Sub-elliptic boundary value problems for quasilinear elliptic operators

Abstract

$C^{2+alpha}(overline{Omega})$ is proved for the oblique derivative problem $$cases{ a^{ij}(x)D_{ij}u + b(x,,u,,Du)=0 & in $Omega$,cr partial u/partial ell =varphi(x) & on $partial Omega$cr} $$ in the case when the vector field $ell(x)=(ell^1(x),ldots,ell^n(x))$ is tangential to the boundary $partial Omega$ at the points of some non-empty set $Ssubsetpartial Omega$, and the nonlinear term $b(x,,u,,Du)$ grows quadratically with respect to the gradient $Du$.

Keywords

• Quasilinear elliptic operator
• degenerate oblique derivative problem
• sub-elliptic estimates.