Maximum Principle for Elliptic Problems












1














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







share|cite|improve this question



























    1














    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







    share|cite|improve this question

























      1












      1








      1







      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







      share|cite|improve this question













      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






      share|cite|improve this question













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      asked Jan 4 at 12:49









      EpsilonDeltaEpsilonDelta

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