Entire solutions in R2 for a class of Allen-Cahn equations by Alessio F., Montecchiari P.

By Alessio F., Montecchiari P.

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Extra resources for Entire solutions in R2 for a class of Allen-Cahn equations

Example text

19) does not hold and so that there exist [ζ1 , ζ2 ] ⊂ (y1 , y2 ), µ > 0, a sequence (yn ) ⊂ [ζ1 , ζ2 ], yn → y¯ ∈ [ζ1 , ζ2 ], and a sequence (xn ) ⊂ R, |xn | → ∞ such that 1 − |u(xn , yn )| ≥ µ. Since u C 2 (S(ζ ,ζ ) ) < +∞, one obtains that there exists ρ > 0 such that 1 2 1 − |u(xn , y)| ≥ µ2 for any y ∈ [ζ1 , ζ2 ] such that |y − y¯| ≤ ρ whenever n is sufficiently large, a contradiction since we already know that 1 − |u(x, y)| → 0 as |x| → +∞ for any y ∈ (ζ1 , ζ2 ). 2). 2) only on the half plane R × (−∞, y0,u ).

Let [ζ1 , ζ2 ] ⊂ (ζ¯1 , ζ¯2 ) ⊂ [ζ¯1 , ζ¯2 ] ⊂ (y1 , y2 ) and θ ∈ C 2 (R) be such that θ(y) = 0 if y ∈ / (ζ¯1 , ζ¯2 ) 1 and θ(y) = 1 for any y ∈ [ζ1 , ζ2 ]. 20) S(ζ¯ ¯ 1 ,ζ2 ) ¯ 1 ,ζ2 ) for any ψ ∈ H01 (S(ζ¯1 ,ζ¯2 ) ). Then one plainly recognizes that f = −aε W (u)θ + θ∂x2 z0 − ∂y2 θ(u − z0 ) − ∂y u∂y θ ∈ L2 (S(ζ¯1 ,ζ¯1 ) ) and by classical elliptic argument recovers that v ∈ H 2 (S(ζ¯1 ,ζ¯2 ) ) and so that u − z0 ∈ H 2 (S(ζ1 ,ζ2 ) ). Then −∆u+aε W (u) = 0 as element of ∩[ζ1 ,ζ2 ]⊂(y1 ,y2 ) L2 (S(y1 ,y2 ) ) and since u L∞ (R2 ) = 1, by a bootstrap argument we obtain that u verifies the equation in a classical sense with u C 2 (S(ζ ,ζ ) ) < +∞ 1 2 for any [ζ1 , ζ2 ] ⊂ (y1 , y2 ).

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