圆柱面上阿尔顿-冈方程的古老解
偏微分方程分析
2024-06-18 v1
摘要
设 N ≥ 2 N\geq 2 N ≥ 2 F : R N → R F:\mathbb{R}^N\to \mathbb{R} F : R N → R F ( 1 ) = 0 F(1)=0 F ( 1 ) = 0 lim r → 1 F r ( r ) = ∞ \lim_{r\to 1}F_r(r)=\infty lim r → 1 F r ( r ) = ∞ r = ∣ x ∣ r=|x| r = ∣ x ∣ M × ( − ∞ , 0 ) M\times(-\infty,0) M × ( − ∞ , 0 ) u ~ t = Δ M u ~ + ( 1 − u ~ 2 ) u ~ \tilde u_t = \Delta_{M} \tilde u + (1-{\tilde u}^2)\tilde u u ~ t = Δ M u ~ + ( 1 − u ~ 2 ) u ~ M = { ( x , ± F ( ∣ x ∣ ) ) : x ∈ R N , ∣ x ∣ ≥ 1 } M=\{(x, \pm F(|x|)):\;x\in\mathbb{R}^N,\;|x|\geq1\} M = {( x , ± F ( ∣ x ∣ )) : x ∈ R N , ∣ x ∣ ≥ 1 } N N N R N + 1 \mathbb{R}^{N+1} R N + 1 Δ M \Delta_{M} Δ M M M M u ( t , r , F ( r ) ) = u ( t , r , − F ( r ) ) u(t,r,F(r))=u(t,r,-F(r)) u ( t , r , F ( r )) = u ( t , r , − F ( r )) u ( t , r , F ( r ) ) ≈ ∑ j = 1 k ( − 1 ) j − 1 w ( r − ρ j ( t ) ) − 1 2 ( 1 + ( − 1 ) k ) as t → − ∞ , u(t,r,F(r)) \approx \sum_{j=1}^k (-1)^{j-1}w(r-\rho_j(t)) - \frac 12 (1+ (-1)^{k}) \quad \hbox{ as } t\to -\infty, u ( t , r , F ( r )) ≈ j = 1 ∑ k ( − 1 ) j − 1 w ( r − ρ j ( t )) − 2 1 ( 1 + ( − 1 ) k ) as t → − ∞ , w ( s ) w(s) w ( s ) w ′ ′ + ( 1 − w 2 ) w = 0 w'' + (1-w^2)w=0 w ′′ + ( 1 − w 2 ) w = 0 w ( ± ∞ ) = ± 1 w(\pm \infty)= \pm 1 w ( ± ∞ ) = ± 1 w ( s ) = tanh ( s 2 ) w(s) = \tanh \left(\frac s{\sqrt{2}} \right) w ( s ) = tanh ( 2 s ) ρ j ( t ) = − 2 ( n − 1 ) t + 1 2 ( j − k + 1 2 ) log ( ∣ t ∣ log ∣ t ∣ ) + O ( 1 ) , j = 1 , … , k . \rho_j(t)=\sqrt{-2(n-1)t}+\frac{1}{\sqrt{2}}\left(j-\frac{k+1}{2}\right)\log\left(\frac {|t|}{\log |t| }\right)+ O(1),\quad j=1,\ldots ,k. ρ j ( t ) = − 2 ( n − 1 ) t + 2 1 ( j − 2 k + 1 ) log ( log ∣ t ∣ ∣ t ∣ ) + O ( 1 ) , j = 1 , … , k .
引用
@article{arxiv.2406.10530,
title = {Ancient solutions to the Allen Cahn equation in catenoids},
author = {Konstantinos T. Gkikas},
journal= {arXiv preprint arXiv:2406.10530},
year = {2024}
}