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Research Article

A class of prescribed Weingarten curvature equations in Euclidean space

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Pages 1326-1343 | Received 18 Jan 2020, Accepted 31 Dec 2020, Published online: 21 Jan 2021

References

  • Krylov, N. (1995). On the general notion of fully nonlinear second order elliptic equation. Trans. Am. Math. Soc. 347(3):857–895. DOI: 10.1090/S0002-9947-1995-1284912-8.
  • Guan, P., Zhang, X. A class of curvature type equations. arXiv:1909.03645.
  • Schneider, R. (2013). Convex Bodies: The Brunn-Minkowski Theory. 2nd ed. Vol. 151. New York, NY: Cambridge University Press.
  • Chen, L., Guo, X., He, Y. A class of fully nonlinear equations arising in conformal geometry. Int. Math. Res. Not. DOI: 10.1093/imrn/rnaa253.
  • Fu, J., Yau, S. (2007). A Monge-Ampère type equation motivated by string theory. Commun. Anal. Geom. 15(1):29–76. DOI: 10.4310/CAG.2007.v15.n1.a2.
  • Fu, J., Yau, S. (2008). The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation. J. Differ. Geom. 78(3):369–428. DOI: 10.4310/jdg/1207834550.
  • Phong, D., Picard, S., Zhang, X. (2017). The Fu-Yau equation with negative slope parameter. Invent. Math. 209(2):541–576. DOI: 10.1007/s00222-016-0715-z.
  • Phong, D., Picard, S., Zhang, X. (2019). On estimates for the Fu-Yau generalization of a Strominger system. J. Reine Angew. Math. 2019(751):243–274. DOI: 10.1515/crelle-2016-0052.
  • Phong, D., Picard, S., Zhang, X. Fu-Yau Hessian Equations. J. Differ. Geom. arXiv:1801.09842.
  • Harvey, R., Lawson, H. (1982). Calibrated geometries. Acta Math. 148(1):47–157. DOI: 10.1007/BF02392726.
  • Bakelman, I., Kantor, B. (1973). Estimates of the solutions of quasilinear elliptic equations that are connected with problems of geometry in the large. Mat. Sb. (N.S.). 91:336–349, 471.
  • Bakelman, I., Kantor, B. (1974). Existence of a hypersurface homeomorphic to the sphere in Euclidean space with a given mean curvature. Geom. Topol. 1:3–10.
  • Treibergs, A., Wei, W. (1983). Embedded hypersurfaces with prescribed mean curvature. J. Differ. Geom. 18(3):513–521. DOI: 10.4310/jdg/1214437786.
  • Oliker, V. (1984). Hypersurfaces in Rn+1 with prescribed Gaussian curvature and related equations of Monge-Ampère type. Commun. Partial Differ. Equ. 9:807–838.
  • Aleksandrov, A., (1956). Uniqueness theorems for surfaces in the large. Vestnik Leningrad Univ. 11:5–17; 12 (1957), 15–44; 13 (1958), 5–8; 13 (1958), 14–34; 14 (1959), 5–13; 15 (1960), 5–13
  • Firey, W. (1968). Christoffel problem for general convex bodies. Mathematik. 15(1):7–21. DOI: 10.1112/S0025579300002321.
  • Caffarelli, L., Nirenberg, L., Spruck, J. (1986). Nonlinear second order elliptic equations, IV. Starshaped compact Weingarten hypersurfaces. In: Ohya, Y., Kasahara, K., Shimaku, N., eds. Current Topics in PDEs. Tokyo: Kinokunize, pp. 1–26.
  • Li, Y., Oliker, V. (2002). Starshaped compact hypersurfaces with prescribed m-th mean curvature in elliptic space. J. Partial Differ. Equ. 15:68–80.
