A counterexample to the "hot spots" conjecture on nested - - PowerPoint PPT Presentation
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 Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li)
Zhejiang University
Cornell University, June 13-17, 2017
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
The “hot spots” conjecture was posed by Rauch in 1974. 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) = u0(x), x ∈ D.
D zt
Informally speaking: Suppose that u(zt, t) = max{u(x, t) : t ∈ D}. Then, we conjecture that for “most" initial conditions, lim
t→∞ d(zt, ∂D) = 0.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
The “hot spots” conjecture was posed by Rauch in 1974. 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) = u0(x), x ∈ D.
D zt
Informally speaking: Suppose that u(zt, t) = max{u(x, t) : t ∈ D}. Then, we conjecture that for “most" initial conditions, lim
t→∞ d(zt, ∂D) = 0.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
The “hot spots” conjecture was posed by Rauch in 1974. 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) = u0(x), x ∈ D.
D zt
Informally speaking: Suppose that u(zt, t) = max{u(x, t) : t ∈ D}. Then, we conjecture that for “most" initial conditions, lim
t→∞ d(zt, ∂D) = 0.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
The “hot spots” conjecture was posed by Rauch in 1974. 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) = u0(x), x ∈ D.
D zt
Informally speaking: Suppose that u(zt, t) = max{u(x, t) : t ∈ D}. Then, we conjecture that for “most" initial conditions, lim
t→∞ d(zt, ∂D) = 0.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Let 0 = µ1 < µ2 ≤ µ3 ≤ · · · be the spectrum of ∆N(D). N2: the set of all N-eigenfunctions corresponding to µ2. “Typically", ∃a1 ∈ R and ϕ2 ∈ N2 with ϕ2 ≡ 0 , s.t. u(t, x) = a1 + ϕ2(x)e−µ2t + R(t, x), where R(t, x) goes to 0 faster than e−µ2t, as t → ∞. (HSC) ∀ϕ2 ∈ N2 which is not identically 0, ϕ2 attains its maximum and minimum on ∂D (only). Conjecture (Rauch, 1974) (HSC) is ture for every domain D ⊂ Rd.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Let 0 = µ1 < µ2 ≤ µ3 ≤ · · · be the spectrum of ∆N(D). N2: the set of all N-eigenfunctions corresponding to µ2. “Typically", ∃a1 ∈ R and ϕ2 ∈ N2 with ϕ2 ≡ 0 , s.t. u(t, x) = a1 + ϕ2(x)e−µ2t + R(t, x), where R(t, x) goes to 0 faster than e−µ2t, as t → ∞. (HSC) ∀ϕ2 ∈ N2 which is not identically 0, ϕ2 attains its maximum and minimum on ∂D (only). Conjecture (Rauch, 1974) (HSC) is ture for every domain D ⊂ Rd.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Let 0 = µ1 < µ2 ≤ µ3 ≤ · · · be the spectrum of ∆N(D). N2: the set of all N-eigenfunctions corresponding to µ2. “Typically", ∃a1 ∈ R and ϕ2 ∈ N2 with ϕ2 ≡ 0 , s.t. u(t, x) = a1 + ϕ2(x)e−µ2t + R(t, x), where R(t, x) goes to 0 faster than e−µ2t, as t → ∞. (HSC) ∀ϕ2 ∈ N2 which is not identically 0, ϕ2 attains its maximum and minimum on ∂D (only). Conjecture (Rauch, 1974) (HSC) is ture for every domain D ⊂ Rd.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Let 0 = µ1 < µ2 ≤ µ3 ≤ · · · be the spectrum of ∆N(D). N2: the set of all N-eigenfunctions corresponding to µ2. “Typically", ∃a1 ∈ R and ϕ2 ∈ N2 with ϕ2 ≡ 0 , s.t. u(t, x) = a1 + ϕ2(x)e−µ2t + R(t, x), where R(t, x) goes to 0 faster than e−µ2t, as t → ∞. (HSC) ∀ϕ2 ∈ N2 which is not identically 0, ϕ2 attains its maximum and minimum on ∂D (only). Conjecture (Rauch, 1974) (HSC) is ture for every domain D ⊂ Rd.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Let 0 = µ1 < µ2 ≤ µ3 ≤ · · · be the spectrum of ∆N(D). N2: the set of all N-eigenfunctions corresponding to µ2. “Typically", ∃a1 ∈ R and ϕ2 ∈ N2 with ϕ2 ≡ 0 , s.t. u(t, x) = a1 + ϕ2(x)e−µ2t + R(t, x), where R(t, x) goes to 0 faster than e−µ2t, as t → ∞. (HSC) ∀ϕ2 ∈ N2 which is not identically 0, ϕ2 attains its maximum and minimum on ∂D (only). Conjecture (Rauch, 1974) (HSC) is ture for every domain D ⊂ Rd.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
(HSC) holds for following domains: (Well known) balls, annulus; (Kawohl, LNM, 1985) D = D1 × (0, a), where ∂D1 ∈ C0,1; (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) : f1(x) < y < f2(x)}, where f1, f2: Lipschitz functions with Lipschitz constant 1; (Miyamoto, J Math Phy, 2009) convex planar domains D with diam(D)2/Area(D) < 1.378.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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(λ2) = 1, and ϕ2 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(λ2) = 1, and ϕ2 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 ⊂ Rd? Does (HSC) hold for all acute triangles?
