wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> medium >> Sobolev space 
        
(Message started by: MonicaMath on Apr 13^th, 2010, 6:25pm)

Title: Sobolev space
Post by MonicaMath on Apr 13^th, 2010, 6:25pm

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

Title: Re: Sobolev space
Post by Obob on Apr 13^th, 2010, 7:46pm

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.

Title: Re: Sobolev space
Post by MonicaMath on Apr 13^th, 2010, 8:27pm

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 ??

Title: Re: Sobolev space
Post by Obob on Apr 14^th, 2010, 9:02am

The function

[hide]f(x,y) = exp(-(y^4+1)x^2)/sqrt(y^4+1)[/hide]

should do the trick. It's instructive to try and construct your own example, though.

Title: Re: Sobolev space
Post by MonicaMath on Apr 15^th, 2010, 9:23am

you mean
[hide]f(x,y) = ( exp(-(y^4+1)x^2) ) / sqrt(y^4+1)[/hide]

Title: Re: Sobolev space
Post by Obob on Apr 15^th, 2010, 9:50am

...that's exactly what I wrote.

Title: Re: Sobolev space
Post by MonicaMath on Apr 15^th, 2010, 10:28am

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... 8)

thank you soo much

Title: Re: Sobolev space
Post by Obob on Apr 15^th, 2010, 10:38am

Yes, for instance exp(x)/x means (e^x)/x, not e^(x/x), just as sin(x)/x is not sin(x/x).

Title: Re: Sobolev space
Post by MonicaMath on Apr 17^th, 2010, 3:49pm

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

Title: Re: Sobolev space
Post by Obob on Apr 17^th, 2010, 4:49pm

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.

Title: Re: Sobolev space
Post by MonicaMath on Apr 17^th, 2010, 8:53pm

yes, you are right.... I made a mistake in my calculations...

thanks a lot... your brilliant ;)