Maximum Principle for Poisson
$begingroup$
Let $Omega subset mathbb{R}^n$ be open $f,ginmathcal{C}(overlineOmega)$. If $uinmathcal{C}^2(overlineOmega)$ satisfies
$-Delta u=f text{on} Omega, u=g text{on} partialOmega$
then there is a constant $C_Omega$ depending only on $Omega$ such that
$max_{overlineOmega}|u|le C_{Omega} left (max_{partialOmega}g+max_{overlineOmega} fright)$
My book suggests using
$tilde u(x)=u(x)+frac{|f|_{L^infty(Omega)}}{2n}|x|^2$
in connection with the maximum principle.
The problem is, I don't think this is enough. This auxiliary function might have to be modified.
My attempt so far:
$-Delta tilde u(x)=f -|f|_{L^infty(overlineOmega)}C_Omegale 0\
tilde u|_{partial Omega} le g - |f|_{L^infty(Omega)}$
where I defined $C_Omega=text{diam}, Omega /n$
Now I would have $tilde ule 0$ by comparison principle, but I am unable to reach the conclusion, because the auxiliary function misses a term for $max_{partialOmega}g$.
ordinary-differential-equations pde
asked Dec 6 '18 at 21:49
EpsilonDeltaEpsilonDelta
6701615
$endgroup$
add a comment |
$begingroup$
Let $Omega subset mathbb{R}^n$ be open $f,ginmathcal{C}(overlineOmega)$. If $uinmathcal{C}^2(overlineOmega)$ satisfies
$-Delta u=f text{on} Omega, u=g text{on} partialOmega$
then there is a constant $C_Omega$ depending only on $Omega$ such that
$max_{overlineOmega}|u|le C_{Omega} left (max_{partialOmega}g+max_{overlineOmega} fright)$
My book suggests using
$tilde u(x)=u(x)+frac{|f|_{L^infty(Omega)}}{2n}|x|^2$
in connection with the maximum principle.
The problem is, I don't think this is enough. This auxiliary function might have to be modified.
My attempt so far:
$-Delta tilde u(x)=f -|f|_{L^infty(overlineOmega)}C_Omegale 0\
tilde u|_{partial Omega} le g - |f|_{L^infty(Omega)}$
where I defined $C_Omega=text{diam}, Omega /n$
Now I would have $tilde ule 0$ by comparison principle, but I am unable to reach the conclusion, because the auxiliary function misses a term for $max_{partialOmega}g$.
ordinary-differential-equations pde
asked Dec 6 '18 at 21:49
EpsilonDeltaEpsilonDelta
6701615
$endgroup$
add a comment |
$begingroup$
Let $Omega subset mathbb{R}^n$ be open $f,ginmathcal{C}(overlineOmega)$. If $uinmathcal{C}^2(overlineOmega)$ satisfies
$-Delta u=f text{on} Omega, u=g text{on} partialOmega$
then there is a constant $C_Omega$ depending only on $Omega$ such that
$max_{overlineOmega}|u|le C_{Omega} left (max_{partialOmega}g+max_{overlineOmega} fright)$
My book suggests using
$tilde u(x)=u(x)+frac{|f|_{L^infty(Omega)}}{2n}|x|^2$
in connection with the maximum principle.
The problem is, I don't think this is enough. This auxiliary function might have to be modified.
My attempt so far:
$-Delta tilde u(x)=f -|f|_{L^infty(overlineOmega)}C_Omegale 0\
tilde u|_{partial Omega} le g - |f|_{L^infty(Omega)}$
where I defined $C_Omega=text{diam}, Omega /n$
Now I would have $tilde ule 0$ by comparison principle, but I am unable to reach the conclusion, because the auxiliary function misses a term for $max_{partialOmega}g$.
ordinary-differential-equations pde
asked Dec 6 '18 at 21:49
EpsilonDeltaEpsilonDelta
6701615
$endgroup$
Let $Omega subset mathbb{R}^n$ be open $f,ginmathcal{C}(overlineOmega)$. If $uinmathcal{C}^2(overlineOmega)$ satisfies
$-Delta u=f text{on} Omega, u=g text{on} partialOmega$
then there is a constant $C_Omega$ depending only on $Omega$ such that
$max_{overlineOmega}|u|le C_{Omega} left (max_{partialOmega}g+max_{overlineOmega} fright)$
My book suggests using
$tilde u(x)=u(x)+frac{|f|_{L^infty(Omega)}}{2n}|x|^2$
in connection with the maximum principle.
The problem is, I don't think this is enough. This auxiliary function might have to be modified.
My attempt so far:
$-Delta tilde u(x)=f -|f|_{L^infty(overlineOmega)}C_Omegale 0\
tilde u|_{partial Omega} le g - |f|_{L^infty(Omega)}$
where I defined $C_Omega=text{diam}, Omega /n$
Now I would have $tilde ule 0$ by comparison principle, but I am unable to reach the conclusion, because the auxiliary function misses a term for $max_{partialOmega}g$.
ordinary-differential-equations pde
ordinary-differential-equations pde
asked Dec 6 '18 at 21:49
EpsilonDeltaEpsilonDelta
6701615
asked Dec 6 '18 at 21:49
EpsilonDeltaEpsilonDelta
6701615
asked Dec 6 '18 at 21:49
EpsilonDeltaEpsilonDelta
6701615
asked Dec 6 '18 at 21:49
EpsilonDeltaEpsilonDelta
6701615
asked Dec 6 '18 at 21:49
EpsilonDeltaEpsilonDelta
6701615
6701615
add a comment |
add a comment |
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