Inequalities of Faber-Krahn type on metric graphs
Jonathan Rohleder
Wien, 17. December 2016
Metric graphs
Γ = (V , E , L) metric graph
V – finite vertex set
E – finite edge set
L : E → (0, ∞) – length function
Γ
Metric graphs
Γ = (V , E , L) metric graph
V – finite vertex set
E – finite edge set
L : E → (0, ∞) – length function
e∼
= [0, L(e)]
Γ
Metric graphs
Γ = (V , E , L) metric graph
V – finite vertex set
E – finite edge set
L : E → (0, ∞) – length function
P
L(Γ) := e∈E L(e)
e∼
= [0, L(e)]
Γ
Metric graphs
Γ = (V , E , L) metric graph
V – finite vertex set
E – finite edge set
L : E → (0, ∞) – length function
P
L(Γ) := e∈E L(e)
For f : Γ → C define fe := f |e .
e∼
= [0, L(e)]
Γ
Metric graphs
Γ = (V , E , L) metric graph
V – finite vertex set
E – finite edge set
L : E → (0, ∞) – length function
P
L(Γ) := e∈E L(e)
For f : Γ → C define fe := f |e .
∂Γ
Γ
Laplacians in L2 (Γ)
If fe ∈ H 2 (e) ∀e ∈ E we define (−∆f )e := −fe00 .
Laplacians in L2 (Γ)
If fe ∈ H 2 (e) ∀e ∈ E we define (−∆f )e := −fe00 .
Standard/Kirchhoff VC:
f continuous at v ,
X
e∼v
∂fe (v ) = 0
(d)
Laplacians in L2 (Γ)
If fe ∈ H 2 (e) ∀e ∈ E we define (−∆f )e := −fe00 .
Standard/Kirchhoff VC:
f continuous at v ,
X
∂fe (v ) = 0
e∼v
Neumann Laplacian
−∆N f = −∆f ,
dom(−∆N ) = f ∈ ⊕e∈E H 2 (e) : (d) ∀v ∈ V
(d)
Laplacians in L2 (Γ)
If fe ∈ H 2 (e) ∀e ∈ E we define (−∆f )e := −fe00 .
Standard/Kirchhoff VC:
f continuous at v ,
X
∂fe (v ) = 0
(d)
e∼v
Neumann Laplacian
−∆N f = −∆f ,
dom(−∆N ) = f ∈ ⊕e∈E H 2 (e) : (d) ∀v ∈ V
Dirichlet Laplacian
−∆D f = −∆f ,
dom(−∆D ) = f ∈ ⊕H 2 (e) : (d) ∀v ∈ V \ ∂Γ, f (v ) = 0 ∀v ∈ ∂Γ
Dirichlet and Neumann spectra
−∆N = −∆∗N , discrete spectrum, nonnegative
R
µ2 (Γ)
Dirichlet and Neumann spectra
−∆N = −∆∗N , discrete spectrum, nonnegative
R
µ2 (Γ)
−∆D = −∆∗D , discrete spectrum, positive
R
λ1 (Γ)
Classical results
For Ω ⊂ Rn bounded domain let
λ1 (Ω) smallest EV of −∆ + Dirichlet BC
µ2 (Ω) smallest positive EV of −∆ + Neumann BC.
Classical results
For Ω ⊂ Rn bounded domain let
λ1 (Ω) smallest EV of −∆ + Dirichlet BC
µ2 (Ω) smallest positive EV of −∆ + Neumann BC.
Theorem [Faber ’23; Krahn ’25; Szegö ’54; Weinberger ’56]
Let Ω ⊂ Rn bounded. Then
λ1 (Ω) ≥ λ1 (B)
and µ2 (Ω) ≤ µ2 (B),
where B is any ball with |B| = |V |. Equality holds iff Ω is a ball.
Counterparts for graphs?
Counterparts for graphs?
Analog of volume?
Counterparts for graphs?
Analog of volume?
Analog of ball?
Counterparts for graphs?
Analog of volume? total length, average length, diameter
Analog of ball?
Counterparts for graphs?
Analog of volume? total length, average length, diameter
Analog of ball?
Estimates involving other parameters: [Band, Lévy ’16; Ariturk ’16]
Extrema under fixed total length I
Theorem [Nicaise ’87; Friedlander ’05; Kurasov, Naboko ’14]
Let Γ finite graph with total length L(Γ). Then
µ2 (Γ) ≥
Equality holds iff Γ is an interval.
