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A Liouville-type theorem for some Weingarten hypersurfaces
1. | Department of Applied Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan |
References:
[1] |
O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl., 76 (1997), 265-288.
doi: doi:10.1016/S0021-7824(97)89952-7. |
[2] |
B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math., 608 (2007), 17-33.
doi: doi:10.1515/CRELLE.2007.051. |
[3] |
L. Caffarelli, P. Guan and X.-N. Ma, A constant rank theorem for solutions of fully nonlinear elliptic equations, Comm. Pure Appl. Math., 60 (2007), 1769-1791.
doi: doi:10.1002/cpa.20197. |
[4] |
M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.
doi: doi:10.1090/S0273-0979-1992-00266-5. |
[5] |
Y. Giga and M. Ohnuma, On strong comparison principle for semicontinuous viscosity solutions of some nonlinear elliptic equations, Int. J. Pure Appl. Math., 22 (2005), 165-184. |
[6] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer-Verlag, Berlin, Heidelberg, New York, 1983. |
[7] |
H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equation, J. Differential Equations, 83 (1990), 26-78.
doi: doi:10.1016/0022-0396(90)90068-Z. |
[8] |
R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Anal., 101 (1988), 1-27.
doi: doi:10.1007/BF00281780. |
[9] |
G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific, 1996. |
[10] |
R. Magnanini and S. Sakaguchi, Stationary isothermic surfaces and some characterizations of the hyperplane in the $N$-dimensional Euclidean space, J. Differential Equations, 248 (2010), 1112-1119.
doi: doi:10.1016/j.jde.2009.11.017. |
[11] |
J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591.
doi: doi:10.1002/cpa.3160140329. |
[12] |
N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal., 111 (1990), 153-179.
doi: doi:10.1007/BF00375406. |
[13] |
J. I. E. Urbas, On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations, Indiana Univ. Math. J., 39 (1990), 355-382.
doi: doi:10.1512/iumj.1990.39.39020. |
show all references
References:
[1] |
O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl., 76 (1997), 265-288.
doi: doi:10.1016/S0021-7824(97)89952-7. |
[2] |
B. Andrews, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math., 608 (2007), 17-33.
doi: doi:10.1515/CRELLE.2007.051. |
[3] |
L. Caffarelli, P. Guan and X.-N. Ma, A constant rank theorem for solutions of fully nonlinear elliptic equations, Comm. Pure Appl. Math., 60 (2007), 1769-1791.
doi: doi:10.1002/cpa.20197. |
[4] |
M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.
doi: doi:10.1090/S0273-0979-1992-00266-5. |
[5] |
Y. Giga and M. Ohnuma, On strong comparison principle for semicontinuous viscosity solutions of some nonlinear elliptic equations, Int. J. Pure Appl. Math., 22 (2005), 165-184. |
[6] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," 2nd edition, Springer-Verlag, Berlin, Heidelberg, New York, 1983. |
[7] |
H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equation, J. Differential Equations, 83 (1990), 26-78.
doi: doi:10.1016/0022-0396(90)90068-Z. |
[8] |
R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Anal., 101 (1988), 1-27.
doi: doi:10.1007/BF00281780. |
[9] |
G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific, 1996. |
[10] |
R. Magnanini and S. Sakaguchi, Stationary isothermic surfaces and some characterizations of the hyperplane in the $N$-dimensional Euclidean space, J. Differential Equations, 248 (2010), 1112-1119.
doi: doi:10.1016/j.jde.2009.11.017. |
[11] |
J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591.
doi: doi:10.1002/cpa.3160140329. |
[12] |
N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal., 111 (1990), 153-179.
doi: doi:10.1007/BF00375406. |
[13] |
J. I. E. Urbas, On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations, Indiana Univ. Math. J., 39 (1990), 355-382.
doi: doi:10.1512/iumj.1990.39.39020. |
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