Method of the unknown trial function: sharp
lower bounds on Laplace eigenvalues
Richard Laugesen
(joint with Bartłomiej Siudeja)
University of Illinois
Queen Dido Conference, March 27, 2010
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
1 / 13
Dirichlet eigenvalues
0 < λ1 < λ2 ≤ λ3 ≤ · · · → ∞
Faber–Krahn inequality
Among all plane domains,
λ1 A is minimal for disk
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
2 / 13
Dirichlet eigenvalues
0 < λ1 < λ2 ≤ λ3 ≤ · · · → ∞
Faber–Krahn inequality
Among all plane domains,
λ1 A is minimal for disk
Pólya–Szegő inequality
Among all triangles,
λ1 A is minimal for equilateral triangle
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
2 / 13
Dirichlet eigenvalues
0 < λ1 < λ2 ≤ λ3 ≤ · · · → ∞
Faber–Krahn inequality
Among all plane domains,
λ1 A is minimal for disk
Pólya–Szegő inequality
Among all triangles,
λ1 A is minimal for equilateral triangle
Want inequalities for higher eigenvalues too, for example
(λ1 + · · · + λn ).
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
2 / 13
Dirichlet eigenvalues
0 < λ1 < λ2 ≤ λ3 ≤ · · · → ∞
Faber–Krahn inequality
Among all plane domains,
λ1 A is minimal for disk
Pólya–Szegő inequality
Among all triangles,
λ1 A is minimal for equilateral triangle
Want inequalities for higher eigenvalues too, for example
(λ1 + · · · + λn ).
First attempt:
(λ1 + λ2 )A
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
2 / 13
Dirichlet eigenvalues
0 < λ1 < λ2 ≤ λ3 ≤ · · · → ∞
Faber–Krahn inequality
Among all plane domains,
λ1 A is minimal for disk
Pólya–Szegő inequality
Among all triangles,
λ1 A is minimal for equilateral triangle
Want inequalities for higher eigenvalues too, for example
(λ1 + · · · + λn ).
First attempt:
(λ1 + λ2 )A
R. S. Laugesen (University of Illinois)
is not minimal for disk. . .
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
2 / 13
Second attempt:
to extend from fundamental tone to higher eigenvalues, weaken the
geometric normalization from area to diameter. . .
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
3 / 13
Second attempt:
to extend from fundamental tone to higher eigenvalues, weaken the
geometric normalization from area to diameter. . .
Faber–Krahn for diameter
Among plane domains, λ1 D 2 is minimal for disk.
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
3 / 13
Second attempt:
to extend from fundamental tone to higher eigenvalues, weaken the
geometric normalization from area to diameter. . .
Faber–Krahn for diameter
Among plane domains, λ1 D 2 is minimal for disk.
Proof. λ1 D 2 = (λ1 A)(D 2 /A), where D 2 /A is minimal for disk by
isodiametric inequality
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
3 / 13
Second attempt:
to extend from fundamental tone to higher eigenvalues, weaken the
geometric normalization from area to diameter. . .
Faber–Krahn for diameter
Among plane domains, λ1 D 2 is minimal for disk.
Proof. λ1 D 2 = (λ1 A)(D 2 /A), where D 2 /A is minimal for disk by
isodiametric inequality
Pólya–Szegő for diameter
Among triangles, λ1 D 2 is minimal for equilateral.
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
3 / 13
Second attempt:
to extend from fundamental tone to higher eigenvalues, weaken the
geometric normalization from area to diameter. . .
Faber–Krahn for diameter
Among plane domains, λ1 D 2 is minimal for disk.
Proof. λ1 D 2 = (λ1 A)(D 2 /A), where D 2 /A is minimal for disk by
isodiametric inequality
Pólya–Szegő for diameter
Among triangles, λ1 D 2 is minimal for equilateral.
Theorem (Laugesen–Siudeja 2010)
Among triangles, the higher eigenvalue sum
(λ1 + · · · + λn )D 2
is minimal for equilateral, for each n = 1, 2, 3, . . .
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
3 / 13
Theorem (Laugesen–Siudeja 2010)
Among triangles, (λ1 + · · · + λn )D 2 is minimal for equilateral.
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
4 / 13
Theorem (Laugesen–Siudeja 2010)
Among triangles, (λ1 + · · · + λn )D 2 is minimal for equilateral.
Comments
Geometrically sharp, for each fixed n.
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
4 / 13
Theorem (Laugesen–Siudeja 2010)
Among triangles, (λ1 + · · · + λn )D 2 is minimal for equilateral.
Comments
Geometrically sharp, for each fixed n.