π2
.
L(Γ)2
Extrema under fixed total length I
Theorem [Nicaise ’87; Friedlander ’05; Kurasov, Naboko ’14]
Let Γ finite graph with total length L(Γ). Then
µ2 (Γ) ≥
π2
.
L(Γ)2
Equality holds iff Γ is an interval.
No maximizer!
For any equilateral star graph: µ2 (Γ) =
|E |2 π 2
4L(Γ)2
Extrema under fixed total length II: trees
Observation
Let Γ finite tree with total length L(Γ). Then
λ1 (Γ) ≥
Equality holds iff Γ is an interval.
π2
.
L(Γ)2
Extrema under fixed total length II: trees
Observation
Let Γ finite tree with total length L(Γ). Then
λ1 (Γ) ≥
π2
.
L(Γ)2
Equality holds iff Γ is an interval.
. . . follows from λ1 (Γ) ≥ µ2 (Γ) [Band, Berkolaiko, Weyand ’15].
Extrema under fixed total length II: trees
Observation
Let Γ finite tree with total length L(Γ). Then
λ1 (Γ) ≥
π2
.
L(Γ)2
Equality holds iff Γ is an interval.
. . . follows from λ1 (Γ) ≥ µ2 (Γ) [Band, Berkolaiko, Weyand ’15].
No maximizer!
For any equilateral star graph: λ1 (Γ) = µ2 (Γ) =
|E |2 π 2
4L(Γ)2
Extrema under fixed average length I
A(Γ) :=
L(Γ)
|E |
Extrema under fixed average length I
A(Γ) :=
L(Γ)
|E |
Theorem [Kennedy, Kurasov, Malenová, Mugnolo ’16]
Let Γ finite graph with total length L(Γ) and |E | ≥ 2. Then
µ2 (Γ) ≤
|E |2 π 2
π2
=
.
L(Γ)2
A(Γ)2
Equality holds iff Γ is an equilateral flower or pumpkin.
Extrema under fixed average length I
A(Γ) :=
L(Γ)
|E |
Theorem [Kennedy, Kurasov, Malenová, Mugnolo ’16]
Let Γ finite graph with total length L(Γ) and |E | ≥ 2. Then
µ2 (Γ) ≤
|E |2 π 2
π2
=
.
L(Γ)2
A(Γ)2
Equality holds iff Γ is an equilateral flower or pumpkin.
No minimizer!
Extrema under fixed average length II: trees
Theorem [R. ’16]
Let Γ finite tree with |E | ≥ 2. Then
µ2 (Γ) ≤
π2
|E |2 π 2
=
.
4L(Γ)2
4A(Γ)2
Equality holds iff Γ is any equilateral star.
Extrema under fixed average length II: trees
Theorem [R. ’16]
Let Γ finite tree with |E | ≥ 2. Then
µ2 (Γ) ≤
π2
|E |2 π 2
=
.
4L(Γ)2
4A(Γ)2
Equality holds iff Γ is any equilateral star.
Sharp upper estimate for λ1 (Γ): open
Extrema under fixed average length II: trees
Theorem [R. ’16]
Let Γ finite tree with |E | ≥ 2. Then
µ2 (Γ) ≤
π2
|E |2 π 2
=
.
4L(Γ)2
4A(Γ)2
Equality holds iff Γ is any equilateral star.
Sharp upper estimate for λ1 (Γ): open
No minimizers!
Extrema under fixed diameter I
diam(Γ) := sup {dist(x, y ) : x, y ∈ Γ}
Extrema under fixed diameter I
diam(Γ) := sup {dist(x, y ) : x, y ∈ Γ}
Theorem [Kennedy, Kurasov, Malenová, Mugnolo ’16]
(a) No minimizer for µ2 (Γ).
Extrema under fixed diameter I
diam(Γ) := sup {dist(x, y ) : x, y ∈ Γ}
Theorem [Kennedy, Kurasov, Malenová, Mugnolo ’16]
(a) No minimizer for µ2 (Γ).
(b) No maximizer for µ2 (Γ).
Extrema under fixed diameter II: trees
Theorem [R. ’16]
Let Γ finite tree. Then
µ2 (Γ) ≤
π2
≤ λ1 (Γ).
diam(Γ)2
For λ1 (Γ), equality holds iff Γ is any equilateral star.