Abstract
In this paper the Dirichlet problem for the equation which prescribes the ith principal curvature of the graph of a function u is considered. A Comparison principle is obtained within the class of semiconvex subsolutions by a local perturbation procedure combined with a fine Lipschitz estimate on the elliptic operator. Existence of solutions is stated for the Dirichlet problem with boundary conditions in the viscosity sense; further assumptions guarantee that no loss of boundary data occurs. Some conditions relating the geometry of the domain and the prescribing data which are sufficient for existence and uniqueness of solutions are presented.
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Bardi M., Mannucci P.: On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Commun. Pure Appl. Anal. 5(4), 709–731 (2006)
Bardi M., Mannucci P.: Comparison principles for subelliptic equations of Monge-Ampère type. Boll. Unione Mat. Ital. 9, 1(2), 489–495 (2008)
Barles G., Busca J.: Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Commun. Partial Differ. Equ. 26(11–12), 2323–2337 (2001)
Barron E.N., Goebel R., Jensen R.R.: Quasiconvex functions and nonlinear PDEs. Trans. Am. Math. Soc. 365(8), 4229–4255 (2013)
Caffarelli L., Nirenberg L., Spruck J.: The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation. Commun. Pure Appl. Math. 37(3), 369–402 (1984)
Caffarelli, L., Nirenberg, L., Spruck, J.: Nonlinear second order elliptic equations. IV. Starshaped compact Weingarten hypersurfaces. In: Current Topics in Partial Differential Equations, pp. 1–26. Kinokuniya, Tokyo (1986)
Cannarsa, P., Sinestrari, C.: Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Progress in Nonlinear Differential Equations and their Applications, p. 58. Birkhäuser Boston Inc., Boston (2004)
Crandall M.G., Ishii H., Lions P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)
Lio F.D.: Comparison results for quasilinear equations in annular domains and applications. Commun. Partial Differ. Equ. 27(1–2), 283–323 (2002)
Gilbarg, D., Trudinger. N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics, 3rd edn. Springer-Verlag, Berlin (2001)
Guan P., Trudinger N.S., Wang X.-J.: On the Dirichlet problem for degenerate Monge-Ampère equations. Acta Math. 182(1), 87–104 (1999)
Harvey F.R., Lawson H.B. Jr.: Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds. J. Differ. Geom. 88(3), 395–482 (2011)
Horn, R.A., Johnson, C.R.: Matrix analysis. Cambridge University Press, Cambridge 1990. Corrected reprint of the 1985 original
Ishii H., Lions P.-L.: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equ. 83(1), 26–78 (1990)
Ivochkina N., Filimonenkova N.: On the backgrounds of the theory of m-Hessian equations. Commun. Pure Appl. Anal. 12(4), 1687–1703 (2013)
Ivochkina N.M.: Solution of the Dirichlet problem for an equation of curvature of order m. Dokl. Akad. Nauk SSSR 299(1), 35–38 (1988)
Luo Y.: On the uniqueness of solutions of spectral equations. J. Global Optim. 40(1–3), 155–160 (2008)
Luo Y., Eberhard A.: An application of C 1,1 approximation to comparison principles for viscosity solutions of curvature equations. Nonlinear Anal. 64(6), 1236–1254 (2006)
Mannucci P.: The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction. Commun. Pure Appl. Anal. 13(1), 119–133 (2014)
Trudinger N.S.: The Dirichlet problem for the prescribed curvature equations. Arch. Rational Mech. Anal. 111(2), 153–179 (1990)
Trudinger N.S., Urbas J.I.E.: The Dirichlet problem for the equation of prescribed Gauss curvature. Bull. Austral. Math. Soc. 28(2), 217–231 (1983)
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The author is supported by a Post-Doc Fellowship from the Università di Padova.
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Cirant, M. The Dirichlet problem for prescribed principal curvature equations. Nonlinear Differ. Equ. Appl. 22, 427–447 (2015). https://doi.org/10.1007/s00030-014-0290-1
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DOI: https://doi.org/10.1007/s00030-014-0290-1