In this paper, we consider a class of quasilinear elliptic problems in a two-component domain (one connected and the other a disconnected union of small <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2858_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-periodic sets) with a jump of the solution on the interface, proportional to the flux. We prescribe <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2858_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> data and coefficient periodic matrix fields A(y, t) only bounded for bounded t, so that we deal with renormalized solutions, where only their truncated functions are in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2858_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$H^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>. We complete here the homogenization results that we obtained in Donato, P., Fulgencio, R., Guibé, O.: Homogenization of a quasilinear elliptic problem in a two-component domain with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2858_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> data. Annal. Mat. 201, 1097–1137 (2022). <a href="https://doi.org/10.1007/s10231-021-01150-1">https://doi.org/10.1007/s10231-021-01150-1</a>, by the periodic unfolding method. We prove here that the unfolded truncated energies converge to the truncated energy of the homogenized problem. This is the most delicate result, and improves the convergences of the truncated gradients obtained in [<CitationRef CitationID="CR1">1</CitationRef>], which allows us to prove also some corrector results in the linear case, where A does not depend on t.

Convergence of the Energy and Correctors for Some Elliptic Problems in a Two-Component Domain With Weak Data

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发表日期：

2025年6月5日

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Patrizia Donato,
Rheadel Fulgencio,
Olivier Guibé

摘要

In this paper, we consider a class of quasilinear elliptic problems in a two-component domain (one connected and the other a disconnected union of small $\varepsilon $ $ε$ -periodic sets) with a jump of the solution on the interface, proportional to the flux. We prescribe $L^1$ $L^{1}$ data and coefficient periodic matrix fields A(y, t) only bounded for bounded t, so that we deal with renormalized solutions, where only their truncated functions are in $H^1$ $H^{1}$ . We complete here the homogenization results that we obtained in Donato, P., Fulgencio, R., Guibé, O.: Homogenization of a quasilinear elliptic problem in a two-component domain with $L^1$ $L^{1}$ data. Annal. Mat. 201, 1097–1137 (2022). https://doi.org/10.1007/s10231-021-01150-1, by the periodic unfolding method. We prove here that the unfolded truncated energies converge to the truncated energy of the homogenized problem. This is the most delicate result, and improves the convergences of the truncated gradients obtained in [1], which allows us to prove also some corrector results in the linear case, where A does not depend on t.