Not asymptotically sharp as n → ∞, since Weyl asymptotic
depends on area (not diameter).
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
4 / 13
Theorem (Laugesen–Siudeja 2010)
Among triangles, (λ1 + · · · + λn )D 2 is minimal for equilateral.
Comments
Geometrically sharp, for each fixed n.
Not asymptotically sharp as n → ∞, since Weyl asymptotic
depends on area (not diameter).
Different from Li–Yau, whose lower bound on (λ1 + · · · + λn )A is
asymptotically sharp but not geometrically sharp.
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
4 / 13
Theorem (Laugesen–Siudeja 2010)
Among triangles, (λ1 + · · · + λn )D 2 is minimal for equilateral.
Comments
Geometrically sharp, for each fixed n.
Not asymptotically sharp as n → ∞, since Weyl asymptotic
depends on area (not diameter).
Different from Li–Yau, whose lower bound on (λ1 + · · · + λn )A is
asymptotically sharp but not geometrically sharp.
Proof Step 1 — Reduction to isosceles with aperture angle < π/3
D
<D
Stretch arbitrary triangle to
isosceles with same diameter.
D
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
4 / 13
Theorem (Laugesen–Siudeja 2010)
Among triangles, (λ1 + · · · + λn )D 2 is minimal for equilateral.
Comments
Geometrically sharp, for each fixed n.
Not asymptotically sharp as n → ∞, since Weyl asymptotic
depends on area (not diameter).
Different from Li–Yau, whose lower bound on (λ1 + · · · + λn )A is
asymptotically sharp but not geometrically sharp.
Proof Step 1 — Reduction to isosceles with aperture angle < π/3
D
<D
D
R. S. Laugesen (University of Illinois)
Stretch arbitrary triangle to
isosceles with same diameter.
Eigenvalues decrease,
by domain monotonicity.
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
4 / 13
Step 2 — Isosceles with aperture in (π/6, π/3)
Transplant with linear maps
T−
T+
Linearly map to isosceles from
equilateral and from 30-60-90◦
right triangles
(0, h)
Te
(−1/2, 0)
R. S. Laugesen (University of Illinois)
(1/2, 0)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
5 / 13
Step 2 — Isosceles with aperture in (π/6, π/3)
Transplant with linear maps
T−
T+
Linearly map to isosceles from
equilateral and from 30-60-90◦
right triangles
(0, h)
Transplanted eigenfunctions of
isosceles provide trial functions
for equilateral and 30-60-90◦
triangle
Te
(−1/2, 0)
R. S. Laugesen (University of Illinois)
(1/2, 0)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
5 / 13
Step 2 — Isosceles with aperture in (π/6, π/3)
Transplant with linear maps
T−
T+
Linearly map to isosceles from
equilateral and from 30-60-90◦
right triangles
(0, h)
Transplanted eigenfunctions of
isosceles provide trial functions
for equilateral and 30-60-90◦
triangle
Te
Eigenvalues of equilateral and
30-60-90◦ triangle are known
explicitly
(−1/2, 0)
R. S. Laugesen (University of Illinois)
(1/2, 0)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
5 / 13
Step 2 — Isosceles with aperture in (π/6, π/3)
Transplant with linear maps
T−
T+
Linearly map to isosceles from
equilateral and from 30-60-90◦
right triangles
(0, h)
Transplanted eigenfunctions of
isosceles provide trial functions
for equilateral and 30-60-90◦
triangle
Te
Eigenvalues of equilateral and
30-60-90◦ triangle are known
explicitly
Method of the Unknown Trial Function
(−1/2, 0)
R. S. Laugesen (University of Illinois)
(1/2, 0)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
5 / 13
Method of the Unknown Trial Function
Orthonormal eigenfunctions u1 , u2 , . . . of isosceles
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
.
Carthage, May 27, 2010
6 / 13
Method of the Unknown Trial Function
Orthonormal eigenfunctions u1 , u2 , . . . of isosceles
.
Transplant each uj with linear transformation Te . These trial
functions are orthogonal on equilateral
= Te−1 ( ) and satisfy
Dirichlet boundary condition.
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
6 / 13
Method of the Unknown Trial Function
Orthonormal eigenfunctions u1 , u2 , . . . of isosceles
.
Transplant each uj with linear transformation Te . These trial
functions are orthogonal on equilateral
= Te−1 ( ) and satisfy
Dirichlet boundary condition.
Rayleigh principle implies
λ1 + · · · + λn |
≤ R[u1 ◦ Te ] + · · · + R[un ◦ Te ]
Left side is known. Right side is unknown.