  • Barbosa, J., de Lira, J., Oliker, V. (2002). A priori estimates for starshaped compact hypersurfaces with prescribed mth curvature function in space forms. In: Nonlinear Problems in Mathematical Physics and Related Topics I, International Mathematical Series (N.Y.). Vol. 1. Boston: Springer, pp. 35–52.
  • Jin, Q., Li, Y. (2006). Starshaped compact hypersurfaces with prescribed k-th mean curvature in hyperbolic space. Discrete Continuous Dyn. Syst. 15(2):367–377. DOI: 10.3934/dcds.2006.15.367.
  • Andrade, F., Barbosa, J., de Lira, J. (2009). Closed Weingarten hypersurfaces in warped product manifolds. Indiana Univ. Math. J. 58(4):1691–1718. DOI: 10.1512/iumj.2009.58.3631.
  • Li, Q., Sheng, W. (2013). Closed hypersurfaces with prescribed Weingarten curvature in Riemannian manifolds. Calc. Var. 48(1–2):41–66. DOI: 10.1007/s00526-012-0540-1.
  • Ivochkina, N. (1990). Solution of the Dirichlet problem for curvature equations of order m. Math. USSR Sb. 67(2):317–339. DOI: 10.1070/SM1990v067n02ABEH002089.
  • Ivochkina, N. (1991). The Dirichlet problem for the equations of curvature of order m. Leningrad Math. J. 2:631–654.
  • Guan, P., Li, J., Li, Y. (2012). Hypersurfaces of prescribed curvature measure. Duke Math. J. 161(10):1927–1942. DOI: 10.1215/00127094-1645550.
  • Guan, P., Lin, C., Ma, X. (2009). The existence of convex body with prescribed curvature measures. Int. Math. Res. Not. 2009(11):1947–1975.
  • Guan, B., Guan, P. (2002). Convex hypersurfaces of prescribed curvatures. Ann. Math. 156(2):655–673. DOI: 10.2307/3597202.
  • Guan, P., Ren, C., Wang, Z. (2015). Global C2 estimates for convex solutions of curvature equations. Commun. Pure Appl. Math. 68(8):1287–1325. DOI: 10.1002/cpa.21528.
  • Li, M., Ren, C., Wang, Z. (2016). An interior estimate for convex solutions and a rigidity theorem. J. Funct. Anal. 270(7):2691–2714. DOI: 10.1016/j.jfa.2016.01.008.
  • Ren, C., Wang, Z. (2019). On the curvature estimates for Hessian equations. Am. J. Math. 141(5):1281–1315. DOI: 10.1353/ajm.2019.0033.
  • Spruck, J., Xiao, L. (2017). A note on starshaped compact hypersurfaces with prescribed scalar curvature in space form. Rev. Mat. Iberoam. 33(2):547–554. DOI: 10.4171/RMI/948.
  • Chen, D., Li, H., Wang, Z. (2018). Starshaped compact hypersurfaces with prescribed Weingarten curvature in warped product manifolds. Calc. Var. 57(2):26. DOI: 10.1007/s00526-018-1314-1.
  • Lin, M., Trudinger, N. (1994). On some inequalities for elementary symmetric functions. Bull. Austral. Math. Soc. 50(2):317–326. DOI: 10.1017/S0004972700013770.
  • Trudinger, N. (1990). The Dirichlet problem for the prescribed curvature equations. Arch. Ration. Mech. Anal. 111(2):153–179. DOI: 10.1007/BF00375406.
  • Li, Y. (1989). Degree theory for second order nonlinear elliptic operators and its applications. Commun. Partial Differ. Equ. 14:1541–1578.
  • Evans, L. (1982). Classical solutions of fully nonlinear, convex, second-order elliptic equations. Commun. Pure Appl. Math. 35(3):333–363. DOI: 10.1002/cpa.3160350303.
  • Krylov, N. (1983). Boundedly inhomogeneous elliptic and parabolic equations in a domain. Izv. Akad. Nauk SSSR Ser. Mat. 47:75–108.

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