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Volume of critical points decrease under symmetric decreasing rearrangements?
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If, $u: mathbbR^n to mathbbR$ be a non-negative test function, i.e., $u in mathcalD(mathbbR^n)$ and $u ge 0$, then does it follow that,
$$mathcalH^n(s < u le t cap nabla u = 0) ge mathcalH^n(s < u^ast le t cap nabla u^ast = 0), text for 0 < s < t tag1$$
where, $u^ast$ denotes the symmetric decreasing rearrangement of $u$?
Context: Using the above fact I wish to conclude the inequality, $$int_u = t frac1,dsigma le int_u^ast = t frac1,dsigma tag2$$ for almost every regular value $t$ of $u$. Since, we are supposed to have equality in $(2)$, both quantities equal to the derivative of the distribution function of $u$, I am trying to find a direct argument for $(1)$ that does not use $(2)$.
Any help is appreciated. Thanks!
measure-theory inequality geometric-measure-theory
asked Jul 21 at 10:05
r9m
13k23970
add a comment |Â
up vote
2
down vote
favorite
If, $u: mathbbR^n to mathbbR$ be a non-negative test function, i.e., $u in mathcalD(mathbbR^n)$ and $u ge 0$, then does it follow that,
$$mathcalH^n(s < u le t cap nabla u = 0) ge mathcalH^n(s < u^ast le t cap nabla u^ast = 0), text for 0 < s < t tag1$$
where, $u^ast$ denotes the symmetric decreasing rearrangement of $u$?
Context: Using the above fact I wish to conclude the inequality, $$int_u = t frac1,dsigma le int_u^ast = t frac1,dsigma tag2$$ for almost every regular value $t$ of $u$. Since, we are supposed to have equality in $(2)$, both quantities equal to the derivative of the distribution function of $u$, I am trying to find a direct argument for $(1)$ that does not use $(2)$.
Any help is appreciated. Thanks!
measure-theory inequality geometric-measure-theory
asked Jul 21 at 10:05
r9m
13k23970
1
mathoverflow.net/questions/306334/…
– Michael Greinecker♦
Jul 21 at 10:10
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
If, $u: mathbbR^n to mathbbR$ be a non-negative test function, i.e., $u in mathcalD(mathbbR^n)$ and $u ge 0$, then does it follow that,
$$mathcalH^n(s < u le t cap nabla u = 0) ge mathcalH^n(s < u^ast le t cap nabla u^ast = 0), text for 0 < s < t tag1$$
where, $u^ast$ denotes the symmetric decreasing rearrangement of $u$?
Context: Using the above fact I wish to conclude the inequality, $$int_u = t frac1,dsigma le int_u^ast = t frac1,dsigma tag2$$ for almost every regular value $t$ of $u$. Since, we are supposed to have equality in $(2)$, both quantities equal to the derivative of the distribution function of $u$, I am trying to find a direct argument for $(1)$ that does not use $(2)$.
Any help is appreciated. Thanks!
measure-theory inequality geometric-measure-theory
asked Jul 21 at 10:05
r9m
13k23970
If, $u: mathbbR^n to mathbbR$ be a non-negative test function, i.e., $u in mathcalD(mathbbR^n)$ and $u ge 0$, then does it follow that,
$$mathcalH^n(s < u le t cap nabla u = 0) ge mathcalH^n(s < u^ast le t cap nabla u^ast = 0), text for 0 < s < t tag1$$
where, $u^ast$ denotes the symmetric decreasing rearrangement of $u$?
Context: Using the above fact I wish to conclude the inequality, $$int_u = t frac1,dsigma le int_u^ast = t frac1,dsigma tag2$$ for almost every regular value $t$ of $u$. Since, we are supposed to have equality in $(2)$, both quantities equal to the derivative of the distribution function of $u$, I am trying to find a direct argument for $(1)$ that does not use $(2)$.
Any help is appreciated. Thanks!
measure-theory inequality geometric-measure-theory
asked Jul 21 at 10:05
r9m
13k23970
asked Jul 21 at 10:05
r9m
13k23970
asked Jul 21 at 10:05
r9m
13k23970
asked Jul 21 at 10:05
r9m
13k23970
13k23970
1
mathoverflow.net/questions/306334/…
– Michael Greinecker♦
Jul 21 at 10:10
add a comment |Â
1
mathoverflow.net/questions/306334/…
– Michael Greinecker♦
Jul 21 at 10:10
1
1
mathoverflow.net/questions/306334/…
– Michael Greinecker♦
Jul 21 at 10:10
mathoverflow.net/questions/306334/…
– Michael Greinecker♦
Jul 21 at 10:10
add a comment |Â
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1
mathoverflow.net/questions/306334/…
– Michael Greinecker♦
Jul 21 at 10:10