Weinstock, Inequalities for a classical eigenvalue problem. J. Rational Mech. Anal., 3 (1954), 745 753. σ j 2πj Ω J. Hersch, L. E. Payne, M. M. Schiffer, Some inequalities for Stekloff eigenvalues. Arch. Rational Mech. Anal., 57 (1975), 99 114. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 5 / 27

Geometric inequalities The problem is completely solved for simply connected Ω R 2 : σ 1 2π Ω R. Weinstock, Inequalities for a classical eigenvalue problem. J. Rational Mech. Anal., 3 (1954), 745 753. σ j 2πj Ω J. Hersch, L. E. Payne, M. M. Schiffer, Some inequalities for Stekloff eigenvalues. Arch. Rational Mech. Anal., 57 (1975), 99 114. A. Girouard, I. Polterovich, Upper bounds for Steklov eigenvalues on surfaces. Electron. Res. Announc. Math. Sci., 19 (2012), 77 85. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 5 / 27

Geometric inequalities In higher dimension we have an isoperimetric control of the eigenvalues j 2 N σ j C N 1 N Ω N 1 I (Ω) N 1 luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 6 / 27

Geometric inequalities In higher dimension we have an isoperimetric control of the eigenvalues j 2 N σ j C N 1 N Ω N 1 I (Ω) N 1 j 2 N C N Ω 1 N 1, where I (Ω) = Ω Ω N 1 N B. Colbois, A. El Soufi, A. Girouard, Isoperimetric control of the Steklov spectrum. J. Funct. Anal., 261(5) (2011),1384 1399. We also have isodiametric control σ j C N j 2 N +1 diam(ω) B. Bogosel, D. Bucur, A. Giacomini, Optimal Shapes Maximizing the Steklov Eigenvalues. SIAM J. Math. Anal., 49(2) (2017), 1645 1680. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 6 / 27

The Laplace-Beltrami operator Let Ω R N be a bounded domain of class C 2 such that Ω is connected. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 7 / 27

The Laplace-Beltrami operator Let Ω R N be a bounded domain of class C 2 such that Ω is connected. Let Ω denote the Laplace-Beltrami operator on Ω. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 7 / 27

The Laplace-Beltrami operator Let Ω R N be a bounded domain of class C 2 such that Ω is connected. Let Ω denote the Laplace-Beltrami operator on Ω. The eigenvalue problem Ω u = λu, on Ω luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 7 / 27

The Laplace-Beltrami operator Let Ω R N be a bounded domain of class C 2 such that Ω is connected. Let Ω denote the Laplace-Beltrami operator on Ω. The eigenvalue problem Ω u = λu, on Ω admits a sequence 0 = λ 0 < λ 1 λ 2 λ j + luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 7 / 27

The Laplace-Beltrami operator Let Ω R N be a bounded domain of class C 2 such that Ω is connected. Let Ω denote the Laplace-Beltrami operator on Ω. The eigenvalue problem Ω u = λu, on Ω admits a sequence 0 = λ 0 < λ 1 λ 2 λ j + with the variational characterization λ j = min V H 1 ( Ω), dimv =j+1 max 0 u V Ω Ωu 2 dσ Ω u2 dσ luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 7 / 27

The Laplace-Beltrami operator The eigenvalues satisfy the Weyl s asymptotic law ( ) λ j 4π 2 ω 2 2 N 1 j N 1 N 1 Ω as j +. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 8 / 27

The Laplace-Beltrami operator The eigenvalues satisfy the Weyl s asymptotic law ( ) λ j 4π 2 ω 2 2 N 1 j N 1 N 1 Ω as j +. and Weyl-type bounds of the form λ j (N 2)κ2 4 ( ) 2 j N 1 + C N. Ω P. Buser, Beispiele für λ 1 auf kompakten Mannigfaltigkeiten. Math. Z., 165(2) (1979), 107 133. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 8 / 27

The Laplace-Beltrami operator The eigenvalues satisfy the Weyl s asymptotic law ( ) λ j 4π 2 ω 2 2 N 1 j N 1 N 1 Ω as j +. and Weyl-type bounds of the form λ j (N 2)κ2 4 ( ) 2 j N 1 + C N. Ω P. Buser, Beispiele für λ 1 auf kompakten Mannigfaltigkeiten. Math. Z., 165(2) (1979), 107 133. Asymptotic formulas suggest that for large j σ j λ j luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 8 / 27

Main result We have a comparison of Steklov and Laplace-Beltrami eigenvalues luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 9 / 27

