A counterexample to the "hot spots" conjecture on nested fractals

Transcription

1 A counterexample to the "hot spots" conjecture on nested fractals Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Zhejiang University Cornell University, June 13-17, 2017

2 Motivation The hot spots conjecture was posed by Rauch in D: open connected bounded subset of R d. u(t, x), t 0, x D: the solution of u t (t, x) = 1 2 xu(t, x), x D, t > 0, u n (t, x) = 0, x D, t > 0, u(0, x) = u 0 (x), x D. Informally speaking: Suppose that u(z t, t) = max{u(x, t) : t D}. z t D Then, we conjecture that for most" initial conditions, lim d(z t, D) = 0. t

3 Motivation The hot spots conjecture was posed by Rauch in D: open connected bounded subset of R d. u(t, x), t 0, x D: the solution of u t (t, x) = 1 2 xu(t, x), x D, t > 0, u n (t, x) = 0, x D, t > 0, u(0, x) = u 0 (x), x D. Informally speaking: Suppose that u(z t, t) = max{u(x, t) : t D}. z t D Then, we conjecture that for most" initial conditions, lim d(z t, D) = 0. t

4 Motivation The hot spots conjecture was posed by Rauch in D: open connected bounded subset of R d. u(t, x), t 0, x D: the solution of u t (t, x) = 1 2 xu(t, x), x D, t > 0, u n (t, x) = 0, x D, t > 0, u(0, x) = u 0 (x), x D. Informally speaking: Suppose that u(z t, t) = max{u(x, t) : t D}. z t D Then, we conjecture that for most" initial conditions, lim d(z t, D) = 0. t

5 Motivation The hot spots conjecture was posed by Rauch in D: open connected bounded subset of R d. u(t, x), t 0, x D: the solution of u t (t, x) = 1 2 xu(t, x), x D, t > 0, u n (t, x) = 0, x D, t > 0, u(0, x) = u 0 (x), x D. Informally speaking: Suppose that u(z t, t) = max{u(x, t) : t D}. z t D Then, we conjecture that for most" initial conditions, lim d(z t, D) = 0. t

6 Motivation Let 0 = µ 1 < µ 2 µ 3 be the spectrum of N (D). N 2 : the set of all N-eigenfunctions corresponding to µ 2. Typically", a 1 R and ϕ 2 N 2 with ϕ 2 0, s.t. u(t, x) = a 1 + ϕ 2 (x)e µ 2t + R(t, x), where R(t, x) goes to 0 faster than e µ 2t, as t. (HSC) ϕ 2 N 2 which is not identically 0, ϕ 2 attains its maximum and minimum on D (only). Conjecture (Rauch, 1974) (HSC) is ture for every domain D R d.

7 Motivation Let 0 = µ 1 < µ 2 µ 3 be the spectrum of N (D). N 2 : the set of all N-eigenfunctions corresponding to µ 2. Typically", a 1 R and ϕ 2 N 2 with ϕ 2 0, s.t. u(t, x) = a 1 + ϕ 2 (x)e µ 2t + R(t, x), where R(t, x) goes to 0 faster than e µ 2t, as t. (HSC) ϕ 2 N 2 which is not identically 0, ϕ 2 attains its maximum and minimum on D (only). Conjecture (Rauch, 1974) (HSC) is ture for every domain D R d.

8 Motivation Let 0 = µ 1 < µ 2 µ 3 be the spectrum of N (D). N 2 : the set of all N-eigenfunctions corresponding to µ 2. Typically", a 1 R and ϕ 2 N 2 with ϕ 2 0, s.t. u(t, x) = a 1 + ϕ 2 (x)e µ 2t + R(t, x), where R(t, x) goes to 0 faster than e µ 2t, as t. (HSC) ϕ 2 N 2 which is not identically 0, ϕ 2 attains its maximum and minimum on D (only). Conjecture (Rauch, 1974) (HSC) is ture for every domain D R d.

9 Motivation Let 0 = µ 1 < µ 2 µ 3 be the spectrum of N (D). N 2 : the set of all N-eigenfunctions corresponding to µ 2. Typically", a 1 R and ϕ 2 N 2 with ϕ 2 0, s.t. u(t, x) = a 1 + ϕ 2 (x)e µ 2t + R(t, x), where R(t, x) goes to 0 faster than e µ 2t, as t. (HSC) ϕ 2 N 2 which is not identically 0, ϕ 2 attains its maximum and minimum on D (only). Conjecture (Rauch, 1974) (HSC) is ture for every domain D R d.

