Regularity for quasi-linear elliptic problems

Problem's Description

Conjecture: Any (weak/viscosity) solution $u$ to the semilinear elliptic equation $$ \hbox{div} (f(x,u)\nabla u) = 0, \qquad \hbox{in } B_1 $$ with $f$ being Dini continuous in $x$-variable, and having only finite number of discontinuities in $u$-variable, and uniform $C^\alpha $ regularity to the right and left of the discontinuities, should have universally bounded derivatives in $B_{1/2}$, with norms depending on $\|u\|_\infty$, ellipticity constants and the space dimension.

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