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
Table of Contents Laplacian and Fractional Laplacian 1 Definitions Weyl Asymptotics 2 Can One Hear the Shape of a Drum? Weyl Asymptotics Other Terms in Weyl Asymptotics Some New Directions 3 Other Terms in Weyl Asymptotics Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian
Table of Contents Laplacian and Fractional Laplacian 1 Definitions Weyl Asymptotics 2 Can One Hear the Shape of a Drum? Weyl Asymptotics Other Terms in Weyl Asymptotics Some New Directions 3 Other Terms in Weyl Asymptotics Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian
Eigenvalues/Eigenfunctions of Laplacian Let Ω be a bounded open set in R d . Dirichlet problem: − ∆ u n = λ n u n in Ω; = on ∂ Ω. u n 0 Eigenvalues of Laplacian: 0 < λ 1 < λ 2 ≤ λ 3 ≤ · · · → ∞ Corresponding Eigenfunctions: u 1 , u 2 , u 3 , . . . . Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian
Fractional Laplacian operator Let S be the Schwartz space of rapidly decaying C ∞ functions in R d (i.e., all of whose derivatives are rapidly decreasing). For any u ∈ S and α ∈ ( 0 , 2 ) , fractional Laplacian operator ( − ∆) α/ 2 is defined as u ( y ) − u ( x ) � α 2 u ( x ) = A d , − α lim ( − ∆) | y − x | d + α dy , ǫ → 0 + {| y | >ǫ } where � d + α 2 α Γ � 2 A d , − α = � . � − α d 2 � �� π � Γ 2 Alternatively, for all x ∈ R d , we can write u ( x + y ) + u ( x − y ) − 2 u ( x ) ( − ∆) α/ 2 u ( x ) = − 1 � 2 C ( d , α ) dy . | y | d + α R d Proof idea: Change of variables Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian
Fourier Transform Definition For any ϕ ∈ S ( R d ) , we define the Fourier transform of ϕ as � 1 R d e − i ξ · x ϕ ( x ) dx . F [ ϕ ]( ξ ) = ( 2 π ) d / 2 Fractional Laplacian ( − ∆) α/ 2 can be viewed as a pseudo-differential operator of symbol (or multiplier) | ξ | α . For any u ∈ S and for all ξ ∈ R d , Fractional Laplacian ( − ∆) α/ 2 u = F − 1 ( | ξ | α ( F u )). Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian
A connection between PDE and Probability Consider the Cauchy problem 0 in R d × ( 0 , ∞ ) u t − ∆ u = f on R d × { t = 0 } . = u This linear PDE describes the evolution of temperature u , starting from an initial temperature distribution f , as heat flows through R d . The solution can be written as � u ( x , t ) = R d p t ( 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
Table of Contents Laplacian and Fractional Laplacian 1 Definitions Weyl Asymptotics 2 Can One Hear the Shape of a Drum? Weyl Asymptotics Other Terms in Weyl Asymptotics Some New Directions 3 Other Terms in Weyl Asymptotics Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian
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
One can hear the area of a drum! H. Weyl (Around 1912) For a bounded domain Ω in R d , as z → ∞ ω d ( 2 π ) d | Ω | z d / 2 + o ( z d / 2 ). N ( z ) = # { k : λ k < z } = “Two domains with different volumes can never have the same spectrum.” Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian
Weyl’s Law for fractional Laplacian E. Harrell-S.Y.Y. (2009) , for α = 1. As k → ∞ , � 1 / d √ � Γ( 1 + d / 2 ) k λ ( 1 ) ∼ 4 π k | Ω | S.Y.Y., T.Yolcu (2013) for 0 < α ≤ 2, � α/ d � Γ( 1 + d / 2 ) k λ ( α ) ∼ ( 4 π ) α/ 2 k | Ω | 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
