Cn one her the shpe of drum? After M. K, C. Gordon, D. We, nd S. Wolpert Corentin Lén Università Degli Studi di Torino Diprtimento di Mtemti Giuseppe Peno UNITO Mthemtis Ph.D Seminrs Mondy 23 My 2016
Motivtion: the wve eqution Ω R 2 open, ounded, nd pieewise regulr open set (think of memrne fixed t its oundry). We ll it domin. z(t, x, y) vertil displement of the memrne t the point (x, y) Ω for the time t. Ω Ω z stisfies the wve eqution : with the Lplin defined y: 2 z t 2 (t, x, y) = 2 ( z)(t, x, y) for (x, y) Ω, nd the Dirihlet oundry ondition: ( f )(x, y) = 2 f x (x, y) + 2 f (x, y), 2 y 2 z(t, x, y) = 0 for (x, y) Ω. The onstnt depends on the physil properties of the memrne. We hose the units so tht = 1.
Motivtion: sttionry solutions Sttionry solutions, or pure tones: eh point of the memrne goes up nd down in n hrmoni motion. We therefore look for solutions of the form z(t, x, y) = u(x, y) os(ω(t t 0)). The funtion u must stisfy u = λu in Ω; (1) u = 0 on Ω. (2) with λ = ω 2. We sy tht λ is Dirihlet eigenvlue if there exists non-zero solution u of (1) nd (2). We ll u eigenfuntion ssoited with λ, nd we ll eigenspe ssoited with λ the vetor spe of ll eigenfuntions.
Motivtion: sttionry solutions Sttionry solutions, or pure tones: eh point of the memrne goes up nd down in n hrmoni motion. We therefore look for solutions of the form z(t, x, y) = u(x, y) os(ω(t t 0)). The funtion u must stisfy u = λu in Ω; (1) u = 0 on Ω. (2) with λ = ω 2. We sy tht λ is Dirihlet eigenvlue if there exists non-zero solution u of (1) nd (2). We ll u eigenfuntion ssoited with λ, nd we ll eigenspe ssoited with λ the vetor spe of ll eigenfuntions. If we look t memrnes tht re free to move up nd down t the oundry, we otin the Neumnn eigenvlues: u = µu in Ω; (3) u ν = 0 on Ω. (4)
Exmple of squre memrne We ssume Ω = (0, L) 2 with L > 0 (squre memrne of side L). For ny pir (m, n) of positive integers, ( mπx ) ( nπy ) u m,n(x, y) := sin sin L L is n eigenfuntion, ssoited to the eigenvlue λ m,n = π2 L 2 (m2 + n 2 ).
Exmple of squre memrne We ssume Ω = (0, L) 2 with L > 0 (squre memrne of side L). For ny pir (m, n) of positive integers, ( mπx ) ( nπy ) u m,n(x, y) := sin sin L L is n eigenfuntion, ssoited to the eigenvlue λ m,n = π2 L 2 (m2 + n 2 ). With some Fourier nlysis, we n show tht ny eigenvlue is of the form λ m,n for some (m, n) N 2 1, nd tht n eigenfuntion for the Dirihlet prolem ssoited with λ m,n is liner omintion of the funtions u p,q stisfying p 2 + q 2 = m 2 + n 2. In prtiulr, ll the eigenspes re finite dimensionl.
Exmple of squre memrne We ssume Ω = (0, L) 2 with L > 0 (squre memrne of side L). For ny pir (m, n) of positive integers, ( mπx ) ( nπy ) u m,n(x, y) := sin sin L L is n eigenfuntion, ssoited to the eigenvlue λ m,n = π2 L 2 (m2 + n 2 ). With some Fourier nlysis, we n show tht ny eigenvlue is of the form λ m,n for some (m, n) N 2 1, nd tht n eigenfuntion for the Dirihlet prolem ssoited with λ m,n is liner omintion of the funtions u p,q stisfying p 2 + q 2 = m 2 + n 2. In prtiulr, ll the eigenspes re finite dimensionl. For the Neumnn eigenvlue prolem, we find u m,n(x, y) := os with (m, n) N 2 0. The sme results re true. ( mπx ) ( nπy ) os nd µ m,n = π2 L L L 2 (m2 + n 2 )
Some exmples for 50 = 7 2 + 1 2 = 5 2 + 5 2 () m = 7 nd n = 1 () m = 5 nd n = 5 () Liner omintion
Generliztion Theorem For generl domin Ω, there exists ountly mny eigenvlues, nd eh of the ssoited eigenspe is finite dimensionl. If we write the eigenvlues λ 1(Ω) λ 2(Ω) λ k (Ω)..., repeting them ording to the dimensions of the eigenspes, we hve lim k + λ k (Ω) = +. We n find sequene of ssoited eigenfuntions u 1, u 2,..., u k,... suh tht every solution z of the wve eqution n e written z(t, x, y) = + k=1 A k os( λ k t + ϕ k )u k (x, y). We hve the sme result for the Neumnn eigenvlues, with the sequene (µ k (Ω)) k 1. The sequene of ll the eigenvlues for domin is lled its (Dirihlet or Neumnn) spetrum.
