rancid axle
If $\Omega$ is an open, bdd, connected subset of the plane, and $u\in C^2(\overline{\Omega})$ satisfies $\Delta u=-2$ on $\Omega$ and $u=0$ on $\partial\Omega$, then $v:=\abs{\nabla u}^2$ attains its maximum on the boundary.
I know it suffices to show $v$ is subharmonic on $\Omega$ ($\Delta v\geq 0$), but my attempts thus far require further differentiability conditions on $u$, which ofc can't be assumed.
foggy ventureBOT
If $\Omega$ is an open, bdd, connected subset of the plane, and $u\in C^2(\overl...
flint cedarBOT
Omegabet_
foggy ventureBOT
+close
As well, only properties of subharmonics we've discussed so far are the volume mean value inequality and the maximum principle
shell trout
wdym with bdd?