large Quasilinear Boundary Value Problems in Non-Smooth Domains 

(A.M. Sandig, Universitat Stuttgart, Math. 
Institut A, Pfaffenwaldring 57, 70569 Stuttgart)

We consider systems of stationary quasilinear partial differential
 equations of second order in domains with conical points, edges and
 vertices. We introduce weighted Sobolev spaces with attached 
asymptotics generated by the asymptotical expansions of solutions
 of corresponding linearized problems near the boundary singularities.
 Our goal is to apply the local invertibility theorem in the
 neighborhood of the zero-solution  in these spaces and to find
 conditions which guarantee that these solutions of the nonlinear
 problem have the same asymptotical structure as the solutions of
 the linearized problem. We use composition properties and 
multiplication theorems  in weighted Sobolev spaces \cite{AN97}
 in order to prove the mapping properties and the Fr\'echet 
differentiability of the  nonlinear operators acting between 
these spaces.
 We characterize  classes of quasilinear problems  by growth conditions
 and by properties of the singular terms in the asymptotic expansion of
 the linearized problems which allow to describe the asymptotic behaviour
 of small solutions of the nonlinear problems near boundary singularities
 similar to the linear case. This method was already applied in \cite{BS99}
 for semilinear problems in non--smooth domains, where the corresponding
 mappings are considered in usual Sobolev spaces with attached asymptotics.



 
bibliography
{AN97}:
F.Ali Mehmeti, S.Nicaise
Nemytskij's operators and global existence of small solutions of semilinear
 evolution equations on nonsmooth domains, Comm. P.D.E. 22 (1997) pp. 1559-1588. 

{BS99}:
M. Bochniak, A.M. Sandig
Local solvability and regularity results for a class of semilinear
elliptic problems in nonsmooth domains,Mathematica Bohemica 2--3, 124 (1999) pp. 245-254