Homogenization of Dynamic Signorini-Type Problems on Critically Oscillating Boundaries under Time-Periodic Forcing

We study the homogenization, in the critical scaling regime, of a boundary value problem with a nonlinear dynamic Signorini-type condition posed on a rapidly oscillating portion of the boundary. The source term is time-periodic and we look for time-periodic solutions. Using the method of oscillating test functions (Tartar), compactness, and monotonicity arguments, we identify the homogenized problem and the effective nonlinear boundary operator. In contrast with the evolutionary (initial value) setting, the periodic framework eliminates memory effects and yields an instantaneous time-periodic operator defined through a periodic-in-time cell problem.