Main result We have a comparison of Steklov and Laplace-Beltrami eigenvalues: Theorem (P. - Stubbe, 2017) Let Ω R N be a bounded domain with connected boundary Ω of class C 2. Then there exists a constant c Ω such that for all j N λ j σj 2 + 2c Ω σ j, σ j c Ω + cω 2 + λ j. In particular, σj λ j 2cΩ, the constant c Ω depending on the maximal possible size of a tubular neighborhood about Ω and on the mean of the maximal curvatures of Ω. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 9 / 27

Main result We have a comparison of Steklov and Laplace-Beltrami eigenvalues: Theorem (P. - Stubbe, 2017) Let Ω R N be a bounded domain with connected boundary Ω of class C 2. Then there exists a constant c Ω such that for all j N λ j σj 2 + 2c Ω σ j, σ j c Ω + cω 2 + λ j. In particular, σj λ j 2cΩ, the constant c Ω depending on the maximal possible size of a tubular neighborhood about Ω and on the mean of the maximal curvatures of Ω. L. Provenzano, J. Stubbe, Weyl-type bounds for Steklov eigenvalues. arxiv:1611.00929 (2017). To appear on the Journal of Spectral Theory. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 9 / 27

Main result The constant c Ω is given by c Ω = 1 2h + N 1 H, 2 luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 10 / 27

Main result The constant c Ω is given by where c Ω = 1 2h + N 1 H, 2 h is the maximal possible size of a tubular neighborhood about Ω; luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 10 / 27

Main result The constant c Ω is given by where c Ω = 1 2h + N 1 H, 2 h is the maximal possible size of a tubular neighborhood about Ω; H = max x Ω ( 1 N 1 N 1 i=1 κ i (x) and κ i (x), i = 1,..., N 1 are the principal curvatures of Ω at x. ) luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 10 / 27

Main result Remark One side of the estimate can be refined so that σ j λ j 1 h + (N 1)H where H := max x Ω H(x) with H(x) the mean curvature of Ω at x. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 11 / 27

Consequences As a consequence of our main result we have a number of corollaries luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 12 / 27

Consequences As a consequence of our main result we have a number of corollaries: Corollary (Weyl-type upper bounds for Steklov eigenvalues) Let Ω be a bounded domain of class C 2 in R N with connected boundary. Then for all j N it holds ( ) 1 j N 1 σ j A Ω + C N, Ω where A Ω > 0 depends on Ω and C N > 0 depends only on the dimension N. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 12 / 27

Consequences As a consequence of our main result we have a number of corollaries: Corollary (Weyl-type upper bounds for Steklov eigenvalues) Let Ω be a bounded domain of class C 2 in R N with connected boundary. Then for all j N it holds ( ) 1 j N 1 σ j A Ω + C N, Ω where A Ω > 0 depends on Ω and C N > 0 depends only on the dimension N. The corollary follows from the main result and from upper bounds for Laplacian eigenvalues on Ω. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 12 / 27

Consequences Theorem (Asymptotically sharp upper bounds for Steklov Riesz-means) Let Ω R N be a bounded domain with connected boundary Ω of class C 2. Then for all z 0 (z σ j ) 2 + j=0 2 N(N + 1) (2π) (N 1) ω N 1 Ω ( z + c Ω ) N+1. The theorem follows from the main result and from the sharp Weyl-type estimates for Laplacian eigenvalues on hypersurfaces (Harrel-Stubbe). luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 13 / 27

Consequences Theorem (Asymptotically sharp upper bounds for Steklov Riesz-means) Let Ω R N be a bounded domain with connected boundary Ω of class C 2. Then for all z 0 (z σ j ) 2 + j=0 2 N(N + 1) (2π) (N 1) ω N 1 Ω ( z + c Ω ) N+1. The theorem follows from the main result and from the sharp Weyl-type estimates for Laplacian eigenvalues on hypersurfaces (Harrel-Stubbe). Bounds are asymptotically sharp since lim z + 1 z N+1 (z σ j ) 2 + = j=0 2 N(N + 1) (2π) (N 1) ω N 1 Ω. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 13 / 27

Consequences Corollary (Sharp upper bounds for the trace of the Steklov heat kernel) Let Ω be a bounded domain of class C 2 in R N with connected boundary. Then e σ j t j=0 for all t > 0, where Γ(a, b) = 1 N(N + 1) (2π) (N 1) ω N 1 Ω t N 1 e c Ωt Γ(N + 2, c Ω t) b t a 1 e t dt. This corollary by Laplace transforming the inequality on Riesz-means. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 14 / 27

