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Interface foliation near minimal submanifolds in Riemannian manifolds with Positive Ricci curvature
 Interface foliation near minimal submanifolds in Riemannian manifolds with Positive Ricci curvature 教师介绍 本讲教师：杨 军 所属学科：理科 人　　气：617 课程介绍 Abstract：We will talk about the construction of clustering interfaces for Allen-Cahn equation on compact Riemannian manifold by the infinitely dimensional reduction method(from "M. del Pino, M. Kowalczyk, J. Wei, Comm. Pure Appl. Math. 2007"). The interaction of neighbouring interfaces will be derived in a form of Toda-Jacobi System involving a resonance phenomena. The result appeared in "M. del Pino, M. Kowalczyk, J. Wei and J. Yang, Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature, Geometric and Functional Analysis, 20 (2010), no. 4, 918-957". Let \$(M , K)\$ be an \$N\$-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation \$\$epsilon^2Delta_g u,+, (1 - u^2)u ,=,0quad mbox{in } M,\$\$ where \$epsilon\$ is a small parameter. Let \$Ksubset M\$ be an \$(N-1)\$-dimensional smooth minimal submanifold that separates \$M\$ into two disjoint components. Assume that \$K\$ is non-degenerate in the sense that it does not support non-trivial Jacobi fields, and that \$|A_K|^2+mbox{Ric}_g(nu_K, nu_K)\$ is positive along \$K\$. Then for each integer \$mgeq 2\$, we establish the existence of a sequence \$epsilon = epsilon_jto 0\$, and solutions \$u_epsilon\$ with \$m\$-transition layers near \$K\$, with mutual distance \$O(epsilon|ln epsilon|)\$.

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