Author |
Topic: Sobolev space (Read 834 times) |
|
MonicaMath
Newbie
Gender: 
Posts: 43
|
 |
Sobolev space
« on: Apr 13th, 2010, 6:25pm » |
 Quote  Modify
|
could anyone help me please in proving:
(1) If f is in H^1(R2) then df/dr is in H^0(R2),
but:
(2) If f is in H^1(R2) then its not necesary that
df/d-theta is in H^0(R2)
(3) is there a counterexample for the case in (2) ?
/ where dr,d-theta are the polar coordinate partial derivatives of f, and H^s is the Sobolev space on R^2
/
I almost did part (1), but still have no idea about part (2) and (3) ? where is the problem in applying
df/d-thete !!
thanx
|
|
 IP Logged |
|
|
|
Obob
Senior Riddler
Gender: 
Posts: 489
|
|
Re: Sobolev space
« Reply #1 on: Apr 13th, 2010, 7:46pm » |
Quote Modify
|
Replacing theta with t for brevity, recall the formulas
d/dr = cos t (d/dx) + sin t (d/dy)
d/dt = -y (d/dx) + x (d/dy)
If f is in H^1, then by definition df/dx and df/dy are in H^0. But if g is in H^0, then g cos t and g sin t are in H^0. So df/dr is in H^0.
On the other hand, the factors x and y mess things up in the second case: we are given only that df/dx and df/dy are square-integrable, from which it will not usually follow that -y(df/dx) and x(df/dy) are square integrable.
It shouldn't be too difficult to come up with a counterexample, but one is eluding me at the moment.
|
|
IP Logged |
|
|
|
MonicaMath
Newbie
Gender:
Posts: 43
|
|
Re: Sobolev space
« Reply #2 on: Apr 13th, 2010, 8:27pm » |
Quote Modify
|
so we are now looking for a function f in H1 with
y(df/dx) or x(df/dy) not in H^0 !!
 how we could make it ??
|
|
IP Logged |
|
|
|
Obob
Senior Riddler
Gender:
Posts: 489
|
|
Re: Sobolev space
« Reply #3 on: Apr 14th, 2010, 9:02am » |
Quote Modify
|
The function
f(x,y) = exp(-(y^4+1)x^2)/sqrt(y^4+1)
should do the trick. It's instructive to try and construct your own example, though.
|
| « Last Edit: Apr 14th, 2010, 9:03am by Obob » |
IP Logged |
|
|
|
MonicaMath
Newbie
Gender:
Posts: 43
|
|
Re: Sobolev space
« Reply #4 on: Apr 15th, 2010, 9:23am » |
Quote Modify
|
you mean
f(x,y) = ( exp(-(y^4+1)x^2) ) / sqrt(y^4+1)
|
| « Last Edit: Apr 15th, 2010, 9:24am by MonicaMath » |
IP Logged |
|
|
|
Obob
Senior Riddler
Gender:
Posts: 489
|
|
Re: Sobolev space
« Reply #5 on: Apr 15th, 2010, 9:50am » |
Quote Modify
|
...that's exactly what I wrote.
|
|
IP Logged |
|
|
|
MonicaMath
Newbie
Gender:
Posts: 43
|
|
Re: Sobolev space
« Reply #6 on: Apr 15th, 2010, 10:28am » |
Quote Modify
|
so you mean all of the term ( exp(-(y^4+1)x^2) ) is divided by sqrt(y^4+1) not the exponent of the exponential... 
thank you soo much
|
| « Last Edit: Apr 15th, 2010, 10:29am by MonicaMath » |
IP Logged |
|
|
|
Obob
Senior Riddler
Gender:
Posts: 489
|
|
Re: Sobolev space
« Reply #7 on: Apr 15th, 2010, 10:38am » |
Quote Modify
|
Yes, for instance exp(x)/x means (e^x)/x, not e^(x/x), just as sin(x)/x is not sin(x/x).
|
|
IP Logged |
|
|
|
MonicaMath
Newbie
Gender:
Posts: 43
|
|
Re: Sobolev space
« Reply #8 on: Apr 17th, 2010, 3:49pm » |
Quote Modify
|
Hi again,
I just calculated the H0-norm (the L2-norm) of df/dx, df/dy for the above function f(x,y) and the result was finite, but also the H0-norm of -y.df/dx and x.df/dy are finite? did I miss something?! (I used some software in my calculations)
thanks
|
| « Last Edit: Apr 17th, 2010, 3:49pm by MonicaMath » |
IP Logged |
|
|
|
Obob
Senior Riddler
Gender:
Posts: 489
|
|
Re: Sobolev space
« Reply #9 on: Apr 17th, 2010, 4:49pm » |
Quote Modify
|
The L2-norms of f, df/dx, df/dy, and x.df/dy should all be finite, but the L2-norm of -y.df/dx is infinite. In fact the integral of (-y.df/dx)^2 with respect to x as x goes from -infinity to infinity should be sqrt(pi/2)y^2/(sqrt(1+y^4)); the integral of this w.r.t. y is clearly infinite.
|
|
IP Logged |
|
|
|
MonicaMath
Newbie
Gender:
Posts: 43
|
|
Re: Sobolev space
« Reply #10 on: Apr 17th, 2010, 8:53pm » |
Quote Modify
|
yes, you are right.... I made a mistake in my calculations...
thanks a lot... your brilliant 
|
|
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print