The Allen-Cahn equation and minimal surfaces

Allen-Cahn 方程和最小曲面

• 批准号: 521052394
• 负责人: Dr. Matteo Rizzi, Ph.D.
• 金额: --
• 依托单位: Mathematisches Institut
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/521052394?language=en
• 关键词: Allen; Cahn; equation; minimal; surfaces

The main purposes of the project are to construct new solutions to some semilinear elliptic PDEs, with particular interest in the Allen-Cahn equation, and to study the qualitative properties of a given solution. Our analysis will often rely on the theory of minimal (hyper)surfaces, which are crucial to understand the behaviour of the zero level set of solutions to the Allen-Cahn equation. A part of our project will be devoted to the fractional Allen-Cahn equation. We are particularly interested in k-ended solutions, that is entire solutions in the euclidean space whose zero level is given, outside a sufficiently large ball, by the disjoint union of a finite number of connected components. We have already constructed examples of such solutions in dimension N ≥ 8 enjoing some symmetry properties, namely they are O(m)×O(n)-invariant, m+n=N, with infinite Morse index and their energy on the ball has polynomial growth with respect to the radius. Our future goals are: -To investigate the qualitative properties of O(m)×O(n)-invariant solutions to the Allen-Cahn equation, with particular interest in their Morse index and in their zero level set. Their energy on the ball will play an important role. -To construct new O(m)×O(n)-invariant entire solutions to the Allen-Cahn equation in dimension 4 ≤ N ≤ 7 and new non O(m)×O(n)-invariant k-ended solutions in dimension N ≥ 8 (symmetry breaking solutions). Their zero level set will be prescribed, close to some suitable minimal hypersurface or to the union of k ≥ 2 graphs over such a hypersurface. -To prove an abstract result which could provide a general strategy to construct solutions to semilinear PDEs, related to nonvariational methods, such as the Lyapunov-Schmidt reduction. -To construct k-ended solutions to the fractional Allen-Cahn equation in dimension 3 whose zero level set is a union of normal graphs over some suitable minimal hypersurface.

该项目的主要目的是构造一些半线性椭圆型偏微分方程解，特别是Allen-Cahn方程，并研究给定解的定性性质。我们的分析将经常依赖于极小(超)曲面理论，这对于理解Allen-Cahn方程的零水平解集的行为至关重要。我们项目的一部分将致力于分数阶Allen-Cahn方程。我们对k端解特别感兴趣，即欧氏空间中的全部解，其零级由有限个连通分量的不相交并在足够大的球外给出。我们已经在N维≥8上构造了这样的解的例子，它们具有一些对称性质，即它们是O(M)×O(N)-不变的，m+n=N，具有无限的Morse指数，并且它们在球上的能量关于半径具有多项式增长。我们未来的目标是：-研究Allen-Cahn方程O(M)×O(N)-不变解的定性性质，特别是对它们的Morse指数和它们的零水平集感兴趣。他们在球上的能量将发挥重要作用。-构造4维≤N≤7上Allen-Cahn方程新的O(M)×O(N)不变整体解和N≥8维非O(M)×O(N)不变k端解(对称破缺解)。它们的零水平集将被规定，接近于某个合适的极小超曲面或接近于这样一个超曲面上的k≥2图的并。-证明一个抽象的结果，它可以提供构造半线性偏微分方程解的一般策略，与非变分方法有关，例如Lyapunov-Schmidt约化。-构造3维分式Allen-Cahn方程的k端解，其零水平集是某个适当的极小超曲面上的正规图的并。