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
6 / 13
Method of the Unknown Trial Function
Orthonormal eigenfunctions u1 , u2 , . . . of isosceles
.
Transplant each uj with linear transformation Te . These trial
functions are orthogonal on equilateral
= Te−1 ( ) and satisfy
Dirichlet boundary condition.
Rayleigh principle implies
λ1 + · · · + λn |
≤ R[u1 ◦ Te ] + · · · + R[un ◦ Te ]
Left side is known. Right side is unknown.
Also, on right side we really want R[u1 ] + · · · + R[un ].
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
6 / 13
What does right side tell us?
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
7 / 13
What does right side tell us?
λ1 + · · · + λn |
R. S. Laugesen (University of Illinois)
≤ R[u1 ◦ Te ] + · · · + R[un ◦ Te ]
2
h
Y
≤X+ √
3/2
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
7 / 13
What does right side tell us?
λ1 + · · · + λn |
where
X =
n Z
X
j=1
R. S. Laugesen (University of Illinois)
≤ R[u1 ◦ Te ] + · · · + R[un ◦ Te ]
2
h
Y
≤X+ √
3/2
2
uj,x
dA
Y =
n Z
X
2
uj,y
dA
j=1
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
7 / 13
What does right side tell us?
λ1 + · · · + λn |
where
X =
n Z
X
≤ R[u1 ◦ Te ] + · · · + R[un ◦ Te ]
2
h
Y
≤X+ √
3/2
2
uj,x
dA
j=1
Y =
n Z
X
2
uj,y
dA
j=1
Transplantation to right triangles similarly gives
λ1 + · · · + λn | ≤
R. S. Laugesen (University of Illinois)
13
4h2
X+
Y.
12
12
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
7 / 13
What does right side tell us?
λ1 + · · · + λn |
where
X =
n Z
X
≤ R[u1 ◦ Te ] + · · · + R[un ◦ Te ]
2
h
Y
≤X+ √
3/2
2
uj,x
dA
j=1
Y =
n Z
X
2
uj,y
dA
j=1
Transplantation to right triangles similarly gives
λ1 + · · · + λn | ≤
13
4h2
X+
Y.
12
12
We want
λ1 + · · · + λn |
≤ (X + Y )D 2
where D 2 = h2 + ( 12 )2 .
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
7 / 13
Define ratio
def
γn =
(λ1 + · · · + λn )|
(λ1 + · · · + λn )|
Then previous slide says
λ1 + · · · + λn |
λ1 + · · · + λn |
4h2
≤X+
Y
3
.
13
4h2
≤
X+
Y
γn
12
12
We need at least one of right sides to be < (X + Y )(h2 + ( 12 )2 ).
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
8 / 13
Thus we need
3
Y
≥
4
X +Y
R. S. Laugesen (University of Illinois)
or
Y
13 − 3γn (1 + 4h2 )
≥
X +Y
13 − 4h2
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
9 / 13
Thus we need
3
Y
≥
4
X +Y
or
Y
13 − 3γn (1 + 4h2 )
≥
X +Y
13 − 4h2
Bad News: we cannot evaluate this fraction Y /(X + Y ) of the Rayleigh
quotient, on the isosceles triangle.
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
9 / 13
Thus we need
3
Y
≥
4
X +Y
or
Y
13 − 3γn (1 + 4h2 )
≥
X +Y
13 − 4h2
Bad News: we cannot evaluate this fraction Y /(X + Y ) of the Rayleigh
quotient, on the isosceles triangle.
Good News: suffices to prove
13 − 3γn (1 + 4h2 )
3
≥ max
4
h
13 − 4h2
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
9 / 13
Thus we need
3
Y
≥
4
X +Y
or
Y
13 − 3γn (1 + 4h2 )
≥
X +Y
13 − 4h2
Bad News: we cannot evaluate this fraction Y /(X + Y ) of the Rayleigh
quotient, on the isosceles triangle.
Good News: suffices to prove
13 − 3γn (1 + 4h2 )
3
≥ max
4
h
13 − 4h2
Equivalently, prove
γn =
R. S. Laugesen (University of Illinois)
λ1 + · · · + λn |
λ1 + · · · + λn |
≥
11
24
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
9 / 13
Thus we need
3
Y
≥
4
X +Y
or
Y
13 − 3γn (1 + 4h2 )
≥
X +Y
13 − 4h2
Bad News: we cannot evaluate this fraction Y /(X + Y ) of the Rayleigh
quotient, on the isosceles triangle.
