The boundary value problem for the Helmholtz equation in a quadrant with different Robin's or impedance-type conditions is considered by the use of Variational methods and boundary pseudodifferential equations. The coerciveness of the underlying sesquilinear form in dependence of the impedance parameters is proved. In the present paper we derive the first term in the asymptotic expansion of the solution in the vicinity of an edge. Note that, knowing this term, optimal mesh gradings can be designed for the numerical solution of this equation. The other eigenvalues depend linearly on a parameter s linked to the Lame coefficients and the associated eigenmodes are the gradients of the Laplace-Dirichlet eigenfunctions.
Similar properties hold in polygons.
We give numerical results for two L-shaped domains. The contact is modelled with a modified version of Coulomb's law of dry friction and, moreover, the coefficient of friction is assumed to depend either on the total slip or on the current slip. In the first case, the problem depends upon contact history. We present the classical formulations of the problems, the variational formulations and establish the existence and uniqueness of a weak solution to each of them, when the coefficient of friction is sufficiently small. The proofs are based on classical results for elliptic variational inequalities and fixed point arguments.