10 Motivation Let 0 = µ 1 < µ 2 µ 3 be the spectrum of N (D). N 2 : the set of all N-eigenfunctions corresponding to µ 2. Typically", a 1 R and ϕ 2 N 2 with ϕ 2 0, s.t. u(t, x) = a 1 + ϕ 2 (x)e µ 2t + R(t, x), where R(t, x) goes to 0 faster than e µ 2t, as t. (HSC) ϕ 2 N 2 which is not identically 0, ϕ 2 attains its maximum and minimum on D (only). Conjecture (Rauch, 1974) (HSC) is ture for every domain D R d.

11 Motivation (HSC) holds for following domains: (Well known) balls, annulus; (Kawohl, LNM, 1985) D = D 1 (0, a), where D 1 C 0,1 ; (Bañuelos-Burdzy, JFA, 1999; Pascu, TAMS, 2002) convex domain which has a line of symmetry; (Ata-Burdzy, JAMS, 2004) lip domains: bounded Lipschitz planar domain D = {(x, y) : f 1 (x) < y < f 2 (x)}, where f 1, f 2 : Lipschitz functions with Lipschitz constant 1; (Miyamoto, J Math Phy, 2009) convex planar domains D with diam(d) 2 /Area(D) <

12 Motivation The hot spots conjecture fails for some planar domains: Burdzy-Werner (Ann Math, 1999): a bounded connected planar domain D (with two holes) s.t. multi(λ 2 ) = 1, and ϕ 2 attains its strict maximum at an interior point of D. Bass-Burdzy (Duke Math J, 2000): a bounded Lipschitz planar domain D s.t. multi(λ 2 ) = 1, and ϕ 2 attains its strict maximum and strict minimum at interior points of D. Burdzy (Duke Math J, 2005): a domain with one hole with above property. Problem Does (HSC) hold for all bounded convex domains D R d? Does (HSC) hold for all acute triangles?

13 Motivation The hot spots conjecture fails for some planar domains: Burdzy-Werner (Ann Math, 1999): a bounded connected planar domain D (with two holes) s.t. multi(λ 2 ) = 1, and ϕ 2 attains its strict maximum at an interior point of D. Bass-Burdzy (Duke Math J, 2000): a bounded Lipschitz planar domain D s.t. multi(λ 2 ) = 1, and ϕ 2 attains its strict maximum and strict minimum at interior points of D. Burdzy (Duke Math J, 2005): a domain with one hole with above property. Problem Does (HSC) hold for all bounded convex domains D R d? Does (HSC) hold for all acute triangles?

14 Motivation The hot spots conjecture fails for some planar domains: Burdzy-Werner (Ann Math, 1999): a bounded connected planar domain D (with two holes) s.t. multi(λ 2 ) = 1, and ϕ 2 attains its strict maximum at an interior point of D. Bass-Burdzy (Duke Math J, 2000): a bounded Lipschitz planar domain D s.t. multi(λ 2 ) = 1, and ϕ 2 attains its strict maximum and strict minimum at interior points of D. Burdzy (Duke Math J, 2005): a domain with one hole with above property. Problem Does (HSC) hold for all bounded convex domains D R d? Does (HSC) hold for all acute triangles?

15 Motivation The hot spots conjecture fails for some planar domains: Burdzy-Werner (Ann Math, 1999): a bounded connected planar domain D (with two holes) s.t. multi(λ 2 ) = 1, and ϕ 2 attains its strict maximum at an interior point of D. Bass-Burdzy (Duke Math J, 2000): a bounded Lipschitz planar domain D s.t. multi(λ 2 ) = 1, and ϕ 2 attains its strict maximum and strict minimum at interior points of D. Burdzy (Duke Math J, 2005): a domain with one hole with above property. Problem Does (HSC) hold for all bounded convex domains D R d? Does (HSC) hold for all acute triangles?

16 Motivation The hot spots conjecture fails for some planar domains: Burdzy-Werner (Ann Math, 1999): a bounded connected planar domain D (with two holes) s.t. multi(λ 2 ) = 1, and ϕ 2 attains its strict maximum at an interior point of D. Bass-Burdzy (Duke Math J, 2000): a bounded Lipschitz planar domain D s.t. multi(λ 2 ) = 1, and ϕ 2 attains its strict maximum and strict minimum at interior points of D. Burdzy (Duke Math J, 2005): a domain with one hole with above property. Problem Does (HSC) hold for all bounded convex domains D R d? Does (HSC) hold for all acute triangles?

17 Motivation Question How about p.c.f. self-similar sets?

18 on p.c.f. self-similar sets By using the spectral decimation method, we know that (HSC) holds on: Sierpinski gasket (R, Nonl Anal, 2012); Level-3 SG (R-Zheng, Nonl Anal, 2013); Higher dimensional SG (Li-R, CPAA, 2016); q 1 q 2 q 12 q2 q3 Figure: SG q1 q3 Figure: SG 3 q 1

19 Question Does (HSC) hold on all p.c.f. self-similar sets introduced by Kigami? Does (HSC) hold on all nested fractals introduced by Linstrøm?

