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

SLIDE 1

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

SLIDE 2

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

SLIDE 3

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

SLIDE 4

The Robin eigenvalue problem

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 4 / 19

SLIDE 5

The Robin eigenvalue problem

Ω⊂ Rd – bounded domain with sufficiently smooth (at least Lipschitz) ∂Ω.

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 4 / 19

SLIDE 6

The Robin eigenvalue problem

Ω⊂ Rd – 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

SLIDE 7

The Robin eigenvalue problem

Ω⊂ Rd – bounded domain with sufficiently smooth (at least Lipschitz) ∂Ω. β ∈ R – coupling constant.

The spectral problem

−∆u = λu, in Ω, ∂νu + βu = 0,

n ∂Ω,
V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 4 / 19

SLIDE 8

The Robin eigenvalue problem

Ω⊂ Rd – bounded domain with sufficiently smooth (at least Lipschitz) ∂Ω. β ∈ R – coupling constant.

The spectral problem

−∆u = λu, in Ω, ∂νu + βu = 0,

n ∂Ω,

The Robin Laplacian on Ω

H1(Ω)∋u →

Ω

|∇u|2dx + β

∂Ω

|u|2dσ = ⇒ Hβ,Ω = H∗

β,Ω in L2(Ω)

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 4 / 19

SLIDE 9

The Robin eigenvalue problem

Ω⊂ Rd – bounded domain with sufficiently smooth (at least Lipschitz) ∂Ω. β ∈ R – coupling constant.

The spectral problem

−∆u = λu, in Ω, ∂νu + βu = 0,

n ∂Ω,

The Robin Laplacian on Ω

H1(Ω)∋u →

Ω

|∇u|2dx + β

∂Ω

|u|2dσ = ⇒ Hβ,Ω = H∗

β,Ω in L2(Ω)

Purely discrete spectrum

λβ

1(Ω) ≤ λβ 2(Ω) ≤ · · · ≤ λβ k(Ω) ≤ . . .

(β · λβ

1(Ω) ≥ 0)

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 4 / 19

SLIDE 10

Optimization of λβ

1(Ω)

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 5 / 19

SLIDE 11

Optimization of λβ

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

SLIDE 12

Optimization of λβ

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

SLIDE 13

Optimization of λβ

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

SLIDE 14

Optimization of λβ

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

1st main objective

To generalize (2) for 2-manifolds.

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 5 / 19

SLIDE 15

Unbounded cones

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 6 / 19

SLIDE 16

Unbounded cones

m ⊂ S2 – a bounded, simply-connected, domain with C2-smooth boundary.

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 6 / 19

SLIDE 17

Unbounded cones

m ⊂ S2 – a bounded, simply-connected, domain with C2-smooth boundary.

The cone Λm ⊂ R3

Λm := R+ × m (in the spherical coordinates).

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 6 / 19

SLIDE 18

Unbounded cones

m ⊂ S2 – a bounded, simply-connected, domain with C2-smooth boundary.

The cone Λm ⊂ R3

Λm := R+ × m (in the spherical coordinates).

m

Λm

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 6 / 19

SLIDE 19

The Robin Laplacian on unbounded cones

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 7 / 19

SLIDE 20

The Robin Laplacian on unbounded cones

β < 0 – coupling constant.

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 7 / 19

SLIDE 21

The Robin Laplacian on unbounded cones

β < 0 – coupling constant.

The Robin Laplacian on Λm

H1(Λm)∋u →

Λm

|∇u|2dx+β

∂Λm

|u|2dσ ⇒ Hβ,Λm =H∗

β,Λm in L2(Λm)

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 7 / 19

SLIDE 22

The Robin Laplacian on unbounded cones

β < 0 – coupling constant.

The Robin Laplacian on Λm

H1(Λm)∋u →

Λm

|∇u|2dx+β

∂Λm

|u|2dσ ⇒ Hβ,Λm =H∗

β,Λm in L2(Λm)

Hβ,Λm ≃ β2H−1,Λm = ⇒ HΛm := H−1,Λm.

