Recent progress on a sharp lower bound for first (nonzero) Steklov - PowerPoint PPT Presentation

Recent progress on a sharp lower bound for first (nonzero) Steklov eigenvalue Chao Xia (Xiamen University) (joint with Changwei Xiong) Asia-Pacific Analysis and PDE Seminar May 11th, 2020 1 / 30

Table of Contents Review of eigenvalue lower bound Introduction to Steklov eigenvalue Review of Steklov eigenvalue estimates Our result and proof 2 / 30

Eigenvalue Lower Bound Theorem (Lichernowicz 1958, Obata 1962) Let ( M n , g ) be a closed Riemannian n-manifold with Ric g ≥ ( n − 1) K > 0 . Then λ 1 ( M ) ≥ nK . Equality holds if and only if M ∼ = S n ( 1 K ) . √ λ 1 ( M ) is first (nonzero) eigenvalue of ∆ M . Variational characterization ∫ M |∇ f | 2 λ 1 ( M ) = inf . ∫ M f 2 f ∈ C 1 ( M ) , ∫ M f =0 Maximum principle or Integral method on Bochner’s formula. 3 / 30

Eigenvalue Lower Bound Integral method on Bochner’s formula ∫ ∫ (∆ f ) 2 − |∇ 2 f | 2 = Ric g ( ∇ f , ∇ f ) . M M Using |∇ 2 f | 2 ≥ 1 n (∆ f ) 2 and Ric g ≥ ( n − 1) K > 0, n − 1 ∫ n − 1 ∫ λ 2 1 f 2 (∆ f ) 2 = n n M M ∫ ∫ |∇ f | 2 = ( n − 1) K λ 1 f 2 . ≥ ( n − 1) K M M Equality by Obata’s theorem: A closed Riemannian n -manifold which admits a solution to ∇ 2 f = − Kfg must be S n ( 1 K ). √ 4 / 30

Eigenvalue Lower Bound Theorem Let ( M n , g ) be a compact Riemannian n-manifold with boundary Σ . (Reilly ’77) Assume Ric g ≥ ( n − 1) K > 0 and H Σ ≥ 0 (mean convex boundary). Then λ D 1 ( M ) ≥ nK. (C. Y. Xia ’88, Escobar ’90) Assume Ric g ≥ ( n − 1) K > 0 and h Σ ≥ 0 (convex boundary). Then λ N 1 ( M ) ≥ nK. Equality holds if and only if M ∼ + ( 1 = S n K ) . √ First Dirichlet eigenvalue and Neumann eigenvalue of ∆ M M |∇ f | 2 ∫ λ D 1 ( M ) = inf . ∫ M f 2 f ∈ C 1 ( M ) , f | Σ =0 ∫ M |∇ f | 2 λ N 1 ( M ) = inf . ∫ M f 2 f ∈ C 1 ( M ) , ∫ M f =0 5 / 30

Eigenvalue Lower Bound Theorem (Li-Yau ’80, Zhong-Yang ’84, Hang-Wang ’07) Let ( M n , g ) be a compact Riemannian n-manifold possibly with convex boundary Σ . Assume Ric g ≥ 0 . Then 1 ( M ) ≥ π 2 λ N d 2 , where d = diam ( M ) . Equality holds if and only if M is a 1 -dmensional round circle or a segment. 6 / 30

Eigenvalue Lower Bound Theorem (Li-Yau ’80, Zhong-Yang ’84, Hang-Wang ’07) Let ( M n , g ) be a compact Riemannian n-manifold possibly with convex boundary Σ . Assume Ric g ≥ 0 . Then 1 ( M ) ≥ π 2 λ N d 2 , where d = diam ( M ) . Equality holds if and only if M is a 1 -dmensional round circle or a segment. Theorem (Andrews-Clutterbuck ’11) Let Ω ⊂ R n be a bounded convex domain and λ be the Dirichlet eigenvalues for Schr¨ odinger operator ∆ + V with convex V . Then λ 2 − λ 1 ≥ 3 π 2 d 2 . 3 π 2 d 2 is the spectral gap for 1 -dimensional Laplacian on [ − D 2 , D 2 ] . 6 / 30

Steklov Eigenvalue Let ( M n , g ) be a compact Riemannian n -manifold with boundary Σ. For f ∈ C ∞ (Σ), let ˆ f be its harmonic extension in M , ∆ˆ ˆ f = 0 in M , f = f on Σ . Dirichlet-to-Neumann operator L : C ∞ (Σ) → C ∞ (Σ) �→ ∂ ˆ f f ∂ν . ν is outward unit normal to Σ. L is linear, nonnegative, self-adjoint operator with compact inverse, hence its spectrum is given by 0 = σ 0 < σ 1 ≤ σ 2 ≤ · · · → ∞ . σ i is called Steklov eigenvalues, first considered by Steklov 1900 in Euclidean space. 7 / 30

Steklov Eigenvalue Steklov eigenvalues: ∂ f ∆ f = 0 in M , ∂ν = σ f on Σ . Variational characterization: M |∇ f | 2 ∫ σ 1 ( M ) = inf , ∫ Σ f 2 f ∈ C 1 ( M ) , ∫ Σ f =0 M |∇ f | 2 ∫ σ k ( M ) = inf sup . ∫ Σ f 2 S⊂ C 1 ( M ) , 0 ̸ = f ∈S dim S = k +1 8 / 30

Steklov Eigenvalue Steklov eigenvalues for Euclidean unit disk B 1 ⊂ R 2 : 0 , 1 , 1 , 2 , 2 , · · · , k , k , · · · Corresponding Steklov eigenfunctions: 1 , r cos ϕ, r sin ϕ, · · · , r k cos k ϕ, r k sin k ϕ, · · · Steklov eigenvalues for Euclidean unit ball B 1 ⊂ R n : ( n + k − 1 ) ( n + k − 3 ) k ∈ N with multiplicity − n − 1 n − 1 Corresponding Steklov eigenfunctions: homogeneous harmonic polynomials of degree k . 9 / 30

Lower Bound for Steklov Eigenvalue Payne ’70: M 2 ⊂ R 2 , boundary geodesic curvature k g (Σ) ≥ c > 0 ⇒ σ 1 ≥ c . Equality holds iff M = B 2 ( 1 c ). Escobar ’97: ( M 2 , g ), Gauss curvature K ≥ 0 and k g (Σ) ≥ c > 0 ⇒ σ 1 ≥ c . Equality holds iff M ∼ = B 2 ( 1 c ). Escobar ’97: ( M n , g ) , n ≥ 3, Ric g ≥ 0 and all boundary principal curvatures κ (Σ) ≥ c > 0 ⇒ σ 1 > c 2 . Escobar’s Conjecture: ( M n , g ) , n ≥ 3, Ric g ≥ 0 and κ (Σ) ≥ c > 0 ⇒ σ 1 ≥ c . Equality holds iff M ∼ = B n ( 1 c ). (Compare to Lichernowicz-Obata’s theorem) Even unknown for Euclidean case M n ⊂ R n , n ≥ 3. 10 / 30

Isoperimetric upper bound for Steklov Eigenvalue Two dimensions ( M 2 , g ) Weinstock ’54: simply connected, σ 1 L ≤ 2 π = ( σ 1 L )( B 2 ) ( L is boundary length). Equality holds iff ∃ a conformal diffeomorphism ϕ : M → B 2 such that ϕ | Σ is an isometry. Fraser-Schoen ’11: , σ 1 L ≤ 2( g + r ) π , genus g and boundary components r . Fraser-Schoen ’16: annulus type, σ 1 L ≤ ( σ 1 L )( M cc ), M cc is critical catenoid in B 3 . Fraser-Schoen ’16: If ( σ 1 L )( M , g 0 ) = max g ( σ 1 L )( M , g ) , then there exist independent eigenfunction u 1 , · · · , u n which give a conformal free boundary minimal immersion u i : ( M , g 0 ) → B n with u i | Σ is an isometry. Matthiesen-Petrides ’20 (arXiv): any topological type, existence of smooth maximal metric for σ 1 L . 11 / 30

Isoperimetric upper bound for Steklov Eigenvalue Higher dimensions M n ⊂ R n , n ≥ 3 1 1 n ≤ ( σ 1 Vol n )( B n ), Equality holds iff Brock ’01: σ 1 Vol M n = B n ( r ). Bucur-Ferone-Nitsch-Trombetti ’17: convex, 1 1 n − 1 ≤ ( σ 1 Area n − 1 )( B n ), Equality holds iff σ 1 Area M n = B n ( r ). Fraser-Schoen ’17: ∃ smooth contractible domain 1 1 M n ⊂ R n , n ≥ 3 with ( σ 1 Area n − 1 )( M ) > ( σ 1 Area n − 1 )( B n ) 12 / 30