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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(λ2) = 1, and ϕ2 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(λ2) = 1, and ϕ2 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 ⊂ Rd? Does (HSC) hold for all acute triangles?
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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(λ2) = 1, and ϕ2 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(λ2) = 1, and ϕ2 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 ⊂ Rd? Does (HSC) hold for all acute triangles?
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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(λ2) = 1, and ϕ2 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(λ2) = 1, and ϕ2 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 ⊂ Rd? Does (HSC) hold for all acute triangles?
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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(λ2) = 1, and ϕ2 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(λ2) = 1, and ϕ2 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 ⊂ Rd? Does (HSC) hold for all acute triangles?
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Question How about p.c.f. self-similar sets?
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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);
1
q
2
q
3
q
Figure: SG
1
q
3
q
2
q
1
q
2
q
3
q
21
q
23
q
12
q
13
q
31
q
32
q
Figure: SG3
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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?
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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?
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Counterexample:
Figure: Hexagasket (HG)
K1 K2 K3 K4 K5 K6 q1 q2 q3
Figure: V0 and V1
Fk(x) = x
3eikπ/3 + 2 3pk, k = 1, . . . , 6.
V0 = {q1, q2, q3} = {p2, p4, p6}.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
We define discrete Laplacian ∆m on Vm =
|w|=m Fw(V0).
Γm: the graph on the vertex set Vm with edge relation ∼m:
x ∼m y ⇔ x = y and ∃w with |w| = m, s.t. x, y ∈ Fw(V0).
Define ∆mu(x) = 1 #{y : y ∼m x}
(u(y)−u(x)), x ∈ Vm\V0. We call um a discrete N-eigenfunction and λm a discrete N-eigenvalue on Vm if −∆mum(x) = λmum(x), x ∈ Vm \ V0, − 1
2
qi ∈ V0. Λm: the set of all discrete N-eigenvalues of ∆m.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
We define discrete Laplacian ∆m on Vm =
|w|=m Fw(V0).
Γm: the graph on the vertex set Vm with edge relation ∼m:
x ∼m y ⇔ x = y and ∃w with |w| = m, s.t. x, y ∈ Fw(V0).
Define ∆mu(x) = 1 #{y : y ∼m x}
(u(y)−u(x)), x ∈ Vm\V0. We call um a discrete N-eigenfunction and λm a discrete N-eigenvalue on Vm if −∆mum(x) = λmum(x), x ∈ Vm \ V0, − 1
2
qi ∈ V0. Λm: the set of all discrete N-eigenvalues of ∆m.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
We define discrete Laplacian ∆m on Vm =
|w|=m Fw(V0).
Γm: the graph on the vertex set Vm with edge relation ∼m:
x ∼m y ⇔ x = y and ∃w with |w| = m, s.t. x, y ∈ Fw(V0).
Define ∆mu(x) = 1 #{y : y ∼m x}
(u(y)−u(x)), x ∈ Vm\V0. We call um a discrete N-eigenfunction and λm a discrete N-eigenvalue on Vm if −∆mum(x) = λmum(x), x ∈ Vm \ V0, − 1
2
qi ∈ V0. Λm: the set of all discrete N-eigenvalues of ∆m.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
We define discrete Laplacian ∆m on Vm =
|w|=m Fw(V0).
Γm: the graph on the vertex set Vm with edge relation ∼m:
x ∼m y ⇔ x = y and ∃w with |w| = m, s.t. x, y ∈ Fw(V0).
Define ∆mu(x) = 1 #{y : y ∼m x}
(u(y)−u(x)), x ∈ Vm\V0. We call um a discrete N-eigenfunction and λm a discrete N-eigenvalue on Vm if −∆mum(x) = λmum(x), x ∈ Vm \ V0, − 1
2
qi ∈ V0. Λm: the set of all discrete N-eigenvalues of ∆m.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Suppose that λm ∈ {1
2, 3 2, 3± √ 5 4
, 3±
√ 2 4
} and λm−1 = Φ(λm), where Φ(λ) = 2λ(λ−1)(16λ2−24λ+7)
2λ−1
. If u is a discrete N-eigenfunction of ∆m−1 with eigenvalue λm−1, then ∃ an extension u on Vm such that u is a discrete N-eigenfunction of ∆m with eigenvalue λm. The expression u on a typical Vm cell is given by y01 = α(λm)a + β(λm)b + γ(λm)c, z01 = 2
where α(λ) = η(λ)−1(−16λ3 + 36λ2 − 23λ + 4), γ(λ) = η(λ)−1(−λ + 1), β(λ) = η(λ)−1(4λ2 − 7λ + 2), δ(λ) = η(λ)−1, η(λ) = (4λ2 − 6λ + 1)(16λ2 − 24λ + 7).
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
a b c y01 y02 y10 y20 y12 y21 z01 z20 z12
Figure: u on one cell of Vm−1
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Let u: discrete N-eigenfunction of ∆m with eigenvalue λm. Then u|Vm−1: discrete N-eigenfunction of ∆m−1 with eigenvalue λm−1. If λm ∈ Λm, then the multiplicity of λm on ∆m equals that of λm−1 on ∆m−1.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Let u: discrete N-eigenfunction of ∆m with eigenvalue λm. Then u|Vm−1: discrete N-eigenfunction of ∆m−1 with eigenvalue λm−1. If λm ∈ Λm, then the multiplicity of λm on ∆m equals that of λm−1 on ∆m−1.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
∀f ∈ C(HG), we say u ∈ dom ∆ with ∆u = f on HG \ V0 if 6 · 14m∆mu(x) ⇒ f on V∗ \ V0 as m → ∞, where V∗ =
m≥0 Vm.
The normal derivative at qi ∈ V0 of a function u on HG: ∂nu(qi) = lim
m→∞
7 3 m
y∼mqi
(u(qi) − u(y)). u ∈ dom ∆ is called an eigenfunction of Neumann Laplacian with eigenvalue λ if −∆u = λu on HG \ V0, and ∂nu = 0 on V0.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
∀f ∈ C(HG), we say u ∈ dom ∆ with ∆u = f on HG \ V0 if 6 · 14m∆mu(x) ⇒ f on V∗ \ V0 as m → ∞, where V∗ =
m≥0 Vm.
The normal derivative at qi ∈ V0 of a function u on HG: ∂nu(qi) = lim
m→∞
7 3 m
y∼mqi
(u(qi) − u(y)). u ∈ dom ∆ is called an eigenfunction of Neumann Laplacian with eigenvalue λ if −∆u = λu on HG \ V0, and ∂nu = 0 on V0.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
∀f ∈ C(HG), we say u ∈ dom ∆ with ∆u = f on HG \ V0 if 6 · 14m∆mu(x) ⇒ f on V∗ \ V0 as m → ∞, where V∗ =
m≥0 Vm.
The normal derivative at qi ∈ V0 of a function u on HG: ∂nu(qi) = lim
m→∞
7 3 m
y∼mqi
(u(qi) − u(y)). u ∈ dom ∆ is called an eigenfunction of Neumann Laplacian with eigenvalue λ if −∆u = λu on HG \ V0, and ∂nu = 0 on V0.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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 λm0 for a nonnegative integer m0, and then extend u to V∗ by successively using spectral decimation on pre-HG, where λm = Φ(λm+1) for all m ≥ m0 with λm+1 = min{x ≥ 0 : Φ(x) = λm} for all but finitely many times.
Figure: The graph of the function Φ
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Theorem (Lau-Li-R) Let λ1 = 1
4 and λm+1 = min{x > 0 : Φ(x) = λm} for all m ≥ 1.
Then λ = lim
m→∞ 6 · 14mλm
(1) exists, and is the second-smallest Neumann eigenvalue of ∆. Furthmore, the multiplicity of λ equals 2.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
x1= y1 1/ =- x2 y2 2 1/
1/
1/
1/ 3 1/ 3 2/
2/ 3 1/ 2 1/ 2
Figure: u1 on V1
1/ 2 1/ 2 1/
1/
1 1 1 1
A'
Figure: u1 + 2u2 on V1
u1 attains the maximum & minimum on V0 = {p1, . . . , p6}. u1 + 2u2 does not attains its maximum and minimum on V0. A = 1.025, A′ = −1.025.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
x1= y1 1/ =- x2 y2 2 1/
1/
1/
1/ 3 1/ 3 2/
2/ 3 1/ 2 1/ 2
Figure: u1 on V1
1/ 2 1/ 2 1/
1/
1 1 1 1
A'
Figure: u1 + 2u2 on V1
u1 attains the maximum & minimum on V0 = {p1, . . . , p6}. u1 + 2u2 does not attains its maximum and minimum on V0. A = 1.025, A′ = −1.025.
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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? Fk(x) = 1 3(x − pk) + pk, k = 1, . . . , 6. In this case, V0 = {p1, . . . , p6}. How can we do if there is no spectral decimation?
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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? Fk(x) = 1 3(x − pk) + pk, k = 1, . . . , 6. In this case, V0 = {p1, . . . , p6}. How can we do if there is no spectral decimation?
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture