Electronic Journal of Differential Equations,
Vol. 1997(1997), No. 01, pp 1-12.
Title: Sub-elliptic boundary value problems for quasilinear elliptic
operators
Authors: Dian K. Palagachev (Technological Univ. of Sofia, Bulgaria)
Peter R. Popivanov (Bulgarian Academy of Sciences, Bulgaria}
Abstract: Classical solvability and uniqueness in the H\"older space
$C^{2+\alpha}(\overline{\Omega})$
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 $S\subset\partial \Omega$,
and the nonlinear term $b(x,\,u,\,Du)$
grows quadratically with respect to the gradient $Du$.
Submitted October 28, 1996. Published January 8, 1997.
Math Subject Classification: 35J65, 35R25.
Key Words: Quasilinear elliptic operator; degenerate oblique derivative
problem; sub-elliptic estimates.