[PPT] - Extensions of some results about the eigenvalues for the Laplacian PowerPoint Presentation

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Extensions of some results about the eigenvalues for the Laplacian

Selma Yıldırım

The University of Chicago

Joint works with Evans Harrell and John Goldman(T¨ urkay Yolcu)

4th April 2020

Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian

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Eigenvalues/Eigenfunctions of Laplacian

Let Ω be a bounded open set in Rd. Dirichlet problem:

−∆un = λnun

in Ω; un

=

n ∂Ω.

Eigenvalues of Laplacian: 0 < λ1 < λ2 ≤ λ3 ≤ · · · → ∞ Corresponding Eigenfunctions: u1, u2, u3, . . ..

Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian

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Fractional Laplacian operator

Let S be the Schwartz space of rapidly decaying C∞ functions in Rd (i.e., all

f whose derivatives are rapidly decreasing). For any u ∈ S and α ∈ (0, 2),

fractional Laplacian operator (−∆)α/2 is defined as

(−∆)

α

2 u(x) = Ad,−α lim

ǫ→0+

{|y|>ǫ}

u(y) − u(x)

|y − x|d+α dy,

where

Ad,−α =

2αΓ

d+α

2

π

d 2

Γ −α

2

.

Alternatively, for all x ∈ Rd, we can write

(−∆)α/2u(x) = −1

2C(d, α)

Rd

u(x + y) + u(x − y) − 2u(x)

|y|d+α

dy. Proof idea: Change of variables

Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian

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Fourier Transform Definition

For any ϕ ∈ S(Rd), we define the Fourier transform of ϕ as

F[ϕ](ξ) =

1

(2π)d/2

Rd e−iξ·x ϕ(x) dx.

Fractional Laplacian (−∆)α/2 can be viewed as a pseudo-differential

perator of symbol (or multiplier) |ξ|α. For any u ∈ S and for all

ξ ∈ Rd,

Fractional Laplacian

(−∆)α/2u = F−1(|ξ|α(Fu)).

Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian

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A connection between PDE and Probability

Consider the Cauchy problem ut − ∆u

=

0 in Rd × (0, ∞) u

=

f on Rd × {t = 0}. This linear PDE describes the evolution of temperature u, starting from an initial temperature distribution f, as heat flows through Rd. The solution can be written as u(x, t) =

Rd pt(x − y)f(y)dy.

Such solutions are not unique. But, D. Widder showed in 1944 that the uniqueness hold if one considers only nonnegative solutions u ≥ 0.

Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian

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Kac (1966) Can One Hear the Shape of a Drum?

You Can’t Hear the Shape of a Drum Gordon, Webb, and Wolpert (1992) There exist nonisometric planar regions that have identical Laplace spectra. Can one hear the shape of a graph? B. Gutkin and U. Smilanski (2001) One can hear the corners of a drum Z. Lu and J. Rowlett (2015) ”The presence or absence of corners is uniquely determined by the spectrum” under some assumptions for all planar domains.

Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian

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One can hear the area of a drum!

H. Weyl (Around 1912)

For a bounded domain Ω in Rd, as z → ∞ N(z) = #{k : λk < z} =

ωd (2π)d |Ω|zd/2 + o(zd/2).

“Two domains with different volumes can never have the same spectrum.”

Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian

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Weyl’s Law for fractional Laplacian

E. Harrell-S.Y.Y. (2009) , for α = 1. As k → ∞,

λ(1)

k

∼ √

4π

Γ(1 + d/2)k |Ω| 1/d

S.Y.Y., T.Yolcu (2013) for 0 < α ≤ 2,

λ(α)

k

∼ (4π)α/2 Γ(1 + d/2)k |Ω| α/d

Blumenthal and Getoor (1959), Weyl Law for the eigenvalues of stable processes (probabilistic proof)

Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian

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Berezin-Li-Yau Inequality for Laplacian

P . Li & S.-T.Yau (1983); F . A. Berezin (1972) For an arbitrary bounded domain Ω in Rd,

k

j=1

λj ≥

d d + 2(4π)

Γ(1 + d

2)

|Ω| 2

d

k1+ 2

d .

S.Y.Y.-T. Yolcu (2013):

For 0 < α ≤ 2 and d ≥ 2,

k

j=1

λ(α)

j

≥ (4π)

α

2

d

α + d

Γ
1 + d

2

|Ω|

α

d

k1+ α

d

For 0 < a ≤ 1, 0 < α ≤ 2 and d ≥ 2,

k

j=1
λ(α)

j

a ≥ (4π)

aα 2

d d + aα

Γ
1 + d

2

|Ω|

aα

d

k1+ aα

d .

