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accepted Existence and regularity of 2D stream function.
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revised Existence and regularity of 2D stream function.
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May
12
comment Existence and regularity of 2D stream function.
Yes I forgot to assume $\Omega$ is also simply connected. I've refined the statement I want to: if $u$ is $C^1(\overline{\Omega})$ then $\psi$ is $C^2(\overline{\Omega})$. May as well assume $\partial \Omega$ is $C^2$ as well.
May
12
asked Existence and regularity of 2D stream function.
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10
accepted If $u_n$ is bounded and pointwise convergent, then $u_n$ convges in $W_{2,p}$.
May
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comment If $u_n$ is bounded and pointwise convergent, then $u_n$ convges in $W_{2,p}$.
$\Omega$ is a constant. Just assume $f$ doesn't depend on $x$, i.e. $f=f(u)$. and $f$ is $C^\infty$ positive increasing convex if it helps. You can read the first four pages of the paper I linked if you want more specifics on the assumptions, but I hope this is enough.
May
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comment If $u_n$ is bounded and pointwise convergent, then $u_n$ convges in $W_{2,p}$.
$\Omega$ is a bounded smooth domain (the paper says $C^{2,\alpha}$, but this can be weakened). The author concludes that $\{u_k\}$ is bounded in $W^{2,p}$ from knowing it is uniformly bounded pointwise. And he claims the statement for all $p \in [1,\infty)$, no relationship to thhe dimension of the space.