For connected C^2 domain \Omega such that |\Omega|=|B_1(0)|,
let u be the soluiton of -\Delta u=1 in \Omega, u=0 on \partial \Omega,
when the value \int_\Omega u dx takes the minimum? is it the unit ball?
There is no minimum. Let a and b denote two positive constants with ab=1. Let u(x,y)=c(1-(x/a)2-(y/b)2). Let \Omega denote the interior of the ellipse (x/a)2+(y/b)2=1. Adapt the constant so that -\Delta u=1 in \Omega . Then the integral tends to zero as a tends to infinity and b=1/a tends to zero.
There is no minimum. Let a and b denote two positive constants with ab=1. Let u(x,y)=c(1-(x/a)2-(y/b)2). Let \Omega denote the interior of the ellipse (x/a)2+(y/b)2=1. Adapt the constant so that -\Delta u=1 in \Omega . Then the integral tends to zero as a tends to infinity and b=1/a tends to zero.
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