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Mathematics版 - 带surface energy问题的existence of minimizer
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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
2
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.


【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
4
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.

【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
6
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
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
7
Thanks 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)
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