Abstract
We studied global regularity properties for the Dirichlet problem concerning divergence form nonlinear Schrödinger equations with partially BMO coefficients in Reifenberg flat domains, and the corresponding \(W^{1, p}\)-regularity estimates for their weak solutions are obtained.
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References
Acerbi, E., Mingione, G.: Gradient estimates for a class of parabolic systems. Duke Math. J. 136, 285–320 (2007)
Auscher, P., Ben Ali, B.: Maximal inequality and Riesz transform estimates on \(L^{p}\) spaces for Schrödinger operator with nonnegative potentials. Ann. Inst. Fourier (Grenoble) 57, 1975–2013 (2007)
Badr, N., Ben Ali, B.: \( L^p\) boundedness of the Riesz transform related to Schrödinger operators on a manifold. Ann. Sc. Norm. Super. Pisa Cl. Sci. 8, 725–765 (2009)
Bramanti, M., Brandolini, L., Harboure, E., Viviani, B.: Global \(W^{2, p}\) estimates for nondivergence elliptic operators with potentials satisfying a reverse Hölder condition. Ann. Mat. Pura Appl. 191, 339–362 (2012)
Byun, S., Kim, Y.: Elliptic equations with measurable nonlinearlities in nonsmooth domains. Adv. Math. 288, 152–200 (2016)
Byun, S., Wang, L.: Elliptic equations with BMO coefficients in Reifenberg domains. Commun. Pure Appl. Math. 57, 1283–1310 (2004)
Byun, S., Wang, L.: The conormal derivative problem for elliptic equation with BMO coefficients on Reifenberg flat domains. Proc. Lond. Math. Soc. 90, 245–272 (2005)
Byun, S., Wang, L.: Elliptic equations with measurable coefficients in Reifenberg domains. Adv. Math. 225, 2648–2673 (2010)
Byun, S., Wang, L.: \(W^{1, p}\) regularity for the conormal derivative problem with parabolic BMO nonlinearity in Reifenberg domains. Discrete Contin. Dyn. Syst. 20, 617–637 (2008)
Chiarenza, F., Frasca, M., Longo, P.: \(W^{2, p}\)-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. Am. Math. Soc. 336, 841–853 (1993)
Di Fazio, G.: \(L^p\) estimates for divergence form elliptic equations with discontinuous coefficients. Boil Unione Mat. Ita. 10, 409–420 (1996)
Dong, H., Kim, D.: Elliptic equations in divergence form with partially \(BMO\) coefficients. Arch. Ration. Mech. Anal. 196, 25–75 (2010)
Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order, Classics in Mathematics, Reprint of the 1998 edition. Springer, Berlin (2001)
Giusti, E.: Direct methods in the calculus of variations. World Scientic Publishing Co., Inc., River Edge (2003)
Han, Q., Lin, F.: Elliptic partial differential equations. Courant Lecture Notes in Mathematics I, New YorK. AMS, Providence (1997)
Hajlas, P., Martio, O.: Trace of sobolev functions on fractal type sets and characterization of extension domain. J. Funct. Anal. 143, 221–246 (1997)
Horgan, C.: Anti-plane shear deformations in linear and nonlinear solid mechanics. SIAM Rev. 37, 53–81 (1995)
Lee, M., Ok, J.: Nonlinear Calderón-Zygmund theory involving dual data. Rev. Mat. Iberoam 35, 1053–1078 (2019)
Lee, M., Ok, J.: Interior and boundary \(W^{1, q}\)-estimates for elliptic quasi-linear equations of Schrödinger type. J. Differ. Equ. 269, 4406–4439 (2020)
Li, Y., Vogelius, M.: Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients. Arch. Ration. Mech. Anal. 153, 91–151 (2000)
Lieberman, G.M.: Oblique derivative problems for elliptic equations. World Scientific Publishing Co. Pte. Ltd., Hackensack (2013)
Pan, G., Tang, L.: Solvability for Schrödinger equation with discontinuous coefficients. J. Funct. Anal. 270, 88–133 (2016)
Phuc, N.: Nonlinear Munckenhoupt–Wheeden type bounds on Reifenberg flat domains with applications to quasilinear Riccati type equations. Adv. Math. 250, 387–419 (2014)
Shen, Z.: \(L^{p}\) estimates for Schrödinger operator with certain potentials. Ann. Inst. Fourier(Grenoble) 45, 513–546 (1995)
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The research was supported by the NNSF (11771023) of China.
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Tang, L., Zhang, G. \(W^{1,p}\) estimates for nonlinear Schrödinger equations with partially BMO coefficients in Reifenberg domains. Math. Z. 304, 41 (2023). https://doi.org/10.1007/s00209-023-03300-y
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DOI: https://doi.org/10.1007/s00209-023-03300-y