Electron. J. Diff. Eqns., Vol. 1997(1997), No. 01, pp 1-12.

Sub-elliptic boundary value problems for quasilinear elliptic operators

Dian K. Palagachev & Peter R. Popivanov

Abstract:
Classical solvability and uniqueness in the Holder 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.

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Dian K. Palagachev
Department of Mathematics
Technological University of Sofia
8 Kl. Okhridski blvd., 1756 Sofia, Bulgaria.
email: dian@bgcict.acad.bg   dian@pascal.dm.uniba.it
Peter R. Popivanov
Institute of Mathematics, Bulgarian Academy of Sciences
G. Bonchev str., bl. 8, 1113 Sofia, Bulgaria
e-mail: popivano@bgearn.acad.bg

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