Poisson problem, minimun value of subharmonic function in the interior.
$begingroup$
Consider the problem:
$$ -Delta u = - f(x_1,x_2), text{in } Omega$$
$$ u = f(x_1,x_2), text{in } partialOmega$$
Where $Omega=B(0;1) subset mathbb{R}^2$ and $f(x_1,x_2)=1-x_1$.
Is it possible to know. without any calculations,if u is positive in the interior of $Omega$? By any calculations I mean without actually solving the problem.
All I can deduce is that since $u$ is subharmonic (since $f(x_1,x_2) geq 0$), the maximum principle applies and:
$$ max_{overline{Omega}} u = max_{partial Omega} u = 2$$
But nothing about whether it's positive or it can reach negative values in the interior.
I'm tempted to use this:
$$ min_{overline{Omega}} u = min_{partial Omega} u = 0 Rightarrow u geq 0$$
But since it's subharmonic I'm not sure I can use that. So, can I say anything? Or it may very well be negative in the interior?
harmonic-functions poissons-equation
asked Jan 15 at 10:56
RafaRafa
19710
$endgroup$
add a comment |
$begingroup$
Consider the problem:
$$ -Delta u = - f(x_1,x_2), text{in } Omega$$
$$ u = f(x_1,x_2), text{in } partialOmega$$
Where $Omega=B(0;1) subset mathbb{R}^2$ and $f(x_1,x_2)=1-x_1$.
Is it possible to know. without any calculations,if u is positive in the interior of $Omega$? By any calculations I mean without actually solving the problem.
All I can deduce is that since $u$ is subharmonic (since $f(x_1,x_2) geq 0$), the maximum principle applies and:
$$ max_{overline{Omega}} u = max_{partial Omega} u = 2$$
But nothing about whether it's positive or it can reach negative values in the interior.
I'm tempted to use this:
$$ min_{overline{Omega}} u = min_{partial Omega} u = 0 Rightarrow u geq 0$$
But since it's subharmonic I'm not sure I can use that. So, can I say anything? Or it may very well be negative in the interior?
harmonic-functions poissons-equation
asked Jan 15 at 10:56
RafaRafa
19710
$endgroup$
add a comment |
$begingroup$
Consider the problem:
$$ -Delta u = - f(x_1,x_2), text{in } Omega$$
$$ u = f(x_1,x_2), text{in } partialOmega$$
Where $Omega=B(0;1) subset mathbb{R}^2$ and $f(x_1,x_2)=1-x_1$.
Is it possible to know. without any calculations,if u is positive in the interior of $Omega$? By any calculations I mean without actually solving the problem.
All I can deduce is that since $u$ is subharmonic (since $f(x_1,x_2) geq 0$), the maximum principle applies and:
$$ max_{overline{Omega}} u = max_{partial Omega} u = 2$$
But nothing about whether it's positive or it can reach negative values in the interior.
I'm tempted to use this:
$$ min_{overline{Omega}} u = min_{partial Omega} u = 0 Rightarrow u geq 0$$
But since it's subharmonic I'm not sure I can use that. So, can I say anything? Or it may very well be negative in the interior?
harmonic-functions poissons-equation
asked Jan 15 at 10:56
RafaRafa
19710
$endgroup$
Consider the problem:
$$ -Delta u = - f(x_1,x_2), text{in } Omega$$
$$ u = f(x_1,x_2), text{in } partialOmega$$
Where $Omega=B(0;1) subset mathbb{R}^2$ and $f(x_1,x_2)=1-x_1$.
Is it possible to know. without any calculations,if u is positive in the interior of $Omega$? By any calculations I mean without actually solving the problem.
All I can deduce is that since $u$ is subharmonic (since $f(x_1,x_2) geq 0$), the maximum principle applies and:
$$ max_{overline{Omega}} u = max_{partial Omega} u = 2$$
But nothing about whether it's positive or it can reach negative values in the interior.
I'm tempted to use this:
$$ min_{overline{Omega}} u = min_{partial Omega} u = 0 Rightarrow u geq 0$$
But since it's subharmonic I'm not sure I can use that. So, can I say anything? Or it may very well be negative in the interior?
harmonic-functions poissons-equation
harmonic-functions poissons-equation
asked Jan 15 at 10:56
RafaRafa
19710
asked Jan 15 at 10:56
RafaRafa
19710
asked Jan 15 at 10:56
RafaRafa
19710
asked Jan 15 at 10:56
RafaRafa
19710
asked Jan 15 at 10:56
RafaRafa
19710
19710
add a comment |
add a comment |
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