Comparison of Steklov Eigenvalue with Boundary Eigenvalue Q.L.Wang-C.Y.Xia ’09: ( M n , g ) , n ≥ 3, Ric g ≥ 0 and κ (Σ) ≥ c > 0, then √ λ 1 √ √ λ 1 − ( n − 1) c 2 ) . σ 1 ≤ ( n − 1) c ( λ 1 + where λ 1 is first closed eigenvalue of (Σ , g Σ ). ( λ 1 ≥ ( n − 1) c 2 was proved by C.Y.Xia ’07.) 13 / 30

Comparison of Steklov Eigenvalue with Boundary Eigenvalue Q.L.Wang-C.Y.Xia ’09: ( M n , g ) , n ≥ 3, Ric g ≥ 0 and κ (Σ) ≥ c > 0, then √ λ 1 √ √ λ 1 − ( n − 1) c 2 ) . σ 1 ≤ ( n − 1) c ( λ 1 + where λ 1 is first closed eigenvalue of (Σ , g Σ ). ( λ 1 ≥ ( n − 1) c 2 was proved by C.Y.Xia ’07.) ( M n , g ) , n ≥ 3, W [2] ≥ 0 and κ (Σ) ≥ c > 0, Karpukhin ’17: then λ k σ k ≤ ( n − 1) c , n ≥ 4 , σ k ≤ 2 λ k 3 c , n = 3 . (Based on Results on Steklov eigenvalue estimates for p -forms by Raulot-Savo ’12, Yang-Yu ’17) 13 / 30

Our results Theorem (Xiong- X. ’19) Let ( M n , g ) , n ≥ 2 be a compact Riemannian n-manifold with boundary Σ . Assume Sect g ≥ 0 and κ (Σ) ≥ c > 0 . Then σ 1 ≥ c. Equality holds if and only if M ∼ = B n ( 1 c ) ⊂ R n . Escobar’s conjecture holds true for manifolds with Sect g ≥ 0. Especially, true for Euclidean domains. 14 / 30

Our results Theorem (Xiong- X. ’19) Let ( M n , g ) , n ≥ 2 be a compact Riemannian n-manifold with boundary Σ . Assume Sect g ≥ 0 and κ (Σ) ≥ c > 0 . Then λ 1 σ 1 ≤ ( n − 1) c with equality holds if and only if M ∼ = B n ( 1 c ) ⊂ R n . Moreover, λ k σ k ≤ ( n − 1) c , ∀ k . Compare with Q.L.Wang-C.Y.Xia ’09, stronger assumption and stronger conclusion; Compare with Karpukhin ’17, different assumption and same conclusion in n ≥ 4 and better conclusion in n = 3. 15 / 30

Review of Payne-Escobar’s method in n = 2. ∆ |∇ f | 2 ≥ 0, then ϕ = |∇ f | 2 attains its maximum at x 0 ∈ ∂ Ω. At x 0 ∈ ∂ Ω, consider Fermi coordinates of ∂ Ω, ∂ Ω is parametrized by arc-length γ ( s ). 0 = ∆ f | Σ = f νν + κ f ν + f ′′ = f νν + κσ 1 f + f ′′ . Then f νν = − κσ 1 f − f ′′ , and 0 ≤ ϕ ν ( s 0 ) = 2( − f ′′ − κσ 1 f ) σ 1 f + 2( σ 1 − κ ) f ′ 2 , ϕ ′ ( s 0 ) = 0 , ϕ ′′ ( s 0 ) ≤ 0 . All inequalities involves only f , f ′ , f ′′ . By simple calculation, one can show σ 1 ≥ κ ( s 0 ) ≥ c . This method fails to handle higher dimensions. 16 / 30

Review of Escobar’s method in n ≥ 3. n ≥ 3, using Reilly’s formula ∫ (∆ f ) 2 − |∇ 2 f | 2 − Ric ( ∇ f , ∇ f ) [ ] M ∫ 2 f ν ∆ Σ f + Hf 2 [ ] = ν + h ( ∇ Σ f , ∇ Σ f ) Σ Using ∆ f = 0, f ν = σ 1 f , Ric ≥ 0, h ≥ cg Σ , one gets ∫ ( c − 2 σ 1 ) |∇ Σ f | 2 + H σ 2 1 f 2 . 0 ≥ Σ Thus σ 1 > c 2 . Σ |∇ Σ f | 2 and Σ f 2 . ∫ ∫ No information between 17 / 30