Optimal regularity for solutions to semilinear problems

Problem's Description

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. 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].

1. ArticleIs an originRegularity issues for semilinear PDE-s (a narrative approach)

   • Henrik Shahgholian

   Algebra i Analiz 27; translation in St. Petersburg Math. J. 27 (2016), no. 3, 577–587 3, 311-325, 2015

2. Article A free boundary problem with log-term singularity

   • de Queiroz,
   • Olivaine S.;,
   • Henrik Shahgholian

   Interfaces Free Bound 19 (3), 351--369, 2017

3. Article Regularity of solutions in semilinear elliptic theory.

   • Emanuel Indre,
   • Andreas Minne,
   • Levon Nurbekyan

   Bull. Math. Sci. 7 (1), 177-200, 2017arXiv

4. Article Partial analyticity and nodal sets for nonlinear elliptic systems.

   • herbert Koch,
   • Nikolai Nadirashvili

   arXiv

5. Article $C^{1,1}$-regularity in semilinear elliptic problems.

   • Henrik Shahgholian

   Comm. Pure Appl. Math. 56 (2003), no. 2, 278–281. 56 (2), 278--281, 2003