Maximum Principle for Elliptic Problems
I am reading Renardy - "An Introduction to PDEs" where I am at the maximum principle for elliptic equations. The strong maximum principle states:
Assume $Luge0 (Lule0)$ in $Omega$ and assume that $u$ is non constant. If $c=0$ then $u$ does not achieve its maximum (minimum) in the interior of $Omega$. If $cle 0$, $u$ cannot achieve a non-negative maximum (non-positive minimum) in the interior. Regardless of the sign of $c$, $u$ cannot be zero at an interior maximum (minimum).
The operator $L$ is elliptic and $Omega$ is the domain. The exact assumptions on the domain and operator will not be given now, but are assumed to hold for my question.
I now want to try it on the equation
$begin{align}
Delta u(x,y)&=u(x,y)+sin(xy), &(x,y)in D_0(1)\
u(x,y)&=1, &(x,y)inpartial D_0(1)
end{align}$
where $D_0(1)$ is the unit disk centered around the origin.
I want to give an estimate of $sup|u|$ using the strong maximum principle (and possibly the weak one, too).
My attempt:
The requirements for the theorem are satisfied, as $Omega = D_0(1)$ and $L$ is elliptic. (But this shall not bother us now)
Since it is also required that $u$ is sufficiently smooth (has to be $mathcal{C}^2$ in the interior) the maximum and supremum coincide.
I obviously have to use the part of the theorem where $cle0$. But here already the first problem occurs, I do not have $Lule 0$ or $Luge 0$.
pde
asked 17 hours ago
EpsilonDelta
6281615
add a comment |
I am reading Renardy - "An Introduction to PDEs" where I am at the maximum principle for elliptic equations. The strong maximum principle states:
Assume $Luge0 (Lule0)$ in $Omega$ and assume that $u$ is non constant. If $c=0$ then $u$ does not achieve its maximum (minimum) in the interior of $Omega$. If $cle 0$, $u$ cannot achieve a non-negative maximum (non-positive minimum) in the interior. Regardless of the sign of $c$, $u$ cannot be zero at an interior maximum (minimum).
The operator $L$ is elliptic and $Omega$ is the domain. The exact assumptions on the domain and operator will not be given now, but are assumed to hold for my question.
I now want to try it on the equation
$begin{align}
Delta u(x,y)&=u(x,y)+sin(xy), &(x,y)in D_0(1)\
u(x,y)&=1, &(x,y)inpartial D_0(1)
end{align}$
where $D_0(1)$ is the unit disk centered around the origin.
I want to give an estimate of $sup|u|$ using the strong maximum principle (and possibly the weak one, too).
My attempt:
The requirements for the theorem are satisfied, as $Omega = D_0(1)$ and $L$ is elliptic. (But this shall not bother us now)
Since it is also required that $u$ is sufficiently smooth (has to be $mathcal{C}^2$ in the interior) the maximum and supremum coincide.
I obviously have to use the part of the theorem where $cle0$. But here already the first problem occurs, I do not have $Lule 0$ or $Luge 0$.
pde
asked 17 hours ago
EpsilonDelta
6281615
add a comment |
I am reading Renardy - "An Introduction to PDEs" where I am at the maximum principle for elliptic equations. The strong maximum principle states:
Assume $Luge0 (Lule0)$ in $Omega$ and assume that $u$ is non constant. If $c=0$ then $u$ does not achieve its maximum (minimum) in the interior of $Omega$. If $cle 0$, $u$ cannot achieve a non-negative maximum (non-positive minimum) in the interior. Regardless of the sign of $c$, $u$ cannot be zero at an interior maximum (minimum).
The operator $L$ is elliptic and $Omega$ is the domain. The exact assumptions on the domain and operator will not be given now, but are assumed to hold for my question.
I now want to try it on the equation
$begin{align}
Delta u(x,y)&=u(x,y)+sin(xy), &(x,y)in D_0(1)\
u(x,y)&=1, &(x,y)inpartial D_0(1)
end{align}$
where $D_0(1)$ is the unit disk centered around the origin.
I want to give an estimate of $sup|u|$ using the strong maximum principle (and possibly the weak one, too).
My attempt:
The requirements for the theorem are satisfied, as $Omega = D_0(1)$ and $L$ is elliptic. (But this shall not bother us now)
Since it is also required that $u$ is sufficiently smooth (has to be $mathcal{C}^2$ in the interior) the maximum and supremum coincide.
I obviously have to use the part of the theorem where $cle0$. But here already the first problem occurs, I do not have $Lule 0$ or $Luge 0$.
pde
asked 17 hours ago
EpsilonDelta
6281615
I am reading Renardy - "An Introduction to PDEs" where I am at the maximum principle for elliptic equations. The strong maximum principle states:
Assume $Luge0 (Lule0)$ in $Omega$ and assume that $u$ is non constant. If $c=0$ then $u$ does not achieve its maximum (minimum) in the interior of $Omega$. If $cle 0$, $u$ cannot achieve a non-negative maximum (non-positive minimum) in the interior. Regardless of the sign of $c$, $u$ cannot be zero at an interior maximum (minimum).
The operator $L$ is elliptic and $Omega$ is the domain. The exact assumptions on the domain and operator will not be given now, but are assumed to hold for my question.
I now want to try it on the equation
$begin{align}
Delta u(x,y)&=u(x,y)+sin(xy), &(x,y)in D_0(1)\
u(x,y)&=1, &(x,y)inpartial D_0(1)
end{align}$
where $D_0(1)$ is the unit disk centered around the origin.
I want to give an estimate of $sup|u|$ using the strong maximum principle (and possibly the weak one, too).
My attempt:
The requirements for the theorem are satisfied, as $Omega = D_0(1)$ and $L$ is elliptic. (But this shall not bother us now)
Since it is also required that $u$ is sufficiently smooth (has to be $mathcal{C}^2$ in the interior) the maximum and supremum coincide.
I obviously have to use the part of the theorem where $cle0$. But here already the first problem occurs, I do not have $Lule 0$ or $Luge 0$.
pde
pde
asked 17 hours ago
EpsilonDelta
6281615
asked 17 hours ago
EpsilonDelta
6281615
asked 17 hours ago
EpsilonDelta
6281615
asked 17 hours ago
EpsilonDelta
6281615
asked 17 hours ago
EpsilonDelta
6281615
6281615
add a comment |
add a comment |
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