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 7 / 19

SLIDE 23

Spectral properties of HΛm

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 8 / 19

SLIDE 24

Spectral properties of HΛm

Proposition (Pankrashkin-16)

σess(HΛm) = [−1, ∞). #σd(HΛm) = ∞ if |m| < 2π

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 8 / 19

SLIDE 25

Spectral properties of HΛm

Proposition (Pankrashkin-16)

σess(HΛm) = [−1, ∞). #σd(HΛm) = ∞ if |m| < 2π The eigenvalues of HΛm λ1(Λm) ≤ λ2(Λm) ≤ · · · ≤ λk(Λm) ≤ · · · ≤ −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

SLIDE 26

Spectral properties of HΛm

Proposition (Pankrashkin-16)

σess(HΛm) = [−1, ∞). #σd(HΛm) = ∞ if |m| < 2π The eigenvalues of HΛm λ1(Λm) ≤ λ2(Λm) ≤ · · · ≤ λk(Λm) ≤ · · · ≤ −1. arise in the leading order of the large coupling asymptotics (β → −∞) for the Robin eigenvalues on a bounded domain with a conical singualarity.

2nd main objective

To obtain an isoperimetric inequality for λ1(Λm).

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 8 / 19

SLIDE 27

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

SLIDE 28

Geometric setting

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 10 / 19

SLIDE 29

Geometric setting

The class of 2-manifolds

M – compact, simply-connected, C∞-smooth two-dimensional manifold with C2-boundary ∂M.

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 10 / 19

SLIDE 30

Geometric setting

The class of 2-manifolds

M – compact, simply-connected, C∞-smooth two-dimensional manifold with C2-boundary ∂M. M is diffeomorphic to the unit disk in R2

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 10 / 19

SLIDE 31

Geometric setting

The class of 2-manifolds

M – compact, simply-connected, C∞-smooth two-dimensional manifold with C2-boundary ∂M. M is diffeomorphic to the unit disk in R2 K : M → R – Gauss curvature of M.

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 10 / 19

SLIDE 32

Geometric setting

The class of 2-manifolds

M – compact, simply-connected, C∞-smooth two-dimensional manifold with C2-boundary ∂M. M is diffeomorphic to the unit disk in R2 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

SLIDE 33

Geometric setting

The class of 2-manifolds

M – compact, simply-connected, C∞-smooth two-dimensional manifold with C2-boundary ∂M. M is diffeomorphic to the unit disk in R2 K : M → R – Gauss curvature of M. −∆ & ∇ – stand for the Laplace-Beltrami operator and the gradient on M

Distance functions (x, y ∈ M)

ρM(x, y) := geodesic distance between x and y. ρ∂M(x) := inf

y∈∂M ρM(x, y).

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 10 / 19

SLIDE 34

Geometric setting

The class of 2-manifolds

M – compact, simply-connected, C∞-smooth two-dimensional manifold with C2-boundary ∂M. M is diffeomorphic to the unit disk in R2 K : M → R – Gauss curvature of M. −∆ & ∇ – stand for the Laplace-Beltrami operator and the gradient on M

Distance functions (x, y ∈ M)

ρM(x, y) := geodesic distance between x and y. ρ∂M(x) := inf

y∈∂M ρM(x, y).

RM := maxx∈M ρ∂M(x) (the in-radius of M)

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 10 / 19

SLIDE 35

Definition of the operator

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 11 / 19

SLIDE 36

Definition of the operator

The spectral problem

−∆u = λu,

n M,

∂νu + βu = 0,

n ∂M,
V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 11 / 19

SLIDE 37

Definition of the operator

The spectral problem

−∆u = λu,

n M,

∂νu + βu = 0,

n ∂M,

The Robin Laplacian on M

H1(M)∋u →

M

|∇u|2dx+β

∂M

|u|2dσ = ⇒ Hβ,M =H∗

β,M in L2(M)

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 11 / 19

SLIDE 38

Definition of the operator

The spectral problem

−∆u = λu,

n M,

∂νu + βu = 0,

n ∂M,

The Robin Laplacian on M

H1(M)∋u →

M

|∇u|2dx+β

∂M

|u|2dσ = ⇒ Hβ,M =H∗

β,M in L2(M)

Purely discrete spectrum

λβ

1(M) ≤ λβ 2(M) ≤ · · · ≤ λβ k(M) ≤ . . .

