Résumé

The solution of elliptic boundary value problems can be written near a corner of a polygonal domain, in the form of a sum u=u_{r}+u_{s}. The regular part ur has the maximum regularity, with respect of he order of he operator and the right hand side of the problem. The singular part us is a linear ombination of singular functions of an explicit form where the explicit development of the singular functions is of the form r^{λ}log^{q}rψ(θ), (r,θ) are the local polar coordinates at the corner. The numbber λ is called the exponent of singularity which is in general the zero of some analytical function F. The equation F(λ)=0 is called the characteristic equation of the problem. The classical method of Kondrat'ev permits to find such an equation only for the invariant problems by rotation, as for the Laplace equation, biharmonic equation, the stokes systems and those of Lamé, In this thesis we give the characteristic equation and the estimate of the exponent of singularity for elliptic problem of order 4, using mainly some results of Costabel and Dauge. This characteristic equation appears explicitly for the first time for problems which are not invariant by rotation.