## EVENT DETAILS AND ABSTRACT

**Special Seminar**
**Title:** $C^{2, \alpha}$ estimates of interfaces for Allen-Cahn equation

**Speaker:** Juncheng Wei

**Speaker Info:**

**Brief Description:**

**Special Note**:

**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.

**Date:** Saturday, April 27, 2019

**Time:** 2:30pm

**Where:** Swift 107

**Contact Person:** Aaron Naber

**Contact email:** anaberi@math.northwestern.edu

**Contact Phone:**

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