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 11 / 19

SLIDE 39

Definition of the operator

The spectral problem

−∆u = λu,

n M,

∂νu + βu = 0,

n ∂M,

The Robin Laplacian on M

H1(M)∋u →

M

|∇u|2dx+β

∂M

|u|2dσ = ⇒ Hβ,M =H∗

β,M in L2(M)

Purely discrete spectrum

λβ

1(M) ≤ λβ 2(M) ≤ · · · ≤ λβ k(M) ≤ . . .

K ≡ 0 corresponds to the flat case.

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 11 / 19

SLIDE 40

Main result

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 12 / 19

SLIDE 41

Main result

Let the constant K◦ ≥ 0 be such that supx∈M K(x) ≤ K◦

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 12 / 19

SLIDE 42

Main result

Let the constant K◦ ≥ 0 be such that supx∈M K(x) ≤ K◦ B◦ – geodesic disk in 2- manifold N◦ (∂N◦ = ∅) of

const. Gauss curv. K◦.

N◦ ≃

  

R2, K◦ = 0, S2

1/√K◦,

K◦ > 0.

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 12 / 19

SLIDE 43

Main result

Let the constant K◦ ≥ 0 be such that supx∈M K(x) ≤ K◦ B◦ – geodesic disk in 2- manifold N◦ (∂N◦ = ∅) of

const. Gauss curv. K◦.

N◦ ≃

  

R2, K◦ = 0, S2

1/√K◦,

K◦ > 0.

Theorem (Khalile-L-19, Cartan-Hadamard manifolds (K◦ = 0))

|∂M| = |∂B◦| = ⇒ λβ

1(M) ≤ λβ 1(B◦),

∀ β < 0.

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 12 / 19

SLIDE 44

Main result

Let the constant K◦ ≥ 0 be such that supx∈M K(x) ≤ K◦ B◦ – geodesic disk in 2- manifold N◦ (∂N◦ = ∅) of

const. Gauss curv. K◦.

N◦ ≃

  

R2, K◦ = 0, S2

1/√K◦,

K◦ > 0.

Theorem (Khalile-L-19, Cartan-Hadamard manifolds (K◦ = 0))

|∂M| = |∂B◦| = ⇒ λβ

1(M) ≤ λβ 1(B◦),

∀ β < 0.

Theorem (Khalile-L-19, K◦ > 0)

|∂M| = |∂B◦|

K◦ · max{|M|, |B◦|} ≤ 2π = ⇒ λβ

1(M) ≤ λβ 1(B◦),

∀ β < 0.

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 12 / 19

SLIDE 45

On the assumption K◦ · max{|M|, |B◦|} ≤ 2π

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 13 / 19

SLIDE 46

On the assumption K◦ · max{|M|, |B◦|} ≤ 2π

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 13 / 19

SLIDE 47

On the assumption K◦ · max{|M|, |B◦|} ≤ 2π

The assumption K◦ · max{|M|, |B◦|} ≤ 2π can not be simply removed.

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 13 / 19

SLIDE 48

On the assumption K◦ · max{|M|, |B◦|} ≤ 2π

The assumption K◦ · max{|M|, |B◦|} ≤ 2π can not be simply removed.

Counterexample on S2 (K◦ = 1): B◦ ⊂ S2, |B◦| < 2π, M := S2 \ B◦

Weak coupling expansions β → 0− λβ

1(M) ∼

β|∂B◦| 4π − |B◦| and λβ

1(B◦) ∼ β|∂B◦|

|B◦| . Hence, λβ

1(M) > λβ 1(B◦) for β < 0 with |β| small.

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 13 / 19

SLIDE 49

Sketch of the proof

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 14 / 19

SLIDE 50

Sketch of the proof

λβ

1(M) =

inf

u∈H1(M)\{0}

∇u2

L2(M;C2) + βu|∂M2 L2(∂M)

u2

L2(M)

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 14 / 19

SLIDE 51

Sketch of the proof

λβ

1(M) =

inf

u∈H1(M)\{0}

∇u2

L2(M;C2) + βu|∂M2 L2(∂M)

u2

L2(M)

Ground-state u◦(x)=ψ(ρ∂B◦(x)) with ψ: [0, RB◦] → R of Hβ,B◦ is radial

n B◦ (Payne-Weinberger-61, Antunes-Freitas-Krejcirik-17)

u⋆(x) := ψ (ρ∂M(x)) ∈ H1(M) (test function)