Consequences Corollary (Sharp upper bounds for the trace of the Steklov heat kernel) Let Ω be a bounded domain of class C 2 in R N with connected boundary. Then e σ j t j=0 for all t > 0, where Γ(a, b) = 1 N(N + 1) (2π) (N 1) ω N 1 Ω t N 1 e c Ωt Γ(N + 2, c Ω t) b t a 1 e t dt. This corollary by Laplace transforming the inequality on Riesz-means. The estimate is sharp as t 0 + since it implies lim sup t N 1 e σ j t (2π) N 1 B N 1 Γ(N) Ω. t 0 + j=0 luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 14 / 27

Consequences Corollary (Weyl-type lower bounds) Let Ω be a bounded domain of class C 2 in R N with connected boundary. Then for all j N: with r N = σ j r N 2πω 1 N 1 N 1 N 1. eγ(n + 1) 1/N ( ) 1 j + 1 N 1 cω Ω luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 15 / 27

Consequences Corollary (Weyl-type lower bounds) Let Ω be a bounded domain of class C 2 in R N with connected boundary. Then for all j N: with r N = σ j r N 2πω 1 N 1 N 1 N 1. eγ(n + 1) 1/N ( ) 1 j + 1 N 1 cω Ω This corollary is an immediate consequence of the bounds on the Steklov heat kernel. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 15 / 27

Strategy of the proof The key point is to prove that for an harmonic function v in Ω the L 2 ( Ω)-norm of the normal derivative and the tangential gradient are equivalent luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 16 / 27

Strategy of the proof The key point is to prove that for an harmonic function v in Ω the L 2 ( Ω)-norm of the normal derivative and the tangential gradient are equivalent, namely i) Ω Ωv 2 dσ Ω ( v ν ) 2 dσ + 2cΩ ( Ω ( v ) ) 1 2 2 ν dσ luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 16 / 27

Strategy of the proof The key point is to prove that for an harmonic function v in Ω the L 2 ( Ω)-norm of the normal derivative and the tangential gradient are equivalent, namely i) ii) Ω Ωv 2 dσ Ω ( ( v ) ) 1 2 2 Ω ν dσ c Ω + ( v ν ) 2 dσ + 2cΩ ( Ω ( v ) ) 1 2 2 ν dσ c 2 Ω + Ω Ωv 2 dσ luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 16 / 27

Strategy of the proof The key point is to prove that for an harmonic function v in Ω the L 2 ( Ω)-norm of the normal derivative and the tangential gradient are equivalent, namely i) ii) Ω Ωv 2 dσ Ω ( ( v ) ) 1 2 2 Ω ν dσ c Ω + ( v ν ) 2 dσ + 2cΩ ( Ω ( v ) ) 1 2 2 ν dσ c 2 Ω + Ω Ωv 2 dσ If we formally substitute λ j ( Ω Ωv 2 ( dσ and σ j v ) 2 ) 1 2 Ω ν dσ, i) and ii) become λ j σj 2 + 2c Ω σ j, σ j c Ω + cω 2 + λ j. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 16 / 27

Strategy of the proof The bounds follow from the min-max principle for σ j and λ j and the fact that both the Laplace-Beltrami eigenfunctions and the restrictions of the Steklov eigenfunction to Ω form a Hilbert basis of L 2 ( Ω). In particular, for λ j : λ j = inf V H 1 ( Ω) dimv =j+1 sup 0 v V Ω v 2 dσ=1 Ω Ω v 2 dσ luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 17 / 27

Strategy of the proof The bounds follow from the min-max principle for σ j and λ j and the fact that both the Laplace-Beltrami eigenfunctions and the restrictions of the Steklov eigenfunction to Ω form a Hilbert basis of L 2 ( Ω). In particular, for λ j : λ j = inf V H 1 ( Ω) dimv =j+1 sup 0 v V Ω v 2 dσ=1 Ω Ω v 2 dσ sup 0 v V S Ω Ω v 2 dσ=1 Ω v 2 dσ luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 17 / 27

Strategy of the proof The bounds follow from the min-max principle for σ j and λ j and the fact that both the Laplace-Beltrami eigenfunctions and the restrictions of the Steklov eigenfunction to Ω form a Hilbert basis of L 2 ( Ω). In particular, for λ j : λ j = where V S = on Ω. inf V H 1 ( Ω) dimv =j+1 sup 0 v V Ω v 2 dσ=1 Ω Ω v 2 dσ sup 0 v V S Ω Ω v 2 dσ=1 Ω v 2 dσ u 0 Ω,..., u j Ω, u i are the first j + 1 Steklov eigenfunctions luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 17 / 27

