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Entire solutions in 2 for a class of Allen-Cahn equations

Francesca Alessio, Piero Montecchiari (2005)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a class of semilinear elliptic equations of the form - ε 2 Δ u ( x , y ) + a ( x ) W ' ( u ( x , y ) ) = 0 , ( x , y ) 2 where ε > 0 , a : is a periodic, positive function and W : is modeled on the classical two well Ginzburg-Landau potential W ( s ) = ( s 2 - 1 ) 2 . We look for solutions to (1) which verify the asymptotic conditions u ( x , y ) ± 1 as x ± uniformly with respect to y . We show via variational methods that if ε is sufficiently small and a is not constant, then (1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.

Entire solutions in 2 for a class of Allen-Cahn equations

Francesca Alessio, Piero Montecchiari (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a class of semilinear elliptic equations of the form 15.7cm - ε 2 Δ u ( x , y ) + a ( x ) W ' ( u ( x , y ) ) = 0 , ( x , y ) 2 where ε > 0 , a : is a periodic, positive function and W : is modeled on the classical two well Ginzburg-Landau potential W ( s ) = ( s 2 - 1 ) 2 . We look for solutions to ([see full textsee full text]) which verify the asymptotic conditions u ( x , y ) ± 1 as x ± uniformly with respect to y . We show via variational methods that if ε is sufficiently small and a is not constant, then ([see full textsee full text]) admits infinitely many of such solutions, distinct...

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