OpenYear of origin: 2003
Posted online: 2018-06-17 20:50:07Z by Henrik Shahgholian
Cite as: P-180617.1
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The following, and several similar problems, was discussed in paper [1].
Let $u$ be a solution to the semilinear problem $$ \Delta u = f(u) , $$ with $f(u)$ continuous. Would this be enough to show that $u$ has continuous second derivatives? The answer to this is no, and a proof was given by [4]. Now, adding one more condition such as $f'\geq -C$, would first of all imply that $u$ is $C^{1,1}$ as proven in [5]. So the question then is whether we can push the regularity all the way to $C^2$, given that $u$ is already $C^{1,1}$, for $u$ solving the above equation, and with $f$ being continuous.
It is quite tempting to stretch the above question further and ask whether $u$ has higher regularity whenever $f$ does it; e.g. what are the conditions on $f(u)$ (beyond classical thresholds of $f^{(m-2)}$ being Dini) to ensure $D^m u$ ($m\geq 2$) to be bounded, or continuous?
In this regard, other aspects to look at are the case of systems of the type $\Delta {\bf u}= {\bf f} (|{\bf u}|)$ (bold text denotes vectors), or equations with lower regularities of the right hand side like $\Delta u =\pm \log |u|$; the optimal regularity of solutions to the latter equation should be below $C^{1,1}$, due to the log-term (see [2]).
In the lines of the above discussions, it is noteworthy, that semilinear problems, involving the gradient, have also been partly in focus. The well-known gradient constraint problem [Ev], as well as the so-called superconductivity problem [CSal], [CSS] treats two different types of such problems. One may very generally ask for problems of the form $\Delta u =f(|\nabla u|)$, and how solutions behave?
For some recent results see [3].
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