Mixing
Open set in $C^1_0(overlineOmega)$
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Let $Omega$ be a bounded domain in $mathbbR^N$ with $partial Omega$ of class $C^2$ and denote $C^1_0(overlineOmega):=uin C^1(overlineOmega): u=0 text on partialOmega$. For $u,v:overlineOmegatomathbbR$ we write $ull v$ if $u(x)<v(x)$ for every $xin Omega$ and, if $u(x)=v(x)$ for some $xinpartial Omega$, then $fracpartial vpartial nu(x)<fracpartial upartial nu(x)$, where $nu=nu(x)$ denotes the unit outer normal to $Omega$ at $xinpartialOmega$.
I need a proof for the fact that the following set $S=uin C^1_0(overlineOmega): all ull b$ is open in $C^1_0(overlineOmega)$.
Above $a,bin C^1,0(overlineOmega)$ with $aleq 0$ on $partialOmega$ and
$bgeq 0$ on $partialOmega$.
Many thanks!
calculus real-analysis derivatives
asked Jul 19 at 18:30
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up vote
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Let $Omega$ be a bounded domain in $mathbbR^N$ with $partial Omega$ of class $C^2$ and denote $C^1_0(overlineOmega):=uin C^1(overlineOmega): u=0 text on partialOmega$. For $u,v:overlineOmegatomathbbR$ we write $ull v$ if $u(x)<v(x)$ for every $xin Omega$ and, if $u(x)=v(x)$ for some $xinpartial Omega$, then $fracpartial vpartial nu(x)<fracpartial upartial nu(x)$, where $nu=nu(x)$ denotes the unit outer normal to $Omega$ at $xinpartialOmega$.
I need a proof for the fact that the following set $S=uin C^1_0(overlineOmega): all ull b$ is open in $C^1_0(overlineOmega)$.
Above $a,bin C^1,0(overlineOmega)$ with $aleq 0$ on $partialOmega$ and
$bgeq 0$ on $partialOmega$.
Many thanks!
calculus real-analysis derivatives
asked Jul 19 at 18:30
Silverlight
11
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $Omega$ be a bounded domain in $mathbbR^N$ with $partial Omega$ of class $C^2$ and denote $C^1_0(overlineOmega):=uin C^1(overlineOmega): u=0 text on partialOmega$. For $u,v:overlineOmegatomathbbR$ we write $ull v$ if $u(x)<v(x)$ for every $xin Omega$ and, if $u(x)=v(x)$ for some $xinpartial Omega$, then $fracpartial vpartial nu(x)<fracpartial upartial nu(x)$, where $nu=nu(x)$ denotes the unit outer normal to $Omega$ at $xinpartialOmega$.
I need a proof for the fact that the following set $S=uin C^1_0(overlineOmega): all ull b$ is open in $C^1_0(overlineOmega)$.
Above $a,bin C^1,0(overlineOmega)$ with $aleq 0$ on $partialOmega$ and
$bgeq 0$ on $partialOmega$.
Many thanks!
calculus real-analysis derivatives
asked Jul 19 at 18:30
Silverlight
11
Let $Omega$ be a bounded domain in $mathbbR^N$ with $partial Omega$ of class $C^2$ and denote $C^1_0(overlineOmega):=uin C^1(overlineOmega): u=0 text on partialOmega$. For $u,v:overlineOmegatomathbbR$ we write $ull v$ if $u(x)<v(x)$ for every $xin Omega$ and, if $u(x)=v(x)$ for some $xinpartial Omega$, then $fracpartial vpartial nu(x)<fracpartial upartial nu(x)$, where $nu=nu(x)$ denotes the unit outer normal to $Omega$ at $xinpartialOmega$.
I need a proof for the fact that the following set $S=uin C^1_0(overlineOmega): all ull b$ is open in $C^1_0(overlineOmega)$.
Above $a,bin C^1,0(overlineOmega)$ with $aleq 0$ on $partialOmega$ and
$bgeq 0$ on $partialOmega$.
Many thanks!
calculus real-analysis derivatives
asked Jul 19 at 18:30
Silverlight
11
asked Jul 19 at 18:30
Silverlight
11
asked Jul 19 at 18:30
Silverlight
11
asked Jul 19 at 18:30
Silverlight
11
11
add a comment |Â
add a comment |Â
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