Homogenization of the brush problem for a nonlinear monotone problem with $L^1$ data
Silvio Bove · HAL (Le Centre pour la Communication Scientifique Directe) · 2025
In this paper we consider a domain Ωε of $\mathbb{R}^N$ having highly oscillating boundary with respect to ε. In the 3D case, the domain is a brush: the teeth of the brush are assumed to be vertical, and the characteristic function of the teeth converges weakly-⋆ in L∞, as ε goes to zero, to a limit volume fraction which is measurable and bounded from below away from zero. We study the asymptotic behavior of the following nonlinear problem. When $f$ belongs to $L^{p'}$ we recall the well-known homogenization results and we give a corrector result which is new to our knowledge. We then address the homogenization question when the source term $f$ belongs to $L^1$. Using the notion of renormalized solution we are able to identify the homogenized problem and to show a corrector result.