20 Question Does (HSC) hold on all p.c.f. self-similar sets introduced by Kigami? Does (HSC) hold on all nested fractals introduced by Linstrøm?

21 Counterexample: q 1 K 2 K 3 K 1 K 4 K 6 q 2 q 3 K 5 Figure: Hexagasket (HG) Figure: V 0 and V 1 F k (x) = x 3 eikπ/ p k, k = 1,..., 6. V 0 = {q 1, q 2, q 3 } = {p 2, p 4, p 6 }.

22 Spectral decimation The key tool to prove (HSC) is the spectral decimation developed by Fukushima, Rammal, Shima and Toulouse etc. Basic idea: If we want to the know the eigenfunctions and eigenvalues of N (or D ), we just analyze its discrete form, and take limit. In fact, it coincides the idea which Kigami define the Laplacian on SG and general p.c.f. self-similar sets.

23 Discrete Laplacian m We define discrete Laplacian m on V m = w =m F w(v 0 ). Γ m : the graph on the vertex set V m with edge relation m : x m y x y and w with w = m, s.t. x, y F w (V 0 ). Define m u(x) = 1 #{y : y m x} (u(y) u(x)), x V m \V 0. y mx We call u m a discrete N-eigenfunction and λ m a discrete N-eigenvalue on V m if { m u m (x) = λ m u m (x), x V m \ V 0, y mq i (u(y) u(q i )) = λ m u m (q i ), q i V Λ m : the set of all discrete N-eigenvalues of m.

24 Discrete Laplacian m We define discrete Laplacian m on V m = w =m F w(v 0 ). Γ m : the graph on the vertex set V m with edge relation m : x m y x y and w with w = m, s.t. x, y F w (V 0 ). Define m u(x) = 1 #{y : y m x} (u(y) u(x)), x V m \V 0. y mx We call u m a discrete N-eigenfunction and λ m a discrete N-eigenvalue on V m if { m u m (x) = λ m u m (x), x V m \ V 0, y mq i (u(y) u(q i )) = λ m u m (q i ), q i V Λ m : the set of all discrete N-eigenvalues of m.

25 Discrete Laplacian m We define discrete Laplacian m on V m = w =m F w(v 0 ). Γ m : the graph on the vertex set V m with edge relation m : x m y x y and w with w = m, s.t. x, y F w (V 0 ). Define m u(x) = 1 #{y : y m x} (u(y) u(x)), x V m \V 0. y mx We call u m a discrete N-eigenfunction and λ m a discrete N-eigenvalue on V m if { m u m (x) = λ m u m (x), x V m \ V 0, y mq i (u(y) u(q i )) = λ m u m (q i ), q i V Λ m : the set of all discrete N-eigenvalues of m.

26 Discrete Laplacian m We define discrete Laplacian m on V m = w =m F w(v 0 ). Γ m : the graph on the vertex set V m with edge relation m : x m y x y and w with w = m, s.t. x, y F w (V 0 ). Define m u(x) = 1 #{y : y m x} (u(y) u(x)), x V m \V 0. y mx We call u m a discrete N-eigenfunction and λ m a discrete N-eigenvalue on V m if { m u m (x) = λ m u m (x), x V m \ V 0, y mq i (u(y) u(q i )) = λ m u m (q i ), q i V Λ m : the set of all discrete N-eigenvalues of m.

27 Spectral decimation on pre-hexagasket (Lau-Li-R) Suppose that λ m { 1 2, 3 2, 3± 5 4, 3± 2 4 } and λ m 1 = Φ(λ m ), where Φ(λ) = 2λ(λ 1)(16λ2 24λ+7) 2λ 1. If u is a discrete N-eigenfunction of m 1 with eigenvalue λ m 1, then an extension ũ on V m such that ũ is a discrete N-eigenfunction of m with eigenvalue λ m. The expression ũ on a typical V m cell is given by where y 01 = α(λ m )a + β(λ m )b + γ(λ m )c, z 01 = 2 ( β(λ m ) γ(λ m ) ) (a + b) + δ(λ m )(a + b + c), α(λ) = η(λ) 1 ( 16λ λ 2 23λ + 4), γ(λ) = η(λ) 1 ( λ + 1), β(λ) = η(λ) 1 (4λ 2 7λ + 2), δ(λ) = η(λ) 1, η(λ) = (4λ 2 6λ + 1)(16λ 2 24λ + 7).

