Abstract: We present some numerical results for the solution of non-convex variational problems. In general, the problems of interest do not attain a minimum energy. Functions that generate a minimizing sequence of energies develop infinitely fine oscillations, and it is believed that these oscillations model the fine scale structures that are ubiquitously observed in metallurgy, ferromagnetism, etc. Direct simulation of these variational problems on discrete meshes is plagued with practical problems. We present a simple 1-D example that exhibits problems typical of those encountered with such an approach. Many of these problems can be traced to the fundamental problem that the variational problem doesn't have a solution. An alternative is to consider the generalized solutions of L. C. Young. We present some numerical experiments using this algorithm for variational problems that involve vector valued functions. !17
展开▼
机译：摘要：我们提供了一些数值结果来解决非凸变分问题。通常，所关注的问题没有达到最小的能量。产生最小化能量序列的函数会产生无限精细的振荡，并且据信，这些振荡模拟了冶金，铁磁性等中普遍存在的精细尺度结构。在离散网格上直接模拟这些变体问题受到实际问题的困扰。 。我们提供了一个简单的一维示例，该示例展示了使用这种方法遇到的典型问题。这些问题中的许多问题都可以追溯到根本问题，即变分问题没有解决方案。另一种选择是考虑L. C. Young的广义解。我们针对使用矢量值函数的变分问题，提出了使用此算法的一些数值实验。 ！17
展开▼