Laptev (1997) - fractional Laplacian (using symbols and traces of matrices)

Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian

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BLY improvements for Laplacian

Melas (2002)

k

j=1

λj ≥ (4π)

d 2 + d

Γ
1 + d

2

|Ω|

2

d

k1+ 2

d +

1 24(2 + d)

|Ω| I(Ω) k,

where I(Ω), the moment of inertia, is defined by

I(Ω) = min

y∈Rd

Ω

|w − y|2 dw.

Harrell-Hermi (2008): Melas’s bound is dual to the following

j
z − λ(2)

j

+ ≤ (4π)− d

2

2 2 + d

|Ω| Γ(1 + d

2)

z −

|Ω|

24(2 + d)I(Ω)

1+ d

2

.

Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian

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BLY improvements for fractional Laplacian

S.Y.Y- T. Yolcu (2013) For 0 < α ≤ 2, we have

k

j=1

λ(α)

j

≥

d

α + d (4π)

α

2

Γ(1 + d

2)

|Ω| α

d

k1+ α

d +L(α, d)|Ω|1+ 2−α d

I(Ω)

k1− 2−α

d

where L(α, d) =

α

48(α + d)

(4π)d/2Γ
1 + d

2

α−2

d

Melas type bounds and their many variants and extensions have recently attracted a lot of attention, see for instance Weidl (2008), Kovaˆ r´ ık-Vugalter-Weidl (2009), S.Y.Y.(2010), Ilyin (2010), S.Y.Y.-T. Yolcu (2012, JMP), S.Y.Y.-T. Yolcu (2013, CCM), S.Y.Y.-T. Yolcu (2013, JMP),

T. Yolcu(2013), Kovaˆ

r´ ık-Weidl (2014), Wei-Sun-Zheng(2014), S.Y.Y.-T. Yolcu (2014) and many others...

Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian

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Proof idea

Following Melas’s footsteps... Key lemma: For t > 0, s > 0, 2 ≤ d, 0 < α ≤ 2, we have td+α ≥ d + α d tdsα − α d sd+α + α d sd+α−2(t − s)2 Lemma 2: For any 0 < ℓ ≤ α/12,

k

j=1

λ(α)

j

≥ dw−α/d

d

k1+α/d d + α

η(0)−α/d + ℓw

2−α d

d

k1− 2−α

d

m2(d + α) η(0)2+ 2−α

d

Minimize this inequality over η(0). Show that we may replace η(0) = |Ω|(2π)−d for ℓ = α/12 and substitute m = 2(2π)−d

|Ω|I(Ω).

Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian

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A remark in Frank-Larson (Arxiv: 2001.01876v1- 2020)

Let Ω ⊂ R2, a piecewise smooth domain with ∂Ω be the union of smooth curve segments γj, j = 1, . . . m parametrized by arclength and

rdered so that γm meets γ1. Let αj ∈ (0, 2π) denote be the interior

angle formed at the point γj ∩ γj+1. Then

k≥1

e−λk t = |Ω| 4πt − H1(∂Ω) 4(4πt)1/2 + 1 12π

m

j=1
γj

κ(s)ds + π2 − α2

j

2αj

+ o(1)

as t → 0+ and κ(s) denotes the curvature.

M. Kac (1966) - Ω bounded by a broken line ∂Ω

van den Berg (1988) - polygonal boundary Mazzeo-Rowlett (2015) - a heat trace anomaly on polygons

Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian

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McKean-Singer (1967)

Let M be a closed d−dimensional, smooth Riemannian manifold with metric tensor g = (gij), and let ∆ denotes the Laplace-Beltrami operator

∆ =

1

det(g)

∂ ∂xi

gij det(g) ∂

∂xj

then

k≥1

e−λk t =

|Ω| (4πt)d/2 +

t 3(4πt)d/2

M

K + t2 180

M

(10A − B + 2C) + o(t3)

where K=scalar curvature at a point of M and where A, B, C are a particular basis of the space of polynomials of degree 2 in the curvature tensor R which are invariant under the action of the orthogonal group. When d = 2, this reduces to

k≥1

e−λk t = |Ω| 4πt + 1 12π

M

K + tπ 60

M

K 2 + o(t2) as t → 0.

Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian

Extensions of some results about the eigenvalues for the Laplacian - - PowerPoint PPT Presentation

Extensions of some results about the eigenvalues for the Laplacian

Table of Contents

Table of Contents

Eigenvalues/Eigenfunctions of Laplacian

Fractional Laplacian operator

Fourier Transform Definition

A connection between PDE and Probability

Table of Contents

Kac (1966) Can One Hear the Shape of a Drum?

One can hear the area of a drum!

Weyl’s Law for fractional Laplacian

Berezin-Li-Yau Inequality for Laplacian

BLY improvements for Laplacian

BLY improvements for fractional Laplacian

Proof idea

Table of Contents

A remark in Frank-Larson (Arxiv: 2001.01876v1- 2020)

McKean-Singer (1967)