STABILITY THEORY FOR NEUMANN EIGENFUNCTIONS
1. A Sobolev inequality
Lemma 1.1. Let ABCD be a parallelogram, and let u be a C 2 function on ABCD.
Let α be the angle subtended by A. Then
Z
1
|u(A) − u(B) + u(C) − u(D)| 6
|∇2 u|.
sin(α) ABCD
Proof. We may normalise A = 0, so that C = B + D. From two applications of the
fundamental theorem of calculus one has
Z 1Z 1
∂st u(sB + tD) = u(A) − u(B) + u(C) − u(D).
0
0
The left-hand side can be rewritten as
Z
1
(B · ∇)(D · ∇)u.
|ABCD| ABCD
Since
|ABCD| = |B||D| sin(α)
the claim follows.
Lemma 1.2. Let ABC be a triangle with angles α, β, γ, and let u be a C 2 function on
ABC that obeys the Neumann boundary condition n · ∇u = 0 on the boundary of ABC.
Then for any P ∈ ABC, one has
Z
4
|∇2 u|.
|u(P ) − u(Q)| 6
3 min(sin(α), sin(β), sin(γ)) ABC
Proof. Let P ∈ ABC. Let D, E ∈ AB, F, G ∈ BC, H, I ∈ AC be the points such
that ADXI, BF XE, CHXG are parallelograms (thus D, G are the intersections with
AB, BC respectively of the line through X parallel to AC, and so forth. Then from the
preceding lemma one has
Z
1
|u(X) + u(A) − u(D) − u(I)| 6
|∇2 u|
sin(α) ADXI
Z
1
|u(X) + u(B) − u(F ) − u(E)| 6
|∇2 u|
sin(β) BF XE
Z
1
|u(X) + u(C) − u(H) − u(G)| 6
|∇2 u|.
sin(γ) CHXG
1
2
STABILITY THEORY FOR NEUMANN EIGENFUNCTIONS
Also, by reflecting the triangles DEX, F GX, XHI across the Neumann boundary and
using the previous lemma, we see that
Z
2
|u(X) + u(X) − u(D) − u(E)| 6
|∇2 u|
sin(γ) DEX
Z
2
|u(X) + u(X) − u(F ) − u(G)| 6
|∇2 u|
sin(α) F GX
Z
2
|∇2 u|.
|u(X) + u(X) − u(H) − u(I)| 6
sin(β) XHI
Summing the latter three combinations of u and subtracting the former three using the
triangle inequality, we conclude that
Z
2
|3u(X) − u(A) − u(B) − u(C)| 6
|∇2 u|.
min(sin(α), sin(β), sin(γ)) ABC
A similar inequality holds with X replaced by Y . Subtracting the two inequailties, we
obtain the claim.
Corollary 1.3. Let ABC be a triangle with angles α, β, γ, and let u be a C 2 function
on ABC which is smooth up to the boundary except possibly at the vertices A, B, C, and
which obeys the Neumann boundary condition n · ∇u = 0 on the boundary of ABC, and
has mean zero on ABC. Then
4
kukL∞ (ABC) 6
|ABC|1/2 k∆ukL2 (ABC) .
3 min(sin(α), sin(β), sin(γ))
Proof. If u has mean zero, then kukL∞ (ABC) is bounded by |u(P ) − u(Q)| for some
P, Q ∈ ABC. From the previous lemma we thus have
kukL∞ (ABC) 6
4
|ABC|1/2 k∇2 ukL2 (ABC) .
3 min(sin(α), sin(β), sin(γ))
It will thus suffice to show the Bochner-Weitzenbock identity
Z
Z
2 2
|∇ u| =
|∆u|2 .
ABC
ABC
But this can be accomplished by two integration by parts, using the smoothness and
Neumann boundary hypotheses on u (and a regularisation argument if necessary to cut
away from the vertices) more details needed here.
2. Schwarz-Christoffel
Let 0 < α, β, γ < π be angles adding up to π, then we can define a Schwarz-Christoffel
map Φα,β : H → ABC from the half-plane H := {z : =(z) > 0} to a triangle ABC with
angles α, β, γ by the formula
Z z
dζ
Φα,β (z) :=
,
1−α/π
(1 − ζ)1−β/π
0 ζ
STABILITY THEORY FOR NEUMANN EIGENFUNCTIONS
3
where the integral is over any contour from 0 to z in H, and one chooses the branch cut
to make both factors in the denominator positive real on the interval [0, 1]. Thus the
vertices of the triangle are given by
A := Φα,β (0) = 0
Z
B := Φα,β (1) =
1
dt
Γ(α/π)Γ(β/π)
=
1−α/π (1 − t)1−β/π
Γ((α + β)/π)
0 t
Z −∞
dt
C := Φα,β (∞) = −eiα
1−α/π
|t|
(1 − t)1−β/π
Z ∞0
ds
= eiα
1−α/π
(s − 1)
s1−β/π
1
Z 1
dv
= eiα
1−α/π
v 1−γ/π
0 (v − 1)
Γ(α/π)Γ(γ/π)
= eiα
Γ((α + γ)/π)
where we have used the beta function identity
Z 1
Γ(x)Γ(y)
dt
=
1−x
1−y
(1 − t)
Γ(x + y)
0 t
and the changes of variable s = 1 − t, v = 1/s. In particular, the area of the triangle
ABC can be expressed as
1
Γ(α/π)2 Γ(β/π)Γ(γ/π)
|ABC| = |B||C| sin(α) =
sin(α)
2
2Γ((α + β)/π)Γ((α + γ)/π)
which can be simplified using the formula Γ(z)Γ(1 − z) =
expression
|ABC| =
π
sin(πz)
as the more symmetric
1
Γ(α/π)2 Γ(β/π)2 Γ(γ/π)2 sin(α) sin(β) sin(γ).
2π 2
(2.1)
We write
|Φ0α,β (z)| = eω(z)
where ω = ωα,β is the harmonic function
ω(z) := (
If u : ABC →
H defined by
α
β
− 1) log |z| + ( − 1) log |1 − z|.
π
π
(2.2)
R is a smooth function, and ũ : H → R is its pullback to the half-plane
ũ := u ◦ Φα,β
then we have
Z
Z
u=
ABC
H
e2ω ũ.
4
STABILITY THEORY FOR NEUMANN EIGENFUNCTIONS
In a similar vein we have the conformal invariance of the two-dimensional Dirichlet
energy
Z
Z
2
|∇u| =
|∇ũ|2
ABC
H
and the conformal transformation of the Laplacian:
f
∆ũ(z) = e2ω ∆u.
In particular, the Rayleigh quotient
Z
Z
2
|u|2
|∇u| /
ABC
with mean zero condition
quotient
R
ABC
ABC
u = 0 becomes, when pulled back to
Z
2
Z
|∇ũ| /
H
with mean zero condition H e ũ = 0.
R
H
H, the Rayleigh
e2ω |ũ|2
2ω
Let u2 , u3 , . . . be an L2 -normalised eigenbasis for the Neumann Laplacian −∆ on ABC
with eigenvalues λ2 6 λ3 6 . . ., thus
−∆uk = λk uk
on ABC with Neumann boundary data
n · ∇uk = 0
and orthonormality
Z
uj uk = δjk
ABC
and mean zero condition
Z
uj = 0.
ABC
One can show that when ABC is acute-angled, these eigenfunctions are smooth except
possibly at the vertices A, B, C, and are uniformly C 2 . add details here
Pulling all this back to H, we obtain transformed eigenfunctions ũ2 , ũ3 , . . . on
conformal eigenfunction equation
−∆ũk = λk e2ω ũk
on
H to the
(2.3)
H with Neumann boundary data
n · ∇ũk = 0
(2.4)
e2ω ũj ũk = δjk
(2.5)
e2ω ũj = 0.
(2.6)
and orthonormality
Z
and mean zero condition
H
Z
H
STABILITY THEORY FOR NEUMANN EIGENFUNCTIONS
5
Now suppose that we vary the angle parameters α, β, γ smoothly with respect to some
time parameter t, thus also varying the triangles ABC, eigenfunctions uk and transformed eigenfunctions ũk , eigenvalues λk , and conformal factor ω. We will use dots to
indicate time differentiation, thus for instance α̇ = dtd α. Let us formally suppose that
all of the above data vary smoothly (or at least C 1 ) in time we will eventually need
to justify this, of course. Since α + β + γ = π, we have
α̇ + β̇ + γ̇ = 0.
The variation ω̇ of the conformal factor is explicitly computable from (2.2) as being a
logarithmic weight:
β̇
α̇
ω̇ = log |z| + log |1 − z|.
π
π
Next, by (formally) differentiating (2.3) we obtain an equation for the variation ˜˙uk of
the k th eigenfunction:
−∆ũ˙ k = λ̇k e2ω ũk + 2λk ω̇e2ω ũk + λk e2ω ũ˙ k .
