Homogenization of quasilinear elliptic problems with interfacial resistance and weak data

ABSTRACT

In this paper, we study the homogenization of quasilinear elliptic problems with a jump on the interface between the two components proportional to the flux depending on a real parameter γ. The data is presumed to be an $L^1$ function and consequently, we consider the notion of renormalized solutions.

In the homogenization process, we employ the periodic unfolding method. The homogenized problem was afterwards identified to be a quasilinear elliptic problem with $L^1$ data for the case that the parameter γ is between $-1$ and 1 and for the case where γ is less than $-1$.

KEYWORDS:

• Homogenization
• periodic unfolding method
• renormalized solution

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 35B27

Acknowledgments

The authors would also like to give their thanks to the anonymous referees for their insightful comments and suggestions. The authors would like to express their deepest gratitude to the Office of the Vice President for Academic Affairs of the University of the Philippines (UP) for funding this research.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 A function $B:Y⟶\mathbb{R}⟶{\mathbb{R}}^{\mathbb{N}\times \mathbb{N}}$ is said to be Carathéodory if

1. the map $t\mapsto B(x,t)$ is continuous for a.e. $x\in \mathrm{\Omega }$;

2. the map $x\mapsto B(x,t)$ is measurable for all $t\in \mathbb{R}$

Additional information

Funding

This work was funded by the UP System Enhanced Creative Work and Research Grant (ECWRG-2022-1-2R).