Berezin-Li-Yau Inequality for Laplacian P . Li & S.-T.Yau (1983); F . A. Berezin (1972) For an arbitrary bounded domain Ω in R d , � 2 � k Γ( 1 + d 2 ) d d k 1 + 2 � d . λ j ≥ d + 2 ( 4 π ) | Ω | j = 1 S.Y.Y.-T. Yolcu (2013): For 0 < α ≤ 2 and d ≥ 2 , � α k � � 1 + d � d Γ d � λ ( α ) α k 1 + α 2 ≥ ( 4 π ) 2 d j α + d | Ω | j = 1 For 0 < a ≤ 1 , 0 < α ≤ 2 and d ≥ 2 , � a α � k � 1 + d � d Γ � a d � λ ( α ) � a α k 1 + a α d . 2 ≥ ( 4 π ) 2 j d + a α | Ω | j = 1 Laptev (1997) - fractional Laplacian (using symbols and traces of matrices) Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian
BLY improvements for Laplacian Melas (2002) � 2 k � � 1 + d � Γ d | Ω | d 1 k 1 + 2 � 2 d + λ j ≥ ( 4 π ) I (Ω) k , 2 + d | Ω | 24 ( 2 + d ) j = 1 where I (Ω) , the moment of inertia, is defined by � | w − y | 2 dw . I (Ω) = min y ∈ R d Ω Harrell-Hermi (2008): Melas’s bound is dual to the following � 1 + d | Ω | � | Ω | 2 2 � � � z − λ ( 2 ) + ≤ ( 4 π ) − d z − . 2 j 2 + d Γ( 1 + d 24 ( 2 + d ) I (Ω) 2 ) j Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian
BLY improvements for fractional Laplacian S.Y.Y- T. Yolcu (2013) For 0 < α ≤ 2, we have � α k � d + L ( α , d ) | Ω | 1 + 2 − α Γ( 1 + d d 2 ) d d λ ( α ) k 1 − 2 − α � α k 1 + α ≥ α + d ( 4 π ) 2 d j | Ω | I (Ω) j = 1 where �� α − 2 α � � 1 + d d ( 4 π ) d / 2 Γ L ( α , d ) = 48 ( α + d ) 2 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
Proof idea Following Melas’s footsteps... Key lemma: For t > 0, s > 0, 2 ≤ d , 0 < α ≤ 2, we have t d + α ≥ d + α t d s α − α d s d + α + α d s d + α − 2 ( t − s ) 2 d Lemma 2: For any 0 < ℓ ≤ α/ 12, 2 − α k ≥ dw − α/ d k 1 − 2 − α k 1 + α/ d η ( 0 ) − α/ d + ℓ w d d λ ( α ) m 2 ( d + α ) η ( 0 ) 2 + 2 − α � d d d j d + α j = 1 Minimize this inequality over η ( 0 ) . Show that we may replace η ( 0 ) = | Ω | ( 2 π ) − d for ℓ = α/ 12 and substitute m = 2 ( 2 π ) − d � | Ω | I (Ω) . Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian
Table of Contents Laplacian and Fractional Laplacian 1 Definitions Weyl Asymptotics 2 Can One Hear the Shape of a Drum? Weyl Asymptotics Other Terms in Weyl Asymptotics Some New Directions 3 Other Terms in Weyl Asymptotics Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian
A remark in Frank-Larson (Arxiv: 2001.01876v1- 2020) Let Ω ⊂ R 2 , a piecewise smooth domain with ∂ Ω be the union of smooth curve segments γ j , j = 1 , . . . m parametrized by arclength and ordered so that γ m meets γ 1 . Let α j ∈ ( 0 , 2 π ) denote be the interior angle formed at the point γ j ∩ γ j + 1 . Then κ ( s ) ds + π 2 − α 2 m �� � 4 π t − H 1 ( ∂ Ω) e − λ k t = | Ω | 1 � � j 4 ( 4 π t ) 1 / 2 + + o ( 1 ) 12 π 2 α j γ j k ≥ 1 j = 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
McKean-Singer (1967) Let M be a closed d − dimensional, smooth Riemannian manifold with metric tensor g = ( g ij ) , and let ∆ denotes the Laplace-Beltrami operator ∂ det ( g ) ∂ 1 g ij � ∆ = � ∂ x i ∂ x j det ( g ) then | Ω | t 2 t � � e − λ k t = � ( 10 A − B + 2 C ) + o ( t 3 ) ( 4 π t ) d / 2 + K + 3 ( 4 π t ) d / 2 180 M M k ≥ 1 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 e − λ k t = | Ω | K + t π 1 � � K 2 + o ( t 2 ) � 4 π t + 12 π 60 M M k ≥ 1 as t → 0. Selma Yıldırım Extensions of some results about the eigenvalues for the Laplacian
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