American Institute of Mathematical Sciences

April  2015, 35(4): 1447-1468. doi: 10.3934/dcds.2015.35.1447

On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions

Received  June 2013 Revised  October 2013 Published  November 2014

This paper deals with several qualitative properties of solutions of some stationary equations associated to the Monge--Ampère operator on the set of convex functions which are not necessarily understood in a strict sense. Mainly, we focus our attention on the occurrence of a free boundary (separating the region where the solution $u$ is locally a hyperplane, thus, the Hessian $D^{2}u$ is vanishing, from the rest of the domain). In particular, our results apply to suitable formulations of the Gauss curvature flow and of the worn stones problems intensively studