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
Motivation The “hot spots” conjecture was posed by Rauch in 1974. open connected bounded subset of R d . D : u ( t , x ) , t ≥ 0 , x ∈ D : the solution of D ∂ u ∂ t ( t , x ) = 1 x ∈ D , t > 0 , 2 ∆ x u ( t , x ) , z t ∂ 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 } . Then, we conjecture that for “most" initial conditions, t →∞ d ( z t , ∂ D ) = 0 . lim Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Motivation The “hot spots” conjecture was posed by Rauch in 1974. open connected bounded subset of R d . D : u ( t , x ) , t ≥ 0 , x ∈ D : the solution of D ∂ u ∂ t ( t , x ) = 1 x ∈ D , t > 0 , 2 ∆ x u ( t , x ) , z t ∂ 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 } . Then, we conjecture that for “most" initial conditions, t →∞ d ( z t , ∂ D ) = 0 . lim Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Motivation The “hot spots” conjecture was posed by Rauch in 1974. open connected bounded subset of R d . D : u ( t , x ) , t ≥ 0 , x ∈ D : the solution of D ∂ u ∂ t ( t , x ) = 1 x ∈ D , t > 0 , 2 ∆ x u ( t , x ) , z t ∂ 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 } . Then, we conjecture that for “most" initial conditions, t →∞ d ( z t , ∂ D ) = 0 . lim Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Motivation The “hot spots” conjecture was posed by Rauch in 1974. open connected bounded subset of R d . D : u ( t , x ) , t ≥ 0 , x ∈ D : the solution of D ∂ u ∂ t ( t , x ) = 1 x ∈ D , t > 0 , 2 ∆ x u ( t , x ) , z t ∂ 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 } . Then, we conjecture that for “most" initial conditions, t →∞ d ( z t , ∂ D ) = 0 . lim Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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 2 t + R ( t , x ) , where R ( t , x ) goes to 0 faster than e − µ 2 t , 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 . Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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 2 t + R ( t , x ) , where R ( t , x ) goes to 0 faster than e − µ 2 t , 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 . Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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 2 t + R ( t , x ) , where R ( t , x ) goes to 0 faster than e − µ 2 t , 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 . Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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 2 t + R ( t , x ) , where R ( t , x ) goes to 0 faster than e − µ 2 t , 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 . Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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 2 t + R ( t , x ) , where R ( t , x ) goes to 0 faster than e − µ 2 t , 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 . Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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 ) < 1 . 378 . Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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? Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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? Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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? Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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? Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
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? Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Motivation Question How about p.c.f. self-similar sets? Huo-Jun Ruan (With K.-S. Lau and X.-H. Li) Hot spots conjecture
Recommend
More recommend
