Change of Variable formula for a non-differentiable mapping.
Let $Omega subset Bbb R^n$. For a diffeomorphism (or merely a differentiable bijection) $varphi:Omega to varphi(Omega)$, we have the formula
$$
int_{Omega} fcircvarphi^{-1}(x), dx = int_{varphi^{-1}(Omega)} f(y)|Dvarphi(y)| ,dy.
$$
How much can we generalize the class in which $varphi$ is allowed to lie in? Is it enough that we have, says, a bijection $varphiin W_{text{loc}}^{1,infty}(Omega ;Bbb R^n)$ or even $varphiin W_{text{loc}}^{1,1}(Omega ;Bbb R^n)$?
How much does the result depends on the domain $Omega$? Is there a big difference between a compact and an open domain? Does the regularity of the boundary $partial Omega$ play any role?
I'd also appreciate if you have a good reference to this kind of result so that I can read further into this interesting issue. Happy New Year to all of you.
real-analysis measure-theory differential-geometry reference-request pde
asked yesterday
BigbearZzz
7,32621648
add a comment |
Let $Omega subset Bbb R^n$. For a diffeomorphism (or merely a differentiable bijection) $varphi:Omega to varphi(Omega)$, we have the formula
$$
int_{Omega} fcircvarphi^{-1}(x), dx = int_{varphi^{-1}(Omega)} f(y)|Dvarphi(y)| ,dy.
$$
How much can we generalize the class in which $varphi$ is allowed to lie in? Is it enough that we have, says, a bijection $varphiin W_{text{loc}}^{1,infty}(Omega ;Bbb R^n)$ or even $varphiin W_{text{loc}}^{1,1}(Omega ;Bbb R^n)$?
How much does the result depends on the domain $Omega$? Is there a big difference between a compact and an open domain? Does the regularity of the boundary $partial Omega$ play any role?
I'd also appreciate if you have a good reference to this kind of result so that I can read further into this interesting issue. Happy New Year to all of you.
real-analysis measure-theory differential-geometry reference-request pde
asked yesterday
BigbearZzz
7,32621648
add a comment |
Let $Omega subset Bbb R^n$. For a diffeomorphism (or merely a differentiable bijection) $varphi:Omega to varphi(Omega)$, we have the formula
$$
int_{Omega} fcircvarphi^{-1}(x), dx = int_{varphi^{-1}(Omega)} f(y)|Dvarphi(y)| ,dy.
$$
How much can we generalize the class in which $varphi$ is allowed to lie in? Is it enough that we have, says, a bijection $varphiin W_{text{loc}}^{1,infty}(Omega ;Bbb R^n)$ or even $varphiin W_{text{loc}}^{1,1}(Omega ;Bbb R^n)$?
How much does the result depends on the domain $Omega$? Is there a big difference between a compact and an open domain? Does the regularity of the boundary $partial Omega$ play any role?
I'd also appreciate if you have a good reference to this kind of result so that I can read further into this interesting issue. Happy New Year to all of you.
real-analysis measure-theory differential-geometry reference-request pde
asked yesterday
BigbearZzz
7,32621648
Let $Omega subset Bbb R^n$. For a diffeomorphism (or merely a differentiable bijection) $varphi:Omega to varphi(Omega)$, we have the formula
$$
int_{Omega} fcircvarphi^{-1}(x), dx = int_{varphi^{-1}(Omega)} f(y)|Dvarphi(y)| ,dy.
$$
How much can we generalize the class in which $varphi$ is allowed to lie in? Is it enough that we have, says, a bijection $varphiin W_{text{loc}}^{1,infty}(Omega ;Bbb R^n)$ or even $varphiin W_{text{loc}}^{1,1}(Omega ;Bbb R^n)$?
How much does the result depends on the domain $Omega$? Is there a big difference between a compact and an open domain? Does the regularity of the boundary $partial Omega$ play any role?
I'd also appreciate if you have a good reference to this kind of result so that I can read further into this interesting issue. Happy New Year to all of you.
real-analysis measure-theory differential-geometry reference-request pde
real-analysis measure-theory differential-geometry reference-request pde
asked yesterday
BigbearZzz
7,32621648
asked yesterday
BigbearZzz
7,32621648
edited yesterday
asked yesterday
BigbearZzz
7,32621648
asked yesterday
BigbearZzz
7,32621648
asked yesterday
BigbearZzz
7,32621648
7,32621648
add a comment |
add a comment |
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