Resonance and Interior Layers in an Inhomogeneous Phase Transition Model
Manuel del Pino, Michał Kowalczyk, Juncheng Wei · SIAM Journal on Mathematical Analysis · 2007
We consider the problem $\ve^2 \Delta u + (u-a(x))(1-u^2)=0$ in $\Omega, \frac{\partial u}{\partial u}=0$ on $\partial \Omega$, where Ω is a smooth and bounded domain in $\R^2$, $ -1 0$ on Γ, where n is the outer normal to $\Omega_+$. Fife and Greenlee [Russian Math. Surveys, 29 (1974), pp. 103–131] proved the existence of an interior transition layer solution $u_\ve$ which approaches ‐1 in $\Omega_{-}$ and +1 in $\Omega_{+}$, for all ε sufficiently small. A question open for many years has been whether an interior transition layer solution approaching 1 in $\Omega_{-}$ and ‐1 in $\Omega_{+}$ exists. In this paper, we answer this question affirmatively when $n=2$, provided that ε is small and away from certain critical numbers. A main difficulty is a resonance phenomenon induced by a large number of small critical eigenvalues of the linearized operator.