平面上变系数Allen-Cahn方程的multiple-end解（英文）
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摘要
平面上Allen-Cahn方程具有multiple-end解,进一步推广,变系数Allen-Cahn方程可以构造一类类似的整体解.给定k≥1,可以发现一个解集远离紧集,且它的零点集渐近于2k条直线(称为ends).这些解具有这样的性质:存在θ_0 <…<θ_(2k)=θ_0+2π,j=0,…,2k-1,且θ是(θ_j,θ_(j+1))的紧子集,使得■关于θ一致成立.
The Allen-Cahn equation has multiple-end solutions on the plane,we construct a similar class of entire solutions for the variable coefficient Allen-Cahn equation for further study.Given k ≥1,we can find a family of solutions whose zero level sets are,away from a compact set,asymptotic to 2k straight lines(which are called the ends).These solutions have the property that■ uniformly in θ on compact subsets of(θ_j,θ_(j+1)) when there exist θ_0 <θ_1 < … <θ_(2k) =θ_0 +2π,for j=0,…,2k-1.
The Allen-Cahn equation has multiple-end solutions on the plane,we construct a similar class of entire solutions for the variable coefficient Allen-Cahn equation for further study.Given k ≥1,we can find a family of solutions whose zero level sets are,away from a compact set,asymptotic to 2k straight lines(which are called the ends).These solutions have the property that■ uniformly in θ on compact subsets of(θ_j,θ_(j+1)) when there exist θ_0 <θ_1 < … <θ_(2k) =θ_0 +2π,for j=0,…,2k-1.
引文
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[2]Kohn R V,Sternberg P.Local minimizers and singular perturbations[J].Proc Roy Soc Edinburgh Sect(A),1989(111):69-84.
[3]Ghoussoub N,Gui C.On a conjecture of De Giorgi and some related problems[J].Math Ann,1998,311(3):481-491.
[4]Ambrosio L,Cabre X.Entire solutions of semilinear elliptic equations in R3 and a conjecture of De Giorgi[J].Amer Math Soc,2000,13(4):725-739.
[5]Savin O.Regularity of flat level sets in phase transitions[J].Ann of Math,2009,169(1):41-78.
[6]Del Pino M,Kowalczyk M,Wei J.On De Giorgi’s conjecture in dimension N≥9[J].Ann of Math,2011,174(3):1485-1569.
[7]Del Pino M,Kowalczyk M,Pacard F,et al.Multiple-end solutions to the Allen-Cahn equation in R[J].J Funct Anal,2010,258(2):458-503.
[8]Kostant B.The solution to a generalized Toda lattice and representation theory[J].Adv Math,1979,34:195-338.
[9]Pacard F,RitoréM.From the constant mean curvature hypersurfaces to the gradient theory of phase transitions[J].Differential Geom,2003,64(3):356-423.
[10]Del Pino M,Kowalczyk M,Wei J.The Toda system and clustering interfaces in the Allen-Cahn equation[J].Arch Ration Mech Anal,2008,190:141-187.
[11]Del Pino M,Kowalczyk M,Pacard F,et al.The Toda system and multiple-end solutions of autonomous planar elliptic problems[J].Adv Math,2010,224(4):1462-1516.