Nondegeneracy Of Symmetry Solution To The Allen-Cahn Equation

Partial differential equations has been an important branch of mathematical research. In this paper,we reseach an important class of second order elliptic partial differential equations:Allen-Cahn equation. It is a classical nonlinear equations which originate in phase transitions theory, It have a very wide range of applications in the image edge preserving denoising processing, crystal growth, and so the direction of the mean curvature of the movement. in recent years, more significant progresses have been made in this equation, so more and more. experts are committed to research on it.In this paper, we study the property of entire solutions for Allen-Cahn equation in two-dimensional plane; Firstly, we introduce some basic theory and related research of Allen-Cahn equation at home and abroad, such as monotony and existence of solutions. Secondly, we review non-degeneracy of the classical saddle solution for Allen-Cahn equation. Finally we prove non-degeneracy of a special class 6-end solution for Allen-Cahn equation in function space with symmetry, the main method is that analysis the form of solution for linear equations in infinity; By using this non-degeneracy results and moduli space theory of Allen-Cahn equation, then, we prove existence of 6-end solutions with regular triangle symmetry.