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The buckling of a thin elastic film bound to a compliant substrate is studied: we analyze the different patterns that arise as a function of the biaxial residual compressive stress in the film. We first clarify the boundary conditions to be used at the interface between film and substrate. We carry out the linear stability analysis of the classical pattern made of straight stripes, and point out secondary instabilities leading to the formation of undulating stripes, varicose, checkerboard or hexagonal patterns. Straight stripes are found to be stable in a narrow window of load parameters only. We present a weakly nonlinear post-buckling analysis of these patterns: for equi-biaxial residual compression, straight wrinkles are never stable and square checkerboard patterns are found to be optimal just above threshold; for anisotropic residual compression, straight wrinkles are present above a primary threshold and soon become …