Diret nd onverse spetrl prolems We dedue from the preeding theorem tht One n see the sound of drum. In prtiulr, if two domins Ω 1 nd Ω 2 re isometri, tht is to sy if there exists n isometry τ : R 2 R 2 ( length-preserving trnsformtion) suh tht τ(ω 1) = Ω 2, then Ω 1 nd Ω 2 re (Dirihlet-)isospetrl: λ k (Ω 1) = λ k (Ω 2) for ll k 1.
Diret nd onverse spetrl prolems We dedue from the preeding theorem tht One n see the sound of drum. In prtiulr, if two domins Ω 1 nd Ω 2 re isometri, tht is to sy if there exists n isometry τ : R 2 R 2 ( length-preserving trnsformtion) suh tht τ(ω 1) = Ω 2, then Ω 1 nd Ω 2 re (Dirihlet-)isospetrl: λ k (Ω 1) = λ k (Ω 2) for ll k 1. In its 1966 pper, Mrk K sked the onverse question: if we hve two domins Ω 1 nd Ω 2 tht re Dirihlet-isospetrl, re they isometri? This would men tht the spetrum of domin determines its shpe.
Diret nd onverse spetrl prolems We dedue from the preeding theorem tht One n see the sound of drum. In prtiulr, if two domins Ω 1 nd Ω 2 re isometri, tht is to sy if there exists n isometry τ : R 2 R 2 ( length-preserving trnsformtion) suh tht τ(ω 1) = Ω 2, then Ω 1 nd Ω 2 re (Dirihlet-)isospetrl: λ k (Ω 1) = λ k (Ω 2) for ll k 1. In its 1966 pper, Mrk K sked the onverse question: if we hve two domins Ω 1 nd Ω 2 tht re Dirihlet-isospetrl, re they isometri? This would men tht the spetrum of domin determines its shpe. In other words: Cn one her the shpe of drum?
Weyl s lw We define the ounting funtions ssoited with domin: N D (λ, Ω) := {k : λ k (Ω) < λ}; N N (λ, Ω) := {k : µ k (Ω) < λ}.
Weyl s lw We define the ounting funtions ssoited with domin: In prtiulr, N D (λ, (0, L) 2 ) = N D (λ, Ω) := {k : λ k (Ω) < λ}; N N (λ, Ω) := {k : µ k (Ω) < λ}. { (m, n) N 2 1 : { N N (λ, (0, L) 2 ) = (m, n) N 2 0 : } π 2 L 2 (m2 + n 2 ) < λ ; } π 2 L 2 (m2 + n 2 ) < λ ;
Weyl s lw We define the ounting funtions ssoited with domin: In prtiulr, N D (λ, (0, L) 2 ) = N D (λ, Ω) := {k : λ k (Ω) < λ}; N N (λ, Ω) := {k : µ k (Ω) < λ}. { (m, n) N 2 1 : { N N (λ, (0, L) 2 ) = (m, n) N 2 0 : In tht se we find, when λ +, } π 2 L 2 (m2 + n 2 ) < λ ; } π 2 L 2 (m2 + n 2 ) < λ ; N D (λ, (0, L) 2 ) N N (λ, (0, L) 2 ) L 2 4π λ = Are(Ω) λ. 4π λl/π
Weyl s lw We define the ounting funtions ssoited with domin: In prtiulr, N D (λ, (0, L) 2 ) = N D (λ, Ω) := {k : λ k (Ω) < λ}; N N (λ, Ω) := {k : µ k (Ω) < λ}. { (m, n) N 2 1 : { N N (λ, (0, L) 2 ) = (m, n) N 2 0 : In tht se we find, when λ +, N D (λ, (0, L) 2 ) N N (λ, (0, L) 2 ) L 2 4π λ = Are(Ω) λ. 4π Theorem (Hermnn Weyl, 1911) For generl domin, } π 2 L 2 (m2 + n 2 ) < λ ; } π 2 L 2 (m2 + n 2 ) < λ λl/π N D (λ, Ω) N N (λ, Ω) Are(Ω) λ when λ +. 4π ;
Proof of Weyl s lw : si priniples We now prove Weyl s lw for the Dirihlet eigenvlues. Disjoint union of domins If Ω = Ω Ω, the union eing disjoint (λ k (Ω)) k 1 = (λ k (Ω )) k 1 (λ k (Ω )) k 1 (µ k (Ω)) k 1 = (µ k (Ω )) k 1 (µ k (Ω )) k 1 (with repetition), nd therefore N D/N (λ, Ω) = N D/N (λ, Ω ) + N D/N (λ, Ω ).