(2.7)
To solve this equation for ũ˙ k , we observe from differentiating (2.4), (2.5), (2.6) that
Z
e2ω ũ˙ k = 0
(2.8)
H
and
Z
e2ω ũk ũ˙ k = 0
H
and
n · ∇ũ˙ k = 0.
By eigenfunction expansion, we thus have
XZ
˙ũk =
( e2ω ũl ũ˙ k )ũl
H
l6=k
(2.9)
in a suitable sense (L2 with weight eω ). Now we evaluate the expression in parentheses.
Integrating (2.7) against ũl and using (2.5) reveals that
Z
Z
Z
2ω
˙
− ∆ũk ũl = 2λk
ω̇e ũk ũl + λk
e2ω ũ˙ k ũl .
(2.10)
H
H
H
By Green’s theorem and the Neumann conditions on ũ˙ k and ũl , the left-hand side is
Z
− ũ˙ k ∆ũl
H
which by (2.3) is equal to
Z
λl e2ω ũ˙ k ũl .
H
Inserting this into (2.10) we see that
Z
Z
2λk
2ω ˙
e ũk ũl =
ω̇e2ω ũk ũl
λ
−
λ
l
k H
H
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STABILITY THEORY FOR NEUMANN EIGENFUNCTIONS
and thus by (2.9)
Z
2λ
k
ũ˙ k =
(
ω̇e2ω ũk ũl )ũl .
λ
−
λ
l
k H
l6=k
X
(2.11)
We can take Laplacians and conclude that
X 2λk λl Z
2ω
˙
ω̇e2ω ũk ũl )ũl .
−∆ũk = e
(
λ
−
λ
l
k H
l6=k
2 λ3
k λl
is bounded in magnitude by λ2λ3 −λ
. From the orthonormality (2.5)
Set k = 2, then λ2λl −λ
2
k
and the Bessel inequality, we conclude that
Z
Z
2λ2 λ3
−2ω
2 1/2
˙
( e |∆ũ2 | ) 6
( |ω̇|2 e2ω ũ22 )1/2 .
λ3 − λ2 H
H
If we change coordinates by writing
ũ˙ = u̇ ◦ Φ
we conclude that
Z
Z
2λ2 λ3
(
|∆u̇2 | ) 6
( |ω̇|2 e2ω ũ22 )1/2 .
λ3 − λ2 H
ABC
Also, u̇2 has mean zero on ABC by (2.8). We conclude from Corollary 1.3 that
Z
4
1/2 2λ2 λ3
ku̇2 kL∞ 6
|ABC|
( |ω̇|2 e2ω ũ22 )1/2 .
3 min(sin(α), sin(β), sin(γ))
λ3 − λ2 H
Pulling back to H, and estimating ũ2 in L∞ norm, we conclude that
kũ˙ 2 kL∞ (H) 6 Xkũ2 kL∞ (H)
2 1/2
where X is the explicit (but somewhat messy) quantity
Z
4
1/2 2λ2 λ3
|ABC|
( |ω̇|2 e2ω )1/2 .
X :=
3 min(sin(α), sin(β), sin(γ))
λ3 − λ2 H
This gives stability of the second eigenfunction in L∞ norm, as long as there is an
eigenvalue gap λ3 − λ2 > 0.
One can compute one factor in the quantity X as follows. Observe that
Z
1
e2ω = |ABC| = 2 Γ(α/π)2 Γ(β/π)2 Γ(γ/π)2 sin(α) sin(β) sin(γ).
2π
H
If we view α, β, γ (and hence ω) as varying linearly in time, and differentiate the above
equation under the integral sign twice in time, we conclude that
Z
Z
d2
2 2ω
e2ω
4 |ω̇| e = 2
dt
H
H
2
∂
∂2
∂2
= (α̇2 2 + 2α̇β̇
+ 2α̇γ̇
∂α
∂α∂β
∂α∂γ
2
2
∂
∂
+ β̇ 2 2 + 2β̇ γ̇
+ γ̇ 2 )
∂β
∂βγ
1
( 2 Γ(α/π)2 Γ(β/π)2 Γ(γ/π)2 sin(α) sin(β) sin(γ)).
2π
STABILITY THEORY FOR NEUMANN EIGENFUNCTIONS
7
R
This, in principle, expresses the factor ( H |ω̇|2 e2ω )1/2 in X as an explicit combination of
trigonometric functions, gamma functions, and the first two derivatives of the gamma
function. However, this formula is somewhat messy, to say the least.