By Malchiodi A., Wei J.
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Extra info for Boundary interface for the Allen-Cahn equation
Phys. 235 (2003), 427-466. Ambrosetti, A. -M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part II, Indiana Univ. Math. J. 53(2004), no. 2, 297-329.  L. Bronsard and B. Stoth, On the existence of high multiplicity interfacses, Math. Res. Lett. 3 (1996), 117-131.  I. Chavel, Riemannian Geometry—A Modern Introduction, Cambridge Tracts in Math. 108, Cambridge Univ. Press, Cambridge, 1993.  G. Flores and P. Padilla, Higher energy solutions in the theory of phase trasitions: a variational approach, J.
30 A. 81) (1 − εxn k(z)) = o(ε) + 2ε 0 Note that the main term in ∂u ˜K ε ∂ε K K ˜K 2 ε ∂uε 2 ∂uε K ∂u ) ) (1 − 3(uK φ + 6u φ1 . 1 ε ε ε ∂xn ∂ε ∂xn is ∂ u˜K ε ∂ε (− xn 1 1 1 + √ + o( ))H . 82) 2 ˜K ε 2 K ∂u (1 − 3(uK ) ) + 6u (ε·) φ20 dxn ε ε ε ∂ε (1 − εxn k(εz)) 2 0 ε−τ = o(1) + 2 0 ε−τ = o(1) + 2 0 1 (2(1 − 3H 2 ) + 6(−xn + √ )HH )H φ1 2 2 1 1 1 [L(−(H t) + ( √ log + Rε + √ )H )]φ1 ε 2 2 2 2 1 1 = 2(2H − √ H )(0)φ1 (0) − 2(H − √ H )(0)φ1 (0) − 2 2 2 2 2 √ τε = 8A0 √ √ e− 2R − 2( 2 ε ε−τ (φ1 + (1 − 3H 2 )φ1 )(H t) 0 (H (H t) )γ5 k(z) + o(1) R (H )2 )γ5 k(z) + o(1) = 8A0 k7 (z) − ( R (H )2 k(z) + o(1) > 0.
Kowalczyk, On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions, Annali di Matematica Pura et Aplicata, to appear. V. Kohn and P. Sternberg, Local minimizers and singular perturbations, Proc. Royal Soc. Edinburgh 11A (1989), 69-84.  A. Malchiodi, Solutions concentrating at curves for some singularly perturbed elliptic problems. C. R. Math. Acad. Sci. Paris 338 (2004), no. 10, 775–780.  A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains, Geom.