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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References
Amrouche, C., Ghosh, A., Conca, C., Acevedo, P.: Stokes and Navier–Stokes equations with Navier boundary condition. C. R. Math. 357, 115–119 (2019)
Auscher, P., Qafsaoui, M.: Observations on \(W^{1,p}\) estimates for divergence elliptic equations with VMO coefficients. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 5(2), 487–509 (2002)
Caffarelli, L.A., Peral, I.: On \(W^{1, p}\) estimates for elliptic equations in divergence form. Commun. Pure Appl. Math. 51(1), 1–21 (1998)
Daners, D.: Robin boundary value problems on arbitrary domains. Trans. Am. Math. Soc. 352(9), 4207–4236 (2000)
Dong, H., Kim, D.: Elliptic equations in divergence form with partially BMO coefficients. Arch. Ration. Mech. Anal. 196(1), 25–70 (2010)
Gehring, F.W.: The Lp-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265–277 (1973)
Geng, J.: \(W^{1, p}\) estimates for elliptic problems with Neumann boundary conditions in Lipschitz domains. Adv. Math. 229(4), 2427–2448 (2012)
Gérard-Varet, D., Masmoudi, N.: Relevance of the slip condition for fluid flows near an irregular boundary. Commun. Math. Phys. 295(1), 99–137 (2010)
Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton (1983)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, volume 224 of Fundamental Principles of Mathematical Sciences, 2nd edn. Springer, Berlin (1983)
Girouard, A., Polterovich, I.: Spectral geometry of the Steklov problem (survey article). J. Spectr. Theory 7(2), 321–359 (2017)
Han, Q., Lin, F.: Elliptic Partial Differential Equations, volume 1 of Courant Lecture Notes in Mathematics. American Mathematical Society, Providence (1997)
Jerison, D.S., Kenig, C.E.: The Neumann problem on Lipschitz domains. Bull. Am. Math. Soc. (N.S.) 4(2), 203–207 (1981)
Kenig, C.E., Lin, F., Shen, Z.: Homogenization of elliptic systems with Neumann boundary conditions. J. Am. Math. Soc. 26(4), 901–937 (2013)
Kristensen, J., Mingione, G.: The singular set of minima of integral functionals. Arch. Ration. Mech. Anal. 180(3), 331–398 (2006)
Pal, D., Rudraiah, N., Devanathan, R.: The effects of slip velocity at a membrane surface on blood flow in the microcirculation. J. Math. Biol. 26(6), 705–712 (1988)
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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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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DOI: https://doi.org/10.1007/s00526-020-1713-y