28 a y 01 y z z 20 y 10 y 20 b y 12 y 21 c z 12 Figure: u on one cell of V m 1

29 Spectral decimation on pre-hexagasket Let u: discrete N-eigenfunction of m with eigenvalue λ m. Then u Vm 1 : discrete N-eigenfunction of m 1 with eigenvalue λ m 1. If λ m Λ m, then the multiplicity of λ m on m equals that of λ m 1 on m 1.

30 Spectral decimation on pre-hexagasket Let u: discrete N-eigenfunction of m with eigenvalue λ m. Then u Vm 1 : discrete N-eigenfunction of m 1 with eigenvalue λ m 1. If λ m Λ m, then the multiplicity of λ m on m equals that of λ m 1 on m 1.

31 Definition of Laplacian f C(HG), we say u dom with u = f on HG \ V 0 if 6 14 m m u(x) f on V \ V 0 as m, where V = m 0 V m. The normal derivative at q i V 0 of a function u on HG: ( ) 7 m n u(q i ) = lim (u(q i ) u(y)). m 3 y mq i u dom is called an eigenfunction of Neumann Laplacian with eigenvalue λ if u = λu on HG \ V 0, and n u = 0 on V 0.

32 Definition of Laplacian f C(HG), we say u dom with u = f on HG \ V 0 if 6 14 m m u(x) f on V \ V 0 as m, where V = m 0 V m. The normal derivative at q i V 0 of a function u on HG: ( ) 7 m n u(q i ) = lim (u(q i ) u(y)). m 3 y mq i u dom is called an eigenfunction of Neumann Laplacian with eigenvalue λ if u = λu on HG \ V 0, and n u = 0 on V 0.

33 Definition of Laplacian f C(HG), we say u dom with u = f on HG \ V 0 if 6 14 m m u(x) f on V \ V 0 as m, where V = m 0 V m. The normal derivative at q i V 0 of a function u on HG: ( ) 7 m n u(q i ) = lim (u(q i ) u(y)). m 3 y mq i u dom is called an eigenfunction of Neumann Laplacian with eigenvalue λ if u = λu on HG \ V 0, and n u = 0 on V 0.

34 According to a theorem by Shima, we can exhaust all N-eigenvalues and corresponding eigen-subspaces of as: Start from a discrete N-eigenfunction u of m0 with eigenvalue λ m0 for a nonnegative integer m 0, and then extend u to V by successively using spectral decimation on pre-hg, where λ m = Φ(λ m+1 ) for all m m 0 with λ m+1 = min{x 0 : Φ(x) = λ m } for all but finitely many times. Figure: The graph of the function Φ

35 Theorem (Lau-Li-R) Let λ 1 = 1 4 and λ m+1 = min{x > 0 : Φ(x) = λ m } for all m 1. Then λ = lim 6 m 14m λ m (1) exists, and is the second-smallest Neumann eigenvalue of. Furthmore, the multiplicity of λ equals 2.

36 The way to disprove (HSC) on hexagasket 2/ 3 x 2 0 1/ 3 1/ 2 x 1/ 3 1= 1/ 2 1 1/ 2-1/ A 1-1 A' -1/ 3 y 1 =-1/ 2-1/ 2-1/ 3 1 1/ 2-1/ 2-1 y 2-2/ 3 0 Figure: u 1 on V 1 Figure: u 1 + 2u 2 on V 1 u 1 attains the maximum & minimum on Ṽ0 = {p 1,..., p 6 }. u 1 + 2u 2 does not attains its maximum and minimum on Ṽ0. A = 1.025, A =

37 The way to disprove (HSC) on hexagasket 2/ 3 x 2 0 1/ 3 1/ 2 x 1/ 3 1= 1/ 2 1 1/ 2-1/ A 1-1 A' -1/ 3 y 1 =-1/ 2-1/ 2-1/ 3 1 1/ 2-1/ 2-1 y 2-2/ 3 0 Figure: u 1 on V 1 Figure: u 1 + 2u 2 on V 1 u 1 attains the maximum & minimum on Ṽ0 = {p 1,..., p 6 }. u 1 + 2u 2 does not attains its maximum and minimum on Ṽ0. A = 1.025, A =

38 Further remarks The following problems are what we are doing or wish to do (with K.-S. Lau, X.-H. Li, and H. Qiu): Does (HSC) holds on hexagasket if we choose another IFS? F k (x) = 1 3 (x p k) + p k, k = 1,..., 6. In this case, V 0 = {p 1,..., p 6 }. How can we do if there is no spectral decimation?

39 Further remarks The following problems are what we are doing or wish to do (with K.-S. Lau, X.-H. Li, and H. Qiu): Does (HSC) holds on hexagasket if we choose another IFS? F k (x) = 1 3 (x p k) + p k, k = 1,..., 6. In this case, V 0 = {p 1,..., p 6 }. How can we do if there is no spectral decimation?

40 Thank you for attention!