This paper is devoted to the analysis of two Helmholtz equations in ℝ2 coupled via quasiperiodic transmission conditions on a set of piecewise smooth interfaces. The solution of this system is quasiperiodic in one direction and satisfies outgoing wave conditions with respect to the other direction. It is shown that Maxwell's equations for the diffraction of a time-harmonic oblique incident plane wave by periodic interfaces can be reduced to problems of this kind. The analysis is based on a strongly elliptic variational formulation of the differential problem in a bounded periodic cell involving nonlocal boundary operators. We obtain existence and uniqueness results for solutions corresponding to electromagnetic fields with locally finite energy. Special attention is paid to the regularity and leading asymptotics of solutions near the edges of the interface.

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We are interested in finding the velocity distribution at the wings of an aeroplane.Within the scope of a three-dimensional linear theory we analyse a model which is formulated as a mixed screen boundary value problem for the Helmholtz equation ( A + k 2 ) Qi = 0 in R3\S where Q, denotes the perturbation velocity potential, induced by the presence of the wings and s := L U w with the projection L of the wings onto the (x,y)-plane and the wake W .Not all Cauchy data are given explicitly on L , respectively W . These missing Cauchy data depend on the wing circulation r. r has to be fixed by the Kutta-Joukovskii condition: V@ should be finite near the trailing edge xt of L. To fulfil this condition in a way that all appearing terms can be defined mathematically exactly and belong to spaces which are physically meaningful, we propose to fix r by the condition of vanishing stress intensity factors of Qi near xt up to a certain order such thatIn the two-dimensional case, and if L is the left half-plane in IRz , we have an explicit formula to E W'2(2t) c L2(xt), > 0.calculate r and we can control the regularity of r and a.

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