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Title: $C^{2, \alpha}$ estimates of interfaces for Allen-Cahn equation
Speaker: Juncheng Wei​
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Abstract:

I will discuss the $C^{2,\alpha}$ estimates of interfaces of stable solutions to singularly perturbed Allen-Cahn $$ \epsilon \Delta u =\frac{1}{\epsilon} (u^3-u)$$ We prove $C^{2,\alpha}$ and curvature estimates of the interfaces in dimensions $ n\leq 10$, which is optimal. We show that the obstruction to $C^{2,\alpha}$ estimates is precisely the existence of Toda system (collapsing interfaces). The proof uses the reverse process of infinite dimensional reduction method. We then discuss two applications. The first is the classification of finite Morse index (and hence finite ends) solutions in $R^2$, and the second one is the classification of axially symmetric solutions with finite Morse. Joint work with K. Wang.​