The patchs antennas are intensively used in industry (mobile phone, radar, sonar). Due to the presence of boundary layers near the edges of these devices, the classical technique of numerical computation (like finite elements and finite differences) suffers from a lack of precision and leads therefore to expensive computations. Indeed, they don't take into account the local singular behavior of the scattered field by the antenna. The aim of this talk is the study of this boundary layer in the context of a simplified 2D patch antenna. This study is based on the work on thin layer [3], thin slot [2] and corner problems [1],[4] and use the technique of matching of asymptotic expansions. This method consists in writing the asymptotic expansion of the exact solution on each subdomain of a partition of the physical domain. After having given the forms of the asymptotic expansions on each subdomain, we exhibit the procedure of matching which leads to its construction. A full understanding of the singularities which appears in the boundary layer is essential and can be achieved via the introduction of a conformal mapping. We end this talk with a complete second order expansion which is validated by the associated error estimates. [1] Maz'ya, V. and Nazarov, S. and Plamenevskij, Boris, Asymptotic theory of ellip-tic boundary value problems in singularly perturbed domains. Vol. I, Operator Theory : Advances and Applications, 2000. [2] Tordeux, S, Méthodes Asymptotiques pour la Propagation des Ondes dans les Milieux Comportant des Fentes, Université de Versailles Saint-Quentin-en-Yvelines, thèse de Doctorat 2004. [3] Bendali, A and Lemrabet, K The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation Society for Industrial and Applied Mathematics, 6, 1664–1693, 1996. [4] Caloz, G. and Costabel, M. and Dauge, M. and Vial, G Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer Asymptotic Analysis 50(1,2) (2006) 121-173.