Good News: suffices to prove
13 − 3γn (1 + 4h2 )
3
≥ max
4
h
13 − 4h2
Equivalently, prove
γn =
λ1 + · · · + λn |
λ1 + · · · + λn |
More good news:
γn → 12/24 as n → ∞ by Weyl, since area(
R. S. Laugesen (University of Illinois)
≥
11
24
) = 2 area(
Sharp lower bounds on Laplace eigenvalues
).
Carthage, May 27, 2010
9 / 13
Thus we need
3
Y
≥
4
X +Y
or
Y
13 − 3γn (1 + 4h2 )
≥
X +Y
13 − 4h2
Bad News: we cannot evaluate this fraction Y /(X + Y ) of the Rayleigh
quotient, on the isosceles triangle.
Good News: suffices to prove
13 − 3γn (1 + 4h2 )
3
≥ max
4
h
13 − 4h2
Equivalently, prove
γn =
λ1 + · · · + λn |
λ1 + · · · + λn |
≥
11
24
More good news:
γn → 12/24 as n → ∞ by Weyl, since area( ) = 2 area(
Make rigorous using counting function, explicit formulas.
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
).
Carthage, May 27, 2010
9 / 13
Summary — Method of the Unknown Trial Function
“interpolates” between two “endpoint” domains whose eigenvalues
we know
applies to linear transformations of arbitrary domains, not just
triangles
could be used on nonlinear transformations too?
applies also to Neumann eigenvalues
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
10 / 13
Summary — Method of the Unknown Trial Function
“interpolates” between two “endpoint” domains whose eigenvalues
we know
applies to linear transformations of arbitrary domains, not just
triangles
could be used on nonlinear transformations too?
applies also to Neumann eigenvalues
Second eigenvalue
We have shown eigenvalue sums are minimal for equilateral.
What about individual eigenvalues???
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
10 / 13
Summary — Method of the Unknown Trial Function
“interpolates” between two “endpoint” domains whose eigenvalues
we know
applies to linear transformations of arbitrary domains, not just
triangles
could be used on nonlinear transformations too?
applies also to Neumann eigenvalues
Second eigenvalue
We have shown eigenvalue sums are minimal for equilateral.
What about individual eigenvalues??? True for λ1 D 2 .
Theorem (Laugesen–Siudeja 2010)
Among triangles, λ2 D 2 is minimal for equilateral.
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
10 / 13
Summary — Method of the Unknown Trial Function
“interpolates” between two “endpoint” domains whose eigenvalues
we know
applies to linear transformations of arbitrary domains, not just
triangles
could be used on nonlinear transformations too?
applies also to Neumann eigenvalues
Second eigenvalue
We have shown eigenvalue sums are minimal for equilateral.
What about individual eigenvalues??? True for λ1 D 2 .
Theorem (Laugesen–Siudeja 2010)
Among triangles, λ2 D 2 is minimal for equilateral.
Proof. First reduce to isosceles, by domain monotonicity. Then . . .
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
10 / 13
Numerical plot for isosceles triangles with aperture α
λ2 D 2
λ2 L2
λ2 A
π
6
R. S. Laugesen (University of Illinois)
π
3
acute
π
2
Sharp lower bounds on Laplace eigenvalues
obtuse
2π
3
Carthage, May 27, 2010
α
11 / 13
Numerical plot for isosceles triangles with aperture α
λ2 D 2
λ2 L2
λ2 A
π
6
π
3
acute
π
2
obtuse
2π
3
α
λ2 D 2 is minimal (numerically) for equilateral, α = π/3
λ2 A and λ2 L2 are not minimal for equilateral
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
11 / 13
Numerical plot for isosceles triangles with aperture α
λ2 D 2
λ2 L2
λ2 A
π
6
π
3
acute
π
2
obtuse
2π
3
α
λ2 D 2 is minimal (numerically) for equilateral, α = π/3
λ2 A and λ2 L2 are not minimal for equilateral
(Consistent with general domains (Bucur, Henrot et al):
λ2 A and λ2 L2 minimal for stadium-like sets, not disk
λ2 D 2 conjectured minimal for disk)
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
11 / 13
Let λ2 (α) =second eigenvalue for isosceles with aperture α. Want
λ2 (α) > λ2 (π/3),
R. S. Laugesen (University of Illinois)
π
π
<α< .
4
3
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
12 / 13
Let λ2 (α) =second eigenvalue for isosceles with aperture α. Want
λ2 (α) > λ2 (π/3),
π
π
<α< .
4
3
How to estimate λ2 from below?
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
12 / 13
Let λ2 (α) =second eigenvalue for isosceles with aperture α. Want
λ2 (α) > λ2 (π/3),
π
π
<α< .
