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Uniform \(W^{1,p}\) estimates for an elliptic operator with Robin boundary condition in a \(\mathcal {C}^1\) domain

Abstract

We consider the Robin boundary value problem \({\mathrm {div}}\,(A\nabla u) = {\mathrm {div}}\,\varvec{f}+F\) in \(\Omega \), a \(C^1\) domain, with \((A\nabla u - \varvec{f})\cdot {\varvec{n}}+ \alpha u = g\) on \(\Gamma \), where the matrix A belongs to \(VMO ({\mathbb {R}}^3) \), and discover the uniform estimates on \(\Vert u\Vert _{W^{1,p}(\Omega )}\), with \(1< p < \infty \), independent of \(\alpha \). At the difference with the case \(p = 2,\) which is simpler, we call here the weak reverse Hölder inequality. This estimates show that the solution of the Robin problem converges strongly to the solution of the Dirichlet (resp. Neumann) problem in corresponding spaces when the parameter \(\alpha \) tends to \(\infty \) (resp. 0).

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Acknowledgements

The authors would like to thank Karthik Adimurthi for his valuable suggestions and remarks to improve this manuscript. The second author is partially supported by PFBasal-001 and AFBasal170001 projects, and from the Regional Program STIC-AmSud Project NEMBICA-20-STIC-05.

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Correspondence to C. Amrouche.

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Communicated by J.Ball.

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Amrouche, C., Conca, C., Ghosh, A. et al. Uniform \(W^{1,p}\) estimates for an elliptic operator with Robin boundary condition in a \(\mathcal {C}^1\) domain. Calc. Var. 59, 71 (2020). https://doi.org/10.1007/s00526-020-1713-y

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Mathematics Subject Classification

  • 35J25