We consider a class of periodic Allen-Cahn equations $ -Delta u(x,y)+a(x,y)W'(u(x,y))=0,quad (x,y)inR^{2}$ where $ain C(R^2)$ is an even, periodic, positive function represenitng a doubly periodic media and $W:R oR$ is a classical double well potential such as the Ginzburg-Landau potential $W(s)=(s^{2}-1^{2})^{2}$}. We show the existence and asymptotic behavior of a saddle solution on the entire plane which has odd symmetry with respect to both axises and even symmetry with respect to the line $x=y$. This result generalizes the classic result on saddle solutions of Allen-Cahn equation in a homogeneous media.
Saddle solutions to Allen-Cahn equations in doubly periodic media / Alessio, FRANCESCA GEMMA; C., Gui; Montecchiari, Piero. - In: INDIANA UNIVERSITY MATHEMATICS JOURNAL. - ISSN 0022-2518. - STAMPA. - 65:1(2016), pp. 199-221.
Saddle solutions to Allen-Cahn equations in doubly periodic media
ALESSIO, FRANCESCA GEMMA
;MONTECCHIARI, Piero
2016-01-01
Abstract
We consider a class of periodic Allen-Cahn equations $ -Delta u(x,y)+a(x,y)W'(u(x,y))=0,quad (x,y)inR^{2}$ where $ain C(R^2)$ is an even, periodic, positive function represenitng a doubly periodic media and $W:R oR$ is a classical double well potential such as the Ginzburg-Landau potential $W(s)=(s^{2}-1^{2})^{2}$}. We show the existence and asymptotic behavior of a saddle solution on the entire plane which has odd symmetry with respect to both axises and even symmetry with respect to the line $x=y$. This result generalizes the classic result on saddle solutions of Allen-Cahn equation in a homogeneous media.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.