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 14 / 19

SLIDE 52

Sketch of the proof

λβ

1(M) =

inf

u∈H1(M)\{0}

∇u2

L2(M;C2) + βu|∂M2 L2(∂M)

u2

L2(M)

Ground-state u◦(x)=ψ(ρ∂B◦(x)) with ψ: [0, RB◦] → R of Hβ,B◦ is radial

n B◦ (Payne-Weinberger-61, Antunes-Freitas-Krejcirik-17)

u⋆(x) := ψ (ρ∂M(x)) ∈ H1(M) (test function) u⋆|∂M∂M = u◦|∂B◦∂B◦, u⋆M ≤ u◦B◦ and ∇u⋆M ≤ ∇u◦B◦ co-area formula/isoperimetric inequality/parallel coords./level lines of ρ∂M

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 14 / 19

SLIDE 53

Sketch of the proof

λβ

1(M) =

inf

u∈H1(M)\{0}

∇u2

L2(M;C2) + βu|∂M2 L2(∂M)

u2

L2(M)

Ground-state u◦(x)=ψ(ρ∂B◦(x)) with ψ: [0, RB◦] → R of Hβ,B◦ is radial

n B◦ (Payne-Weinberger-61, Antunes-Freitas-Krejcirik-17)

u⋆(x) := ψ (ρ∂M(x)) ∈ H1(M) (test function) u⋆|∂M∂M = u◦|∂B◦∂B◦, u⋆M ≤ u◦B◦ and ∇u⋆M ≤ ∇u◦B◦ co-area formula/isoperimetric inequality/parallel coords./level lines of ρ∂M λβ

1(M)

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 14 / 19

SLIDE 54

Sketch of the proof

λβ

1(M) =

inf

u∈H1(M)\{0}

∇u2

L2(M;C2) + βu|∂M2 L2(∂M)

u2

L2(M)

Ground-state u◦(x)=ψ(ρ∂B◦(x)) with ψ: [0, RB◦] → R of Hβ,B◦ is radial

n B◦ (Payne-Weinberger-61, Antunes-Freitas-Krejcirik-17)

u⋆(x) := ψ (ρ∂M(x)) ∈ H1(M) (test function) u⋆|∂M∂M = u◦|∂B◦∂B◦, u⋆M ≤ u◦B◦ and ∇u⋆M ≤ ∇u◦B◦ co-area formula/isoperimetric inequality/parallel coords./level lines of ρ∂M λβ

1(M) ≤

∇u⋆2

L2(M;C2) + βu⋆|∂M2 L2(∂M)

u⋆2

L2(M)

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 14 / 19

SLIDE 55

Sketch of the proof

λβ

1(M) =

inf

u∈H1(M)\{0}

∇u2

L2(M;C2) + βu|∂M2 L2(∂M)

u2

L2(M)

Ground-state u◦(x)=ψ(ρ∂B◦(x)) with ψ: [0, RB◦] → R of Hβ,B◦ is radial

n B◦ (Payne-Weinberger-61, Antunes-Freitas-Krejcirik-17)

u⋆(x) := ψ (ρ∂M(x)) ∈ H1(M) (test function) u⋆|∂M∂M = u◦|∂B◦∂B◦, u⋆M ≤ u◦B◦ and ∇u⋆M ≤ ∇u◦B◦ co-area formula/isoperimetric inequality/parallel coords./level lines of ρ∂M λβ

1(M) ≤

∇u⋆2

L2(M;C2) + βu⋆|∂M2 L2(∂M)

u⋆2

L2(M)

≤ ∇u◦2

L2(B◦;C2) + βu◦|∂B◦2 L2(∂B◦)

u◦2

L2(B◦)

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 14 / 19

SLIDE 56

Sketch of the proof

λβ

1(M) =

inf

u∈H1(M)\{0}

∇u2

L2(M;C2) + βu|∂M2 L2(∂M)

u2

L2(M)

