Optimization of lowest Robin eigenvalues on 2-manifolds and - PowerPoint PPT Presentation

Optimization of lowest Robin eigenvalues on 2-manifolds and unbounded cones Vladimir Lotoreichik in collaboration with M. Khalile Czech Academy of Sciences, Řež near Prague OTKR, Vienna, 19.12.2019 V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 1 / 19

Outline 1 Motivation & background 2 Optimization on 2-manifolds 3 Optimization on unbounded cones V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 2 / 19

Outline 1 Motivation & background 2 Optimization on 2-manifolds 3 Optimization on unbounded cones V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 3 / 19

The Robin eigenvalue problem V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 4 / 19

The Robin eigenvalue problem Ω ⊂ R d – bounded domain with sufficiently smooth (at least Lipschitz) ∂ Ω. V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 4 / 19

The Robin eigenvalue problem Ω ⊂ R d – bounded domain with sufficiently smooth (at least Lipschitz) ∂ Ω. β ∈ R – coupling constant. V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 4 / 19

The Robin eigenvalue problem Ω ⊂ R d – bounded domain with sufficiently smooth (at least Lipschitz) ∂ Ω. β ∈ R – coupling constant. The spectral problem − ∆ u = λ u , in Ω , ∂ ν u + β u = 0 , on ∂ Ω , V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 4 / 19

The Robin eigenvalue problem Ω ⊂ R d – bounded domain with sufficiently smooth (at least Lipschitz) ∂ Ω. β ∈ R – coupling constant. The spectral problem − ∆ u = λ u , in Ω , ∂ ν u + β u = 0 , on ∂ Ω , The Robin Laplacian on Ω � � H 1 (Ω) ∋ u �→ |∇ u | 2 d x + β | u | 2 d σ β, Ω in L 2 (Ω) H β, Ω = H ∗ = ⇒ Ω ∂ Ω V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 4 / 19

The Robin eigenvalue problem Ω ⊂ R d – bounded domain with sufficiently smooth (at least Lipschitz) ∂ Ω. β ∈ R – coupling constant. The spectral problem − ∆ u = λ u , in Ω , ∂ ν u + β u = 0 , on ∂ Ω , The Robin Laplacian on Ω � � H 1 (Ω) ∋ u �→ |∇ u | 2 d x + β | u | 2 d σ β, Ω in L 2 (Ω) H β, Ω = H ∗ = ⇒ Ω ∂ Ω Purely discrete spectrum λ β 1 (Ω) ≤ λ β 2 (Ω) ≤ · · · ≤ λ β ( β · λ β k (Ω) ≤ . . . 1 (Ω) ≥ 0) V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 4 / 19

Optimization of λ β 1 (Ω) V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 5 / 19

Optimization of λ β 1 (Ω) The ball (the disk) is: 1 minimizer ( β > 0, | Ω | = const ) Bossel-86, Daners-06 2 maximizer ( β < 0, | ∂ Ω | = const , d =2), Antunes-Freitas-Krejčiřík-17 3 maximizer ( β < 0, | ∂ Ω | = const , Ω – convex, d ≥ 3). Bucur-Ferone-Nitsch-Trombetti-18, Vikulova-19 V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 5 / 19

Optimization of λ β 1 (Ω) The ball (the disk) is: 1 minimizer ( β > 0, | Ω | = const ) Bossel-86, Daners-06 2 maximizer ( β < 0, | ∂ Ω | = const , d =2), Antunes-Freitas-Krejčiřík-17 3 maximizer ( β < 0, | ∂ Ω | = const , Ω – convex, d ≥ 3). Bucur-Ferone-Nitsch-Trombetti-18, Vikulova-19 The ball (the disk) is not an optimizer if ( β < 0, | Ω | = const , d ≥ 2), Freitas-Krejčiřík-15 V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 5 / 19

Optimization of λ β 1 (Ω) The ball (the disk) is: 1 minimizer ( β > 0, | Ω | = const ) Bossel-86, Daners-06 2 maximizer ( β < 0, | ∂ Ω | = const , d =2), Antunes-Freitas-Krejčiřík-17 3 maximizer ( β < 0, | ∂ Ω | = const , Ω – convex, d ≥ 3). Bucur-Ferone-Nitsch-Trombetti-18, Vikulova-19 The ball (the disk) is not an optimizer if ( β < 0, | Ω | = const , d ≥ 2), Freitas-Krejčiřík-15 The analogue of (1) for manifolds: Chami-Ginoux-Habib-19 V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 5 / 19

Optimization of λ β 1 (Ω) The ball (the disk) is: 1 minimizer ( β > 0, | Ω | = const ) Bossel-86, Daners-06 2 maximizer ( β < 0, | ∂ Ω | = const , d =2), Antunes-Freitas-Krejčiřík-17 3 maximizer ( β < 0, | ∂ Ω | = const , Ω – convex, d ≥ 3). Bucur-Ferone-Nitsch-Trombetti-18, Vikulova-19 The ball (the disk) is not an optimizer if ( β < 0, | Ω | = const , d ≥ 2), Freitas-Krejčiřík-15 The analogue of (1) for manifolds: Chami-Ginoux-Habib-19 1 st main objective To generalize (2) for 2-manifolds. V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 5 / 19

Unbounded cones V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 6 / 19

Unbounded cones m ⊂ S 2 – a bounded, simply-connected, domain with C 2 -smooth boundary. V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 6 / 19

Unbounded cones m ⊂ S 2 – a bounded, simply-connected, domain with C 2 -smooth boundary. The cone Λ m ⊂ R 3 Λ m := R + × m (in the spherical coordinates). V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 6 / 19

Unbounded cones m ⊂ S 2 – a bounded, simply-connected, domain with C 2 -smooth boundary. The cone Λ m ⊂ R 3 Λ m := R + × m (in the spherical coordinates). Λ m m V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 6 / 19

The Robin Laplacian on unbounded cones V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 7 / 19

The Robin Laplacian on unbounded cones β < 0 – coupling constant. V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 7 / 19

The Robin Laplacian on unbounded cones β < 0 – coupling constant. The Robin Laplacian on Λ m � � H 1 (Λ m ) ∋ u �→ |∇ u | 2 d x + β | u | 2 d σ β, Λ m in L 2 (Λ m ) H β, Λ m =H ∗ ⇒ Λ m ∂ Λ m V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 7 / 19

The Robin Laplacian on unbounded cones β < 0 – coupling constant. The Robin Laplacian on Λ m � � H 1 (Λ m ) ∋ u �→ |∇ u | 2 d x + β | u | 2 d σ β, Λ m in L 2 (Λ m ) H β, Λ m =H ∗ ⇒ Λ m ∂ Λ m H β, Λ m ≃ β 2 H − 1 , Λ m = ⇒ H Λ m := H − 1 , Λ m . V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 7 / 19

Spectral properties of H Λ m V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 8 / 19

Spectral properties of H Λ m Proposition ( Pankrashkin-16 ) σ ess (H Λ m ) = [ − 1 , ∞ ) . # σ d (H Λ m ) = ∞ if | m | < 2 π V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 8 / 19

Spectral properties of H Λ m Proposition ( Pankrashkin-16 ) σ ess (H Λ m ) = [ − 1 , ∞ ) . # σ d (H Λ m ) = ∞ if | m | < 2 π The eigenvalues of H Λ m λ 1 (Λ m ) ≤ λ 2 (Λ m ) ≤ · · · ≤ λ k (Λ m ) ≤ · · · ≤ − 1 . arise in the leading order of the large coupling asymptotics ( β → −∞ ) for the Robin eigenvalues on a bounded domain with a conical singualarity. V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 8 / 19

Spectral properties of H Λ m Proposition ( Pankrashkin-16 ) σ ess (H Λ m ) = [ − 1 , ∞ ) . # σ d (H Λ m ) = ∞ if | m | < 2 π The eigenvalues of H Λ m λ 1 (Λ m ) ≤ λ 2 (Λ m ) ≤ · · · ≤ λ k (Λ m ) ≤ · · · ≤ − 1 . arise in the leading order of the large coupling asymptotics ( β → −∞ ) for the Robin eigenvalues on a bounded domain with a conical singualarity. 2 nd main objective To obtain an isoperimetric inequality for λ 1 (Λ m ). V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 8 / 19

Outline 1 Motivation & background 2 Optimization on 2-manifolds 3 Optimization on unbounded cones V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 9 / 19

Geometric setting V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 10 / 19

Geometric setting The class of 2-manifolds M – compact, simply-connected, C ∞ -smooth two-dimensional manifold with C 2 -boundary ∂ M . V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 10 / 19

Geometric setting The class of 2-manifolds M – compact, simply-connected, C ∞ -smooth two-dimensional manifold with C 2 -boundary ∂ M . M is diffeomorphic to the unit disk in R 2 V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 10 / 19

Geometric setting The class of 2-manifolds M – compact, simply-connected, C ∞ -smooth two-dimensional manifold with C 2 -boundary ∂ M . M is diffeomorphic to the unit disk in R 2 K : M → R – Gauss curvature of M . V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 10 / 19

Geometric setting The class of 2-manifolds M – compact, simply-connected, C ∞ -smooth two-dimensional manifold with C 2 -boundary ∂ M . M is diffeomorphic to the unit disk in R 2 K : M → R – Gauss curvature of M . − ∆ & ∇ – stand for the Laplace-Beltrami operator and the gradient on M V. Lotoreichik (NPI CAS) Optimization on manifolds 19.12.2019 10 / 19