boards

Mathematics版 - 带surface energy问题的existence of minimizer

HOw to numerically integrate noisy data请教minimization的问题
help on piecewise linear functions请教piecewise linear fitting

number of inflexion points for a Bezier curve一个问题，拓扑高手帮忙看一下

 1 (共1页)
 b******v发帖数: 1493 1假如对于u \in H^1(D, R^2)，定义如下能量泛函 F = \int_D |\nabla u|^2 dx + \int_S f(u) ds 其中D=[0,1]^2, S是D的边界。而f总是非负。 假如不添加其他限制条件。这样的问题是否一定有 能量极小解(existence of minimizer)？ 在哪里可以查到一般的理论？ 多谢！ n***p发帖数: 7668 2Without any restriction on $u$, the minimizer of $F$ will be a constant function $u=c$ where $c$ is the global minimizer of $f$. The problem can be more interesting with some restrictions. 【在 b******v 的大作中提到】: 假如对于u \in H^1(D, R^2)，定义如下能量泛函: F = \int_D |\nabla u|^2 dx + \int_S f(u) ds: 其中D=[0,1]^2, S是D的边界。而f总是非负。: 假如不添加其他限制条件。这样的问题是否一定有: 能量极小解(existence of minimizer)？: 在哪里可以查到一般的理论？: 多谢！ b******v发帖数: 1493 3但是f(u)是个分段定义的函数，在边界不同部分的minimizer不是同一个常数。 【在 n***p 的大作中提到】: Without any restriction on $u$, the minimizer of $F$: will be a constant function $u=c$ where $c$ is the global: minimizer of $f$.: The problem can be more interesting with some restrictions.: n***p发帖数: 7668 4OK, if $f$ depends on the position on the boundary, then in fact it should be written as $f(x,u)$. It becomes a slightly more interesting variational problem. Suppose the minimizer u exists, which is not always true, then it satisfies -\Delta u = 0 in the domain 2 \nabla u\cdot n + f_u (x,u) = 0 on the boundary. Here \nabla u\cdot n is the normal derivative of u. 【在 b******v 的大作中提到】: 但是f(u)是个分段定义的函数，在边界不同部分的minimizer不是同一个常数。 b******v发帖数: 1493 5多谢回复，我之前的记号确实有问题。 如果这种带surface energy问题解的存在性不容易确定的话，和我原问题相关的一个问 题是，在D=[0, 1]^2区域上考虑Ginzburg-Landau的能量泛函，假如给的边条件是u2=0, 而u1在上下边非负，在左右边非正，对应的energy minimizer是否存在，如果存在的 话，它是什么样子的？ 其中Ginzburg-Landau能量泛函的表达式是 E=\int_D |\nabla u|^2 + 1/\varepsilon^2 (1-|u|^2)^2 我的考虑是任给一个边界上符合边条件的Lipshitz continuous的函数g, 根据u=(0, g) 都能对应一个minimized energy, 但是这样的函数g很多，不知道该怎样去找那个最优 的。 【在 n***p 的大作中提到】: OK, if $f$ depends on the position on the boundary, then: in fact it should be written as $f(x,u)$. It becomes a slightly: more interesting variational problem. Suppose the minimizer u: exists, which is not always true, then it satisfies: -\Delta u = 0 in the domain: 2 \nabla u\cdot n + f_u (x,u) = 0 on the boundary.: Here \nabla u\cdot n is the normal derivative of u. n***p发帖数: 7668 6Hope this problem is not your homework. I thought about it and give the following predictions. Nothing is guaranteed to be true though. Apparently what you need to do is to minimize E over the admissible set S={u=(u1,u2)\in H^1(D, R^2): u2=0 on the boundary, u1 u1在上下边非负，在左右边非正 }. Now S is a closed, convex subset of H^1(D, R^2). Using relatively standard arguments in calculus of variations, There exists a minimizer for E on S. But since E is not a convex functional, the uniqueness is more complicated and generally not true. Since E is the Ginzburg-Landau functional, the minimizer u should be piecewise constant with transition layer of width \varepsilon (or 2*\varepsilon?). Those constant pieces should satisfy |u|=1. Now look at the boundary. Since u2=0, u1 will be approximately +1 or -1. I don't see any reason that prevents u2=0 over the whole domain D. So I expect the minimizers u=(u1,0). Next, since E is approximately the total length of the transition layer, and the infimum length of the transition layer is 2, I think there are at least two choices. The first is u1 = 1 in most part of the domain D and u1 \approx -1 in two curved regions on the left and right sides. The second choice apparently will be to rotate the above shape by 90 degrees and let u1=-1 in it. Consequently in the two curved regions on the top and bottom sides u1\approx 1. 0, g) 【在 b******v 的大作中提到】: 多谢回复，我之前的记号确实有问题。: 如果这种带surface energy问题解的存在性不容易确定的话，和我原问题相关的一个问: 题是，在D=[0, 1]^2区域上考虑Ginzburg-Landau的能量泛函，假如给的边条件是u2=0,: 而u1在上下边非负，在左右边非正，对应的energy minimizer是否存在，如果存在的: 话，它是什么样子的？: 其中Ginzburg-Landau能量泛函的表达式是: E=\int_D |\nabla u|^2 + 1/\varepsilon^2 (1-|u|^2)^2: 我的考虑是任给一个边界上符合边条件的Lipshitz continuous的函数g, 根据u=(0, g): 都能对应一个minimized energy, 但是这样的函数g很多，不知道该怎样去找那个最优: 的。 b******v发帖数: 1493 7Thanks a lot! 【在 n***p 的大作中提到】: Hope this problem is not your homework. I thought: about it and give the following predictions. Nothing: is guaranteed to be true though.: Apparently what you need to do is to minimize E over: the admissible set: S={u=(u1,u2)\in H^1(D, R^2): u2=0 on the boundary,: u1 u1在上下边非负，在左右边非正 }.: Now S is a closed, convex subset of H^1(D, R^2). Using: relatively standard arguments in calculus of variations,: There exists a minimizer for E on S. But since E is not
 1 (共1页)