t*******t
发帖数: 1067 | 1 问题如下：
Prove the uniqueness of the Green's function for the boundary value-problem:
(p(x)y')' + q(x)y =-f(x) on (a,b)
a1*y(a) + a2*y'(a)=0
b1*y(b) + b2*y'(b) =0=
I think we just need to prove that the homogeneous problem has only trivial
solution:
(p(x)y')' + q(x)y =0 on (a,b)
a1*y(a) + a2*y'(a)=0
b1*y(b) + b2*y'(b) =0
However it seems not easy. Any comments and hints are greatly appreciated.
Thanks all |
b*********n
发帖数: 56 | 2 You need some kind of assumptions on p(x), and q(x). The usual ones are
p(x)<0, and q(x)>=0. Then the problem seems do-able by energy estimate
method. This is a bit of cheating .... I don't know if there is some
other elementary method. |
t*******t
发帖数: 1067 | 3 Thanks. There is no other assumption on p(x) and q(x), except for that
they are cont. and p(x) is differentialble. Will try again. Thanks
【在 b*********n 的大作中提到】
: You need some kind of assumptions on p(x), and q(x). The usual ones are
: p(x)<0, and q(x)>=0. Then the problem seems do-able by energy estimate
: method. This is a bit of cheating .... I don't know if there is some
: other elementary method.
|
b*********n
发帖数: 56 | 4 In addition to the assumptions on p(x) and q(x), I suspect that there should
also be some assumptions on the coefficient a1,a2,b1 and b2. Just keep
these in your mind when you try again. |
R*********r
发帖数: 1855 | 5 没有任何假设的话是证明不了的。
比如p(x)在(a，b)里有零点就可能变成奇异方程。 |
R*********r
发帖数: 1855 | 6 又比如
y''+y=0
y(0)=y(pi)=0
有非平凡解 sinx |
