Homogenization of degenerate parabolic Signorini problems with vanishing storage and oscillating surface fluxes

Abstract

We study the homogenization of a unilateral quasilinear parabolic problem in a periodically perforated medium. The model combines two degeneracies: the coefficient of the time derivative is non-negative and may vanish, while the directional diffusion weights are positive almost everywhere but need not be bounded away from zero. The exterior boundary carries a Signorini condition, and the holes carry an unscaled zero-mean flux oscillating on the diffusive time scale. For any family of microscopic solutions we prove strong space--time $L^2$ compactness through a weighted detection inequality, which replaces the Aubin--Lions lemma when the storage vanishes, and we recover spatial $H^1$ regularity of the limit by a closed-range argument for the weighted two-scale gradients. The limit solves a coupled two-scale problem in Minty form, and microscopic existence holds under joint monotonicity. Under a strengthened condition, the limit decouples into a macroscopic Signorini inequality whose uniqueness follows from an exponential regularization, without time regularity of the limit, and the whole family converges. The surface datum enters both the effective flux and the exterior boundary condition.