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Generalizing Hopf’s Boundary Point Lemma

Part of: Partial differential equations

Published online by Cambridge University Press:  04 January 2019

Leobardo Rosales
Leobardo Rosales*
Affiliation:
Keimyung University, Department of Mathematics, 1095 Dalgubeol-daero, Daegu, Republic of Korea, 42601 Email: rosales.leobardo@gmail.com
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Abstract

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We give a Hopf boundary point lemma for weak solutions of linear divergence form uniformly elliptic equations, with Hölder continuous top-order coefficients and lower-order coefficients in a Morrey space.

Keywords

partial differential equationdivergence formHopf boundary point lemma

MSC classification

Primary: 35B50: Maximum principles
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 1 , March 2019 , pp. 183 - 197
Copyright
© Canadian Mathematical Society 2018 

Footnotes

This work was partly conducted while the author was an Associate Member at the Korea Institute for Advanced Study.

References

Avelin, B., Kuusi, T., and Mingione, G., Nonlinear Calderón-Zygmund theory in the limiting case . Arch. Rat. Mech. Anal. Online: September 16, 2017. https://doi.org/10.1007/s00205-017-1171-7.Google Scholar
Cianci, P., Cirmi, G. R., D’Asero, S., and Leonardi, S., Morrey estimates for solutions of singular quadratic non linear equations . Ann. Mat. Pura Appl. 196(2017), no. 5, 1739–1758. https://doi.org/10.1007/s10231-017-0636-5.Google Scholar
Cirmi, G. R., D’Asero, S., and Leonardi, S., Gradient estimate for solutions of a class of nonlinear elliptic equations below the duality exponent . Math. Method Appl. Sci. Online: October 5, 2017. https://doi.org/10.1002/mma.4609.Google Scholar
Cirmi, G. R. and Leonardi, S., Regularity results for the gradient of solutions of linear elliptic equations with L 1, 𝜆 data . Ann. Mat. Pura e Appl. 185(2006), no. 4, 537–553. https://doi.org/10.1007/s10231-005-0167-3.Google Scholar
Cirmi, G. R. and Leonardi, S., Higher differentiability for solutions of linear elliptic systems with measure data . Discrete Contin. Dyn. Syst. 26(2010), no. 1, 89–104.Google Scholar
Cirmi, G. R. and Leonardi, S., Higher differentiability for the solutions of nonlinear elliptic systems with lower order terms and L 1, 𝜃-data . Ann. Mat. Pura Appl. 193(2014), no. 1, 115–131. https://doi.org/10.1007/s10231-012-0269-7.Google Scholar
Cirmi, G. R., Leonardi, S., and Stará, J., Regularity results for the gradient of solutions of a class of linear elliptic systems with L 1, 𝜆 data . Nonlinear Anal. 68(2008), no. 12, 3609–3624. https://doi.org/10.1016/j.na.2007.04.004.Google Scholar
Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order. Second ed., Grundlehren der Mathematischen Wissenschaften Springer-Verlag, Berlin, 1983.Google Scholar
Hardt, R. and Simon, L., Boundary regularity and embedded solutions for the oriented Plateau problem . Ann. of Math. 110(1979), no. 3, 439–486. https://doi.org/10.2307/1971233.Google Scholar
Hopf, E., A remark on linear elliptic differential equations of second order . Proc. Amer. Math. Soc. 3(1952), no. 5, 791–793. https://doi.org/10.1090/S0002-9939-1952-0050126-X.Google Scholar
Kristensen, J. and Mingione, G., Boundary regularity in variational problems . Arch. Ration. Mech. Anal. 198(2010), no. 2, 369–455. https://doi.org/10.1007/s00205-010-0294-x.Google Scholar
Leonardi, S., Gradient estimates below duality exponent for a class of linear elliptic systems . Nonlinear Differential Equations Appl. 18(2011), no. 3, 237–254. https://doi.org/10.1007/s00030-010-0093-y.Google Scholar
Leonardi, S., Fractional differentiability for solutions of a class of parabolic systems with L 1, 𝜃-data . NonLinear Anal. 95(2014), 530–542. https://doi.org/10.1016/j.na.2013.10.003.Google Scholar
Leonardi, S. and Stará, J., Regularity results for the gradient of solutions of linear elliptic systems with VMO-coefficients and L 1, 𝜆 data . Forum Math. 22(2010), no. 5, 913–940.Google Scholar
Leonardi, S. and Stará, J., Regularity up to the boundary for the gradient of solutions of linear elliptic systems with VMO coefficients and L 1, 𝜆 . Complex Var. Elliptic Equ. 56(2011), no. 12, 1086–1098. https://doi.org/10.1080/17476933.2010.534152.Google Scholar
Leonardi, S. and Stará, J., Regularity results for solutions of a class of parabolic systems with measure data . Nonlinear Anal. 75(2012), no. 4, 2069–2089. https://doi.org/10.1016/j.na.2011.10.008.Google Scholar
Leonardi, S. and Stará, J., Higher differentiability for solutions of a class of parabolic systems with L 1, 𝜃-data . Q. J. Math. 66(2015), no. 2, 659–676. https://doi.org/10.1093/qmath/hau031.Google Scholar
Mingione, G., The Calderon-Zygmund theory for elliptic problems with measure data . Ann. Sc. Norm. Super, Pisa Cl. Sci. 6(2007), no. 2, 195–261.Google Scholar
Morrey, C. B. Jr, Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, 130, Springer-Verlag, New York, 1966.Google Scholar
Sabina De Lis, J. C., Hopf maximum principle revisited . Electron. J. Differential Equations 115(2015), 1–9.Google Scholar