Show $frac{partial}{partial t} int_{f(x) < t} varphi(x) text{d}x in C_0^infty(mathbb{R}_t )$ if $varphi in...












3












$begingroup$


Let $X$ be an open subset of $mathbb{R}^n$. Suppose that $f in C^infty(X)$ is real-valued, with the property that $nabla f(x) neq 0$ for all $x in X$. For any $varphi in C_0^infty(X)$, and $t in mathbb{R}$, define
$$varphi_f(t) = frac{partial}{partial t} int_{f(x) < t} varphi(x) text{d}x.$$




I would like to show that $varphi_f in C_0^infty(mathbb{R})$.




This is a claim made in Chapter 7 of Friedlander's book on distributions. I am having trouble following his justification.



My attempt at a solution:



My idea is to first show that $frac{partial}{partial t} int_{f(x) < t} varphi(x) text{d}x$ is well-defined (i.e., that the derivative exists) and find it's value, with the hope this will give a clue about how to calculate the higher-order derivatives. For $h > 0$:



$$frac{1}{h} left(int_{f(x) < t} varphi(x) text{d}x - int_{f(x) < t + h} varphi(x) text{d}x right) = frac{1}{h} int mathbf{1}_{t le f < t + h }(x) varphi(x) text{d}x.$$



But it's not clear to me that the right side converges to anything as $h to 0^+$. I think maybe a change of variables (made possible by the implicit function Theorem?) is in order to make the dependence on $f$ more explicit.



Hints or solutions are greatly appreciated!






real-analysis pde partial-derivative







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  • 1




    $begingroup$
    If you apply the coarea formula with $g = varphi / |nabla f|$, you can at least find a nice formula for $varphi_f$.
    $endgroup$
    – Michał Miśkiewicz
    Aug 7 '17 at 19:57
















3












$begingroup$


Let $X$ be an open subset of $mathbb{R}^n$. Suppose that $f in C^infty(X)$ is real-valued, with the property that $nabla f(x) neq 0$ for all $x in X$. For any $varphi in C_0^infty(X)$, and $t in mathbb{R}$, define
$$varphi_f(t) = frac{partial}{partial t} int_{f(x) < t} varphi(x) text{d}x.$$




I would like to show that $varphi_f in C_0^infty(mathbb{R})$.




This is a claim made in Chapter 7 of Friedlander's book on distributions. I am having trouble following his justification.



My attempt at a solution:



My idea is to first show that $frac{partial}{partial t} int_{f(x) < t} varphi(x) text{d}x$ is well-defined (i.e., that the derivative exists) and find it's value, with the hope this will give a clue about how to calculate the higher-order derivatives. For $h > 0$:



$$frac{1}{h} left(int_{f(x) < t} varphi(x) text{d}x - int_{f(x) < t + h} varphi(x) text{d}x right) = frac{1}{h} int mathbf{1}_{t le f < t + h }(x) varphi(x) text{d}x.$$



But it's not clear to me that the right side converges to anything as $h to 0^+$. I think maybe a change of variables (made possible by the implicit function Theorem?) is in order to make the dependence on $f$ more explicit.



Hints or solutions are greatly appreciated!






real-analysis pde partial-derivative







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    If you apply the coarea formula with $g = varphi / |nabla f|$, you can at least find a nice formula for $varphi_f$.
    $endgroup$
    – Michał Miśkiewicz
    Aug 7 '17 at 19:57














3












3








3


1



$begingroup$


Let $X$ be an open subset of $mathbb{R}^n$. Suppose that $f in C^infty(X)$ is real-valued, with the property that $nabla f(x) neq 0$ for all $x in X$. For any $varphi in C_0^infty(X)$, and $t in mathbb{R}$, define
$$varphi_f(t) = frac{partial}{partial t} int_{f(x) < t} varphi(x) text{d}x.$$




I would like to show that $varphi_f in C_0^infty(mathbb{R})$.




This is a claim made in Chapter 7 of Friedlander's book on distributions. I am having trouble following his justification.



My attempt at a solution:



My idea is to first show that $frac{partial}{partial t} int_{f(x) < t} varphi(x) text{d}x$ is well-defined (i.e., that the derivative exists) and find it's value, with the hope this will give a clue about how to calculate the higher-order derivatives. For $h > 0$:



$$frac{1}{h} left(int_{f(x) < t} varphi(x) text{d}x - int_{f(x) < t + h} varphi(x) text{d}x right) = frac{1}{h} int mathbf{1}_{t le f < t + h }(x) varphi(x) text{d}x.$$



But it's not clear to me that the right side converges to anything as $h to 0^+$. I think maybe a change of variables (made possible by the implicit function Theorem?) is in order to make the dependence on $f$ more explicit.



Hints or solutions are greatly appreciated!






real-analysis pde partial-derivative







share|cite|improve this question











$endgroup$




Let $X$ be an open subset of $mathbb{R}^n$. Suppose that $f in C^infty(X)$ is real-valued, with the property that $nabla f(x) neq 0$ for all $x in X$. For any $varphi in C_0^infty(X)$, and $t in mathbb{R}$, define
$$varphi_f(t) = frac{partial}{partial t} int_{f(x) < t} varphi(x) text{d}x.$$




I would like to show that $varphi_f in C_0^infty(mathbb{R})$.




This is a claim made in Chapter 7 of Friedlander's book on distributions. I am having trouble following his justification.



My attempt at a solution:



My idea is to first show that $frac{partial}{partial t} int_{f(x) < t} varphi(x) text{d}x$ is well-defined (i.e., that the derivative exists) and find it's value, with the hope this will give a clue about how to calculate the higher-order derivatives. For $h > 0$:



$$frac{1}{h} left(int_{f(x) < t} varphi(x) text{d}x - int_{f(x) < t + h} varphi(x) text{d}x right) = frac{1}{h} int mathbf{1}_{t le f < t + h }(x) varphi(x) text{d}x.$$



But it's not clear to me that the right side converges to anything as $h to 0^+$. I think maybe a change of variables (made possible by the implicit function Theorem?) is in order to make the dependence on $f$ more explicit.



Hints or solutions are greatly appreciated!






real-analysis pde partial-derivative




real-analysis pde partial-derivative






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 20 '18 at 12:11









Falrach

1,664223




1,664223










asked Aug 7 '17 at 18:11









JZShapiroJZShapiro

1,99511125




1,99511125








  • 1




    $begingroup$
    If you apply the coarea formula with $g = varphi / |nabla f|$, you can at least find a nice formula for $varphi_f$.
    $endgroup$
    – Michał Miśkiewicz
    Aug 7 '17 at 19:57














  • 1




    $begingroup$
    If you apply the coarea formula with $g = varphi / |nabla f|$, you can at least find a nice formula for $varphi_f$.
    $endgroup$
    – Michał Miśkiewicz
    Aug 7 '17 at 19:57








1




1




$begingroup$
If you apply the coarea formula with $g = varphi / |nabla f|$, you can at least find a nice formula for $varphi_f$.
$endgroup$
– Michał Miśkiewicz
Aug 7 '17 at 19:57




$begingroup$
If you apply the coarea formula with $g = varphi / |nabla f|$, you can at least find a nice formula for $varphi_f$.
$endgroup$
– Michał Miśkiewicz
Aug 7 '17 at 19:57










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