Integration by part on a surface
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Let $f$ be a smooth positive function from $mathbb{R}^3$ to $mathbb{R}$. Can we "simplify" the integral
$$
int_{mathbb{S}^2} dfrac{Delta f(x)}{f(x)} d sigma(x),
$$
where $Delta f$ is the three-dimensional Laplacian of $f$. For instance, what nice conditions on $f$ ensure that this integral have a sign?
integration multivariable-calculus
asked Dec 19 '18 at 22:27
dgontierdgontier
262
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add a comment |
$begingroup$
Let $f$ be a smooth positive function from $mathbb{R}^3$ to $mathbb{R}$. Can we "simplify" the integral
$$
int_{mathbb{S}^2} dfrac{Delta f(x)}{f(x)} d sigma(x),
$$
where $Delta f$ is the three-dimensional Laplacian of $f$. For instance, what nice conditions on $f$ ensure that this integral have a sign?
integration multivariable-calculus
asked Dec 19 '18 at 22:27
dgontierdgontier
262
$endgroup$
$begingroup$
What is the meaning of "this integral have a sign"?
$endgroup$
– manooooh
Dec 19 '18 at 23:27
$begingroup$
For, instance, if $f > 0$ and is subharmonic, then $f^{-1} Delta f > 0$, and the integral is positive. If $f < 0$ is superharmonic, then $f^{-1} Delta f < 0$, and the integral is negative. So I rephrase my question as follows: assume $f$ smooth positive and goes to zero at infinity. Is there $R > 0$ large enough so that $$ int_{R mathbb{S}} f^{-1} Delta f le 0.$$
$endgroup$
– dgontier
Dec 20 '18 at 8:45
add a comment |
$begingroup$
Let $f$ be a smooth positive function from $mathbb{R}^3$ to $mathbb{R}$. Can we "simplify" the integral
$$
int_{mathbb{S}^2} dfrac{Delta f(x)}{f(x)} d sigma(x),
$$
where $Delta f$ is the three-dimensional Laplacian of $f$. For instance, what nice conditions on $f$ ensure that this integral have a sign?
integration multivariable-calculus
asked Dec 19 '18 at 22:27
dgontierdgontier
262
$endgroup$
Let $f$ be a smooth positive function from $mathbb{R}^3$ to $mathbb{R}$. Can we "simplify" the integral
$$
int_{mathbb{S}^2} dfrac{Delta f(x)}{f(x)} d sigma(x),
$$
where $Delta f$ is the three-dimensional Laplacian of $f$. For instance, what nice conditions on $f$ ensure that this integral have a sign?
integration multivariable-calculus
integration multivariable-calculus
asked Dec 19 '18 at 22:27
dgontierdgontier
262
asked Dec 19 '18 at 22:27
dgontierdgontier
262
asked Dec 19 '18 at 22:27
dgontierdgontier
262
asked Dec 19 '18 at 22:27
dgontierdgontier
262
asked Dec 19 '18 at 22:27
dgontierdgontier
262
262
$begingroup$
What is the meaning of "this integral have a sign"?
$endgroup$
– manooooh
Dec 19 '18 at 23:27
$begingroup$
For, instance, if $f > 0$ and is subharmonic, then $f^{-1} Delta f > 0$, and the integral is positive. If $f < 0$ is superharmonic, then $f^{-1} Delta f < 0$, and the integral is negative. So I rephrase my question as follows: assume $f$ smooth positive and goes to zero at infinity. Is there $R > 0$ large enough so that $$ int_{R mathbb{S}} f^{-1} Delta f le 0.$$
$endgroup$
– dgontier
Dec 20 '18 at 8:45
add a comment |
$begingroup$
What is the meaning of "this integral have a sign"?
$endgroup$
– manooooh
Dec 19 '18 at 23:27
$begingroup$
For, instance, if $f > 0$ and is subharmonic, then $f^{-1} Delta f > 0$, and the integral is positive. If $f < 0$ is superharmonic, then $f^{-1} Delta f < 0$, and the integral is negative. So I rephrase my question as follows: assume $f$ smooth positive and goes to zero at infinity. Is there $R > 0$ large enough so that $$ int_{R mathbb{S}} f^{-1} Delta f le 0.$$
$endgroup$
– dgontier
Dec 20 '18 at 8:45
$begingroup$
What is the meaning of "this integral have a sign"?
$endgroup$
– manooooh
Dec 19 '18 at 23:27
$begingroup$
What is the meaning of "this integral have a sign"?
$endgroup$
– manooooh
Dec 19 '18 at 23:27
$begingroup$
For, instance, if $f > 0$ and is subharmonic, then $f^{-1} Delta f > 0$, and the integral is positive. If $f < 0$ is superharmonic, then $f^{-1} Delta f < 0$, and the integral is negative. So I rephrase my question as follows: assume $f$ smooth positive and goes to zero at infinity. Is there $R > 0$ large enough so that $$ int_{R mathbb{S}} f^{-1} Delta f le 0.$$
$endgroup$
– dgontier
Dec 20 '18 at 8:45
$begingroup$
For, instance, if $f > 0$ and is subharmonic, then $f^{-1} Delta f > 0$, and the integral is positive. If $f < 0$ is superharmonic, then $f^{-1} Delta f < 0$, and the integral is negative. So I rephrase my question as follows: assume $f$ smooth positive and goes to zero at infinity. Is there $R > 0$ large enough so that $$ int_{R mathbb{S}} f^{-1} Delta f le 0.$$
$endgroup$
– dgontier
Dec 20 '18 at 8:45
add a comment |
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$begingroup$
What is the meaning of "this integral have a sign"?
$endgroup$
– manooooh
Dec 19 '18 at 23:27
$begingroup$
For, instance, if $f > 0$ and is subharmonic, then $f^{-1} Delta f > 0$, and the integral is positive. If $f < 0$ is superharmonic, then $f^{-1} Delta f < 0$, and the integral is negative. So I rephrase my question as follows: assume $f$ smooth positive and goes to zero at infinity. Is there $R > 0$ large enough so that $$ int_{R mathbb{S}} f^{-1} Delta f le 0.$$
$endgroup$
– dgontier
Dec 20 '18 at 8:45