Corrector results for a class of elliptic problems with nonlinear Robin conditions and $ L^1 $ data

Patrizia Donato; Olivier Guibé; Alip Oropeza; Patrizia Donato; Olivier Guibé; Alip Oropeza

doi:10.3934/nhm.2023054

Networks and Heterogeneous Media

2023, Volume 18, Issue 3: 1236-1259. doi: 10.3934/nhm.2023054

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Research article

Corrector results for a class of elliptic problems with nonlinear Robin conditions and $ L^1 $ data

1.
Univ Rouen Normandie, CNRS, LMRS UMR 6085, F-76000, Rouen, France
2.
Institute of Mathematics, University of the Philippines - Diliman, 1101 Diliman, Quezon City, Philippines

Received: 05 December 2022 Revised: 22 March 2023 Accepted: 03 April 2023 Published: 25 April 2023

In this paper, we consider a class of elliptic problems in a periodically perforated domain with $ L^1 $ data and nonlinear Robin conditions on the boundary of the holes. Using the framework of renormalized solutions, which is well adapted to this situation, we show a convergence result for the truncated energy in the quasilinear case. When the operator is linear, we also prove a corrector result. Since we cannot expect to have solutions belonging to $ H^1 $, the main difficulty is to express the corrector result through the truncations of the solutions, together with the fact that the definition of a renormalized solution contains test functions which are nonlinear functions of the solution itself.
- homogenization,
- periodic unfolding,
- nonlinear Robin condition,
- correctors,
- renormalized solutions,
- integrable data
Citation: Patrizia Donato, Olivier Guibé, Alip Oropeza. Corrector results for a class of elliptic problems with nonlinear Robin conditions and $ L^1 $ data[J]. Networks and Heterogeneous Media, 2023, 18(3): 1236-1259. doi: 10.3934/nhm.2023054

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Abstract

In this paper, we consider a class of elliptic problems in a periodically perforated domain with $ L^1 $ data and nonlinear Robin conditions on the boundary of the holes. Using the framework of renormalized solutions, which is well adapted to this situation, we show a convergence result for the truncated energy in the quasilinear case. When the operator is linear, we also prove a corrector result. Since we cannot expect to have solutions belonging to $ H^1 $, the main difficulty is to express the corrector result through the truncations of the solutions, together with the fact that the definition of a renormalized solution contains test functions which are nonlinear functions of the solution itself.

References

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