Ground-state u◦(x)=ψ(ρ∂B◦(x)) with ψ: [0, RB◦] → R of Hβ,B◦ is radial

n B◦ (Payne-Weinberger-61, Antunes-Freitas-Krejcirik-17)

u⋆(x) := ψ (ρ∂M(x)) ∈ H1(M) (test function) u⋆|∂M∂M = u◦|∂B◦∂B◦, u⋆M ≤ u◦B◦ and ∇u⋆M ≤ ∇u◦B◦ co-area formula/isoperimetric inequality/parallel coords./level lines of ρ∂M λβ

1(M) ≤

∇u⋆2

L2(M;C2) + βu⋆|∂M2 L2(∂M)

u⋆2

L2(M)

≤ ∇u◦2

L2(B◦;C2) + βu◦|∂B◦2 L2(∂B◦)

u◦2

L2(B◦)

= λβ

1(B◦)

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 14 / 19

SLIDE 57

Outline

1

Motivation & background

2

Optimization on 2-manifolds

3

Optimization on unbounded cones

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 15 / 19

SLIDE 58

Recall the assumptions

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Optimization on manifolds 19.12.2019 16 / 19

SLIDE 59

Recall the assumptions

m ⊂ S2 – a bounded, simply-connected, domain with C2-smooth boundary.

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Optimization on manifolds 19.12.2019 16 / 19

SLIDE 60

Recall the assumptions

m ⊂ S2 – a bounded, simply-connected, domain with C2-smooth boundary. Λm := R+ × m ⊂ R3 (in the spherical coordinates).

V. Lotoreichik (NPI CAS)

Optimization on manifolds 19.12.2019 16 / 19

SLIDE 61

Recall the assumptions

m ⊂ S2 – a bounded, simply-connected, domain with C2-smooth boundary. Λm := R+ × m ⊂ R3 (in the spherical coordinates).

The Robin Laplacian on Λm

H1(Λm)∋u →

Λm

|∇u|2dx −

∂Λm

|u|2dσ ⇒ HΛm =H∗

Λm in L2(Λm)

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Optimization on manifolds 19.12.2019 16 / 19

SLIDE 62

Recall the assumptions

m ⊂ S2 – a bounded, simply-connected, domain with C2-smooth boundary. Λm := R+ × m ⊂ R3 (in the spherical coordinates).

The Robin Laplacian on Λm

H1(Λm)∋u →

Λm

|∇u|2dx −

∂Λm

|u|2dσ ⇒ HΛm =H∗

Λm in L2(Λm)

|m| < 2π = ⇒ λ1(Λm) < inf σess(HΛm) = −1.

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SLIDE 63

Main result

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SLIDE 64

Main result

b ⊂ S2 – geodesic disk (spherical cap).

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Optimization on manifolds 19.12.2019 17 / 19

SLIDE 65

Main result

b ⊂ S2 – geodesic disk (spherical cap).

m b

Λm Λb

Theorem (Khalile-L-19)

L := |∂m| = |∂b| < 2π

|m|, |b| < 2π = ⇒ λ1(Λm) ≤ λ1(Λb) = −4π2 L2

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SLIDE 66

On the method of the proof

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SLIDE 67

On the method of the proof

An analogous result is known for δ-interactions on ∂Λm (Exner-L-17). The method of the proof used there is not applicable any more.

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Optimization on manifolds 19.12.2019 18 / 19

SLIDE 68

On the method of the proof

An analogous result is known for δ-interactions on ∂Λm (Exner-L-17). The method of the proof used there is not applicable any more.

The proof relies on the min-max principle

The ground-state u◦(x) = φ

|x|, ρ∂b
x

|x|

with φ: R+ × [0, Rb] → R of

HΛb is rotationally invariant u⋆(x) := φ

|x|, ρ∂m

x

|x|

∈ H1(Λm)

(test function)

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SLIDE 69

Thank you

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SLIDE 70

Thank you

M. Khalile and V. L., Spectral isoperimetric inequalities for Robin

Laplacians on 2-manifolds and unbounded cones, arXiv:1909.10842.

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Optimization on manifolds 19.12.2019 19 / 19

SLIDE 71

Thank you

M. Khalile and V. L., Spectral isoperimetric inequalities for Robin

Laplacians on 2-manifolds and unbounded cones, arXiv:1909.10842.

Thank you for your attention!

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Optimization on manifolds 19.12.2019 19 / 19