4
3
How to estimate λ2 from below? Decompose
λ2 = (λ1 + λ2 ) − λ1
and estimate λ1 + λ2 from below and λ1 from above!
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
12 / 13
Let λ2 (α) =second eigenvalue for isosceles with aperture α. Want
λ2 (α) > λ2 (π/3),
π
π
<α< .
4
3
How to estimate λ2 from below? Decompose
λ2 = (λ1 + λ2 ) − λ1
and estimate λ1 + λ2 from below and λ1 from above!
Step 1. λ1 A3 /I is maximal for equilateral by Pólya, so
λ1 (α) < λ1 (π/3) + f (α)
for explicit f (α) > 0, f (π/3) = 0.
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
12 / 13
Let λ2 (α) =second eigenvalue for isosceles with aperture α. Want
λ2 (α) > λ2 (π/3),
π
π
<α< .
4
3
How to estimate λ2 from below? Decompose
λ2 = (λ1 + λ2 ) − λ1
and estimate λ1 + λ2 from below and λ1 from above!
Step 1. λ1 A3 /I is maximal for equilateral by Pólya, so
λ1 (α) < λ1 (π/3) + f (α)
for explicit f (α) > 0, f (π/3) = 0.
Step 2. Refine the Method of Unknown Trial Function to show
(λ1 + λ2 )(α) > (λ1 + λ2 )(π/3) + f (α)
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
12 / 13
Let λ2 (α) =second eigenvalue for isosceles with aperture α. Want
λ2 (α) > λ2 (π/3),
π
π
<α< .
4
3
How to estimate λ2 from below? Decompose
λ2 = (λ1 + λ2 ) − λ1
and estimate λ1 + λ2 from below and λ1 from above!
Step 1. λ1 A3 /I is maximal for equilateral by Pólya, so
λ1 (α) < λ1 (π/3) + f (α)
for explicit f (α) > 0, f (π/3) = 0.
Step 2. Refine the Method of Unknown Trial Function to show
(λ1 + λ2 )(α) > (λ1 + λ2 )(π/3) + f (α)
Step 3. Subtract!
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
12 / 13
Open problems for triangles
λ1 D 2 is minimal for equilateral
λ2 D 2 is minimal for equilateral
λ3 D 2 is minimal for equilateral
Is λn D 2 minimal for equilateral, for each n?
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
13 / 13
Open problems for triangles
λ1 D 2 is minimal for equilateral
λ2 D 2 is minimal for equilateral
λ3 D 2 is minimal for equilateral
Is λn D 2 minimal for equilateral, for each n?
Spectral gap conjecture (Antunes–Freitas):
Is (λ2 − λ1 )D 2 minimal for equilateral?
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
13 / 13
Open problems for triangles
λ1 D 2 is minimal for equilateral
λ2 D 2 is minimal for equilateral
λ3 D 2 is minimal for equilateral
Is λn D 2 minimal for equilateral, for each n?
Spectral gap conjecture (Antunes–Freitas):
Is (λ2 − λ1 )D 2 minimal for equilateral?
Open problems for general domains
Is (λ1 + · · · + λn )D 2 minimal for disk?
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
13 / 13
Open problems for triangles
λ1 D 2 is minimal for equilateral
λ2 D 2 is minimal for equilateral
λ3 D 2 is minimal for equilateral
Is λn D 2 minimal for equilateral, for each n?
Spectral gap conjecture (Antunes–Freitas):
Is (λ2 − λ1 )D 2 minimal for equilateral?
Open problems for general domains
Is (λ1 + · · · + λn )D 2 minimal for disk?
Is λ2 D 2 minimal for disk? (Bucur, Henrot)
Can assume domain is convex (by expanding to convex hull), and
has constant width.
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
13 / 13
Open problems for triangles
λ1 D 2 is minimal for equilateral
λ2 D 2 is minimal for equilateral
λ3 D 2 is minimal for equilateral
Is λn D 2 minimal for equilateral, for each n?
Spectral gap conjecture (Antunes–Freitas):
Is (λ2 − λ1 )D 2 minimal for equilateral?
Open problems for general domains
Is (λ1 + · · · + λn )D 2 minimal for disk?
Is λ2 D 2 minimal for disk? (Bucur, Henrot)
Can assume domain is convex (by expanding to convex hull), and
has constant width.
Spectral gap conjecture (van den Berg):
Is (λ2 − λ1 )D 2 minimal for degenerate rectangle?
R. S. Laugesen (University of Illinois)
Sharp lower bounds on Laplace eigenvalues
Carthage, May 27, 2010
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