Proof of Weyl s lw : si priniples We now prove Weyl s lw for the Dirihlet eigenvlues. Disjoint union of domins If Ω = Ω Ω, the union eing disjoint (λ k (Ω)) k 1 = (λ k (Ω )) k 1 (λ k (Ω )) k 1 (µ k (Ω)) k 1 = (µ k (Ω )) k 1 (µ k (Ω )) k 1 (with repetition), nd therefore N D/N (λ, Ω) = N D/N (λ, Ω ) + N D/N (λ, Ω ). Monotoniity priniple (Rihrd Cournt) If we dd Dirihlet ondition, ll the eigenvlues go up, nd therefore the ounting funtion goes down. If we reple Dirihlet ondition y Neumnn ondition, or if we dd Neumnn ondition, ll the eigenvlues go down, nd therefore the ounting funtion goes up.
Proof of Weyl s lw: inner nd outer pproximtion For given ε > 0, we find l > 0, Ω i nd Ω o suh tht Ω i is the reunion of n i squres of side l nd Ω 0 is the reunion of n o squres of side l; Ω i Ω Ω o; Are(Ω i ) (1 ε)are(ω) nd Are(Ω o) (1 + ε)are(ω). N D (λ, Ω) N D (λ, Ω i ) = n i N(λ, (0, l) 2 ) n il 2 4π (1 ε)are(ω) λ λ 4π
Proof of Weyl s lw: Dirihlet-Neumnn rketing For the outer pproximtion, we put Neumnn oundry ondition on the oundry of the squres. N D (λ, Ω) N N (λ, Ω o) = n on N (λ, (0, l) 2 ) nol2 4π We onlude tht (1 ε) Ω 4π N D (λ, Ω) N D (λ, Ω) lim inf lim sup λ + λ λ + λ We mke ε 0 nd we otin Weyl s lw. (1 + ε) Ω λ λ 4π (1 + ε) Ω. 4π
The Ryleigh-Fer-Krhn inequlity I Lord Ryleigh, 1877, in The theory of sound, volume I Of ll memrnes of equl re tht of irulr form s the grvest pith.
The Ryleigh-Fer-Krhn inequlity II Theorem (Georg Fer 1923, Edgr Krhn 1925-1926) If D Ω is disk with the sme re s the domin Ω, λ 1(D Ω ) λ 1(Ω), with equlity if, nd only if, Ω is isometri to D Ω. This inequlity n e dedued from the isoperimetri inequlity. Consequene: More preise formultion: One n her the shpe of irulr drum. Corollry (Spetrl rigidity of the disk) If domin Ω s the sme eigenvlues s disk D, then Ω is isometri to D.
One nnot her the shpe of drum Conjeture (Mrk K, 1966) One nnot her the shpe of drum. Theorem (Crolyn Gordon, Dvid We, nd Sott Wolpert, 1992) There exists pirs of domins tht re Dirihlet-isospetrl (i.e. λ k (Ω 1) = λ k (Ω 2) for ll k 1), nd lso Neumnn-isopetrl (i.e. µ k (Ω 1) = µ k (Ω 2) for ll k 1). A lot of different exmples hve een found sine then. We will study n exmple due to Peter Buser, John Conwy, nd Peter Doyle (1994), ut the tehniques re very generl.
Building lok C 75 60 45 A B
The reflexion priniple We reflet the tringle with respet to side. If u stisfies u = λu in the tringle, nd Dirihlet oundry ondition on the side, the ntisymmetri extension of u stisfies u = λu in the symmetrized domin. u u σ
The two domins 1 2 3 4 5 6 7 1 2 3 4 5 6 7 Ω 1 Ω 2
Deompositions nd trnsplnttions If u is funtion on Ω 1, we note u i, for i {1,..., 7}, the restrition of u to the tringle n i. We deompose in the sme wy ny funtion v on Ω 2 We n write this U = u 1 u 2 u 3 u 4 u 5 u 6 u 7 nd V = v 1 v 2 v 3 v 4 v 5 v 6 v 7. For ll i, j {1,..., 7}, there is unique isometry τ i,j of the tringle n i in Ω 1 to the tringle n j in Ω 2. There is unique trnsplnttion of u i to the tringle n j: u i τ 1 i,j. The ide of the proof is to strt from n eigenfuntion u in Ω 1, ssoited with λ, nd to onstrut n eigenfuntion v on Ω 2, ssoited with λ, using the trnsplnttions of the funtions u i.
Constrution of the glol trnsplnttion mpping 6 3 4 1 5 2 7 2 Ω 1 Ω 2
Constrution of the glol trnsplnttion mpping 6 3 4 1 5 2 7 2 1 Ω 1 Ω 2
Constrution of the glol trnsplnttion mpping 1 2 3 4 5 6 7 2 1 2 Ω 1 Ω 2
Constrution of the glol trnsplnttion mpping 1 2 3 4 5 6 7 2 1 3 2 Ω 1 Ω 2
Constrution of the glol trnsplnttion mpping 1 2 3 4 5 6 7 2 1 3 1 2 Ω 1 Ω 2
Constrution of the glol trnsplnttion mpping 1 2 3 4 5 6 7 2 +3 1 3 1 2 Ω 1 Ω 2
Constrution of the glol trnsplnttion mpping 1 2 3 4 5 6 7 2 +3 +4 1 3 1 2 Ω 1 Ω 2
Constrution of the glol trnsplnttion mpping 6 3 +6 7 3 4 1 2 7 5 2 5 7 1 4 +7 2 +3 +4 1 3 +5 1 2 +6 4 5 6 Ω 1 Ω 2
Mtrix representtion The glol trnsplnttion mpping n e represented y mtrix T suh tht V = TU. Here we hve: 0 1 1 1 0 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 T = 1 1 0 0 0 1 0. 0 0 1 0 0 1 1 0 1 0 0 1 0 1 0 0 0 1 1 1 0 This mtrix is invertile, so tht U = T 1 V. We find: 4 8 8 8 4 4 4 8 4 4 8 4 8 4 T 1 = 1 8 8 4 4 8 4 4 24 8 4 8 4 4 4 8. 4 8 4 4 4 8 8 4 4 4 8 8 4 8 4 4 8 4 8 8 4
End of the proof The mtrix representtion shows tht the glol trnsplnttion mpping is n invertile liner mpping. By onstrution, it sends n eigenspe for the Dirihlet eigenvlue prolem in Ω 1 into n eigenspe for the Dirihlet eigenvlue prolem in Ω 2. Proposition Any Dirihlet eigenvlue for the domin Ω 1 is lso Dirihlet eigenvlue for the domin Ω 2, with greter or equl multipliity. Using the trnsplnttion mpping defined y the mtrix T 1, we n exhnge the role of Ω 1 nd Ω 2. We onlude tht Ω 1 nd Ω 2 hve the sme Dirihlet eigenvlues, with the sme multipliities, tht is to sy they re Dirihlet-isospetrl. In the sme wy, we ould prove tht Ω 1 nd Ω 2 re Neumnn-isospetrl.
Open prolems Are there sets of more thn two isospetrl domins?
Open prolems Are there sets of more thn two isospetrl domins? Are there smooth isospetrl domins?
Open prolems Are there sets of more thn two isospetrl domins? Are there smooth isospetrl domins? Are there simply onneted isospetrl domins?
Open prolems Are there sets of more thn two isospetrl domins? Are there smooth isospetrl domins? Are there simply onneted isospetrl domins? Are there onvex isospetrl domins? The nswer to the lst question is positive if we llow mixed oundry onditions (Virginie Bonnillie-Noël, Bernrd Helffer, Thoms Hoffmnn-Ostenhof, 2009). Wht out the Dirihlet ondition?
Grzie per l ttenzione!