Geometric bounds for Steklov eigenvalues

Transcription

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

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

3 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), / 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, / 27

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

5 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, / 27

6 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, / 27

7 Geometric inequalities We have considered the issue of finding upper bounds for the Steklov eigenvalues. (EPFL) Steklov eigenvalues June 20, / 27

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

9 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, / 27

10 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, / 27

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

12 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), σ j 2πj Ω J. Hersch, L. E. Payne, M. M. Schiffer, Some inequalities for Stekloff eigenvalues. Arch. Rational Mech. Anal., 57 (1975), luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, / 27

13 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), σ j 2πj Ω J. Hersch, L. E. Payne, M. M. Schiffer, Some inequalities for Stekloff eigenvalues. Arch. Rational Mech. Anal., 57 (1975), A. Girouard, I. Polterovich, Upper bounds for Steklov eigenvalues on surfaces. Electron. Res. Announc. Math. Sci., 19 (2012), luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, / 27

14 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, / 27

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

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

17 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, / 27

18 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, / 27

19 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, / 27

20 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, / 27

21 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, / 27

22 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, / 27

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

24 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), Asymptotic formulas suggest that for large j σ j λ j luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, / 27

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

26 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, / 27

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: (2017). To appear on the Journal of Spectral Theory. luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, / 27

28 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, / 27

29 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, / 27

30 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, / 27

31 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, / 27

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

33 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, / 27

34 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, / 27

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

36 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, / 27

37 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, / 27

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

39 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, / 27

40 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, / 27

41 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, / 27

42 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, / 27

43 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, / 27

44 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 ) ) ν dσ luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, / 27

45 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 ) ) Ω ν dσ c Ω + ( v ν ) 2 dσ + 2cΩ ( Ω ( v ) ) ν dσ c 2 Ω + Ω Ωv 2 dσ luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, / 27

46 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 ) ) Ω ν dσ c Ω + ( v ν ) 2 dσ + 2cΩ ( Ω ( v ) ) 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, / 27

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

48 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, / 27

49 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, / 27

50 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, / 27

51 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, / 27

52 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, / 27

53 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, / 27

54 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, / 27

55 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, / 27

56 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, / 27

57 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, / 27

58 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, / 27

59 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, / 27

60 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, / 27

61 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, / 27

62 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, / 27

63 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, / 27

64 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, / 27

65 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, / 27

66 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, / 27

67 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, / 27

68 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, / 27

69 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 λ j, κ i (x). luigi.provenzano@epfl.ch (EPFL) Steklov eigenvalues June 20, / 27

70 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 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, / 27

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

72 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, / 27

73 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, / 27

74 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, / 27

75 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, / 27

76 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, / 27

77 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, / 27

78 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, / 27

79 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, / 27

80 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, / 27

81 THANK YOU (EPFL) Steklov eigenvalues June 20, / 27