Strategy of the proof The bounds follow from the min-max principle for σ j and λ j and the fact that both the Laplace-Beltrami eigenfunctions and the restrictions of the Steklov eigenfunction to Ω form a Hilbert basis of L 2 ( Ω). In particular, for λ j : λ j = where V S = inf V H 1 ( Ω) dimv =j+1 sup 0 v V Ω v 2 dσ=1 Ω Ω v 2 dσ sup 0 v V S Ω Ω v 2 dσ=1 Ω v 2 dσ u 0 Ω,..., u j Ω, u i are the first j + 1 Steklov eigenfunctions on Ω. The bounds follows then from i). luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 17 / 27

Strategy of the proof For σ j : σ j = inf V H 1 (Ω) dimv =j+1 sup 0 v V Ω v 2 dσ=1 Ω v 2 dx luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 18 / 27

Strategy of the proof For σ j : σ j = inf V H 1 (Ω) dimv =j+1 sup 0 v V Ω v 2 dσ=1 Ω v 2 dx sup v 2 dx 0 v V LB Ω Ω v 2 dσ=1 luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 18 / 27

Strategy of the proof For σ j : σ j = inf V H 1 (Ω) dimv =j+1 sup 0 v V Ω v 2 dσ=1 Ω v 2 dx sup v 2 dx 0 v V LB Ω Ω v 2 dσ=1 where V LB = ϕ 0,..., ϕ j, ϕ i are the harmonic extension in Ω of the first j + 1 Laplace-Beltrami eigenfunctions on Ω. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 18 / 27

Strategy of the proof For σ j : σ j = inf V H 1 (Ω) dimv =j+1 sup 0 v V Ω v 2 dσ=1 Ω v 2 dx sup v 2 dx 0 v V LB Ω Ω v 2 dσ=1 where V LB = ϕ 0,..., ϕ j, ϕ i are the harmonic extension in Ω of the first j + 1 Laplace-Beltrami eigenfunctions on Ω. Since v V LB are harmonic, Ω v 2 dx and the bounds follow from ii). ( Ω 1 ( ) v 2 2 dσ) ν luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 18 / 27

Strategy of the proof For σ j : σ j = inf V H 1 (Ω) dimv =j+1 sup 0 v V Ω v 2 dσ=1 Ω v 2 dx sup v 2 dx 0 v V LB Ω Ω v 2 dσ=1 where V LB = ϕ 0,..., ϕ j, ϕ i are the harmonic extension in Ω of the first j + 1 Laplace-Beltrami eigenfunctions on Ω. Since v V LB are harmonic, Ω v 2 dx and the bounds follow from ii). It remains then to prove i) and ii). ( Ω 1 ( ) v 2 2 dσ) ν luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 18 / 27

Strategy of the proof Inequalities i) and ii) follow by applying a Rellich-Pohozaev identity for harmonic functions v and Lipschitz vector fields F : Ω v ν F vdσ 1 v 2 F νdσ 2 Ω + 1 v 2 divfdx 2 Ω Ω (DF v) vdx = 0, luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 19 / 27

Strategy of the proof Inequalities i) and ii) follow by applying a Rellich-Pohozaev identity for harmonic functions v and Lipschitz vector fields F : Ω v ν F vdσ 1 v 2 F νdσ 2 Ω + 1 v 2 divfdx 2 We need a very specific F in order to obtain i) and ii). Ω Ω (DF v) vdx = 0, luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 19 / 27

Strategy of the proof Inequalities i) and ii) are consequence of the following choice { 0, if x Ω \ ω h, F (x) := η, if x ω h, luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 20 / 27

Strategy of the proof Inequalities i) and ii) are consequence of the following choice { 0, if x Ω \ ω h, F (x) := η, if x ω h, where ω h := {x Ω : dist(x, Ω) < h}, luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 20 / 27

Strategy of the proof Inequalities i) and ii) are consequence of the following choice { 0, if x Ω \ ω h, F (x) := η, if x ω h, where ω h := {x Ω : dist(x, Ω) < h}, the number h is chosen to be the maximal possible tubular radius luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 20 / 27

Strategy of the proof Inequalities i) and ii) are consequence of the following choice { 0, if x Ω \ ω h, F (x) := η, if x ω h, where ω h := {x Ω : dist(x, Ω) < h}, the number h is chosen to be the maximal possible tubular radius, and η(x) := (h dist(x, Ω))2. 2 luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 20 / 27

Strategy of the proof By construction F (x) = hν(x) on Ω and F (x) = 0 if dist(x, Ω) = h. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 21 / 27

Strategy of the proof By construction F (x) = hν(x) on Ω and F (x) = 0 if dist(x, Ω) = h. Plugging F into the Rellich-Pohozaev identity we obtain that for harmonic functions v it holds Ω ( ) v 2 dσ ν Ω Ω v 2 dσ = 1 ( 2(D 2 η v) v v 2 η ) dx. h ω h luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 21 / 27

Strategy of the proof Let ρ 1 (x),..., ρ N (x) the eigenvalues of D 2 η(x) luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 22 / 27

Strategy of the proof Let ρ 1 (x),..., ρ N (x) the eigenvalues of D 2 η(x), we have { (h dist(x, Ω))κi (x ) 1 dist(x, Ω)κ ρ i (x) = i (x ), if i = 1,..., N 1, 1, if i = N, where x is the nearest point to x on Ω. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 22 / 27

Strategy of the proof Let ρ 1 (x),..., ρ N (x) the eigenvalues of D 2 η(x), we have { (h dist(x, Ω))κi (x ) 1 dist(x, Ω)κ ρ i (x) = i (x ), if i = 1,..., N 1, 1, if i = N, where x is the nearest point to x on Ω. 2(D 2 η v) v v 2 ηdx (1 + (N 1) H h) ω h This and suitable estimates imply i) and ii). Ω v 2 dx. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 22 / 27

Example 1: convex domains If Ω is a bounded and convex domain of class C 2 in R N then and where K = max x Ω i=1,...,n 1 λ j σ 2 j + (N 1)K σ j σ j K K 2 + 2 4 + λ j, κ i (x). luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 23 / 27

Example 1: convex domains If Ω is a bounded and convex domain of class C 2 in R N then and where K = max x Ω i=1,...,n 1 λ j σ 2 j + (N 1)K σ j σ j K K 2 + 2 4 + λ j, κ i (x). In particular ( ) 1 j N 1 σ j K + C N. Ω luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 23 / 27

Example 2: balls It is known that if Ω is a ball of radius R in R N, then given a Steklov eigenvalue σ it is of the form σ = l R, for some l N. The corresponding eigenfunction are the restriction to Ω of the harmonic polynomials in R N. Given a Laplace-Beltrami eigenvalue λ, it is of the form l(l + N 2) λ = R 2 for some l N. The corresponding eigenfunctions are the spherical harmonics on Ω. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 24 / 27

Example 2: balls If Ω is a ball, η(x) = x 2 2 so we can take F (x) = x in the Rellich-Pohozaev identity luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 25 / 27

Example 2: balls We obtain that for a function v harmonic in Ω ( ) v 2 Ω v 2 dσ = dσ + N 2 ν R Ω Ω Ω v 2 dσ and Ω ( ) v 2 dσ = Ω v 2 dσ N 2 v 2 dσ, ν Ω R Ω luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 26 / 27

Example 2: balls We obtain that for a function v harmonic in Ω ( ) v 2 Ω v 2 dσ = dσ + N 2 ν R and Ω Ω which imply Ω Ω v 2 dσ ( ) v 2 dσ = Ω v 2 dσ N 2 v 2 dσ, ν Ω R Ω λ j σ 2 j + (N 2) σ j R and σ j (N 2) 2 4R 2 + λ j N 2 2R luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 26 / 27

Example 2: balls We obtain that for a function v harmonic in Ω ( ) v 2 Ω v 2 dσ = dσ + N 2 ν R and Ω Ω which imply λ j σ 2 j + and in particular Ω Ω v 2 dσ ( ) v 2 dσ = Ω v 2 dσ N 2 v 2 dσ, ν Ω R Ω (N 2) σ j R and σ j λ j = σ 2 j + (N 2) 2 4R 2 + λ j N 2 2R (N 2) σ j. R luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 26 / 27

Possible improvements In the constant c Ω replace h with more suitable quantities (Cheeger constants, inradius); luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 27 / 27

Possible improvements In the constant c Ω replace h with more suitable quantities (Cheeger constants, inradius); In the constant c Ω replace max x Ω H(x) with some H L p ( Ω); luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 27 / 27

Possible improvements In the constant c Ω replace h with more suitable quantities (Cheeger constants, inradius); In the constant c Ω replace max x Ω H(x) with some H L p ( Ω); Remove the hypothesis of connected boundary; luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 27 / 27

Possible improvements In the constant c Ω replace h with more suitable quantities (Cheeger constants, inradius); In the constant c Ω replace max x Ω H(x) with some H L p ( Ω); Remove the hypothesis of connected boundary; Less regular domains (piecewise C 1, polygonal) with other choices of F in the Rellich-Pohozaev identity. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 27 / 27

THANK YOU luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, 2017 27 / 27