HIGHLY OSCILLATING BOUNDARIES AND REDUCTION OF DIMENSION: THE CRITICAL CASE

Abstract

In [4], the first two authors studied a nonlinear monotone problem in a multidomain composed of a part [Formula: see text], with a highly oscillating boundary, placed upon an asymptotically flat part of thickness hε. More precisely, [Formula: see text] is a "forest" of cylinders with fixed height and small cross section of size ε, distributed with ε-periodicity upon the flat domain. The analysis was achieved under the assumption εp/hε → 0 (p - 1 is the growth order of the operator at infinity), as ε tends to 0, and for rescaled external forces hε fε converging to 0 in the (rescaled) flat domain. In the present paper, we present the analysis under the assumption εp/hε → l, with l ∈ [0, +∞], and for general limit forces in the flat domain. When l ∈ ]0, +∞[, we show that a discontinuity in the Dirichlet transmission condition may occur between the limit domain filled by the oscillating boundary and the plate. This discontinuity is derived through solving a nonlinear problem (in general for a different monotone operator) in the unit cell of the oscillating boundary. When l = +∞, we show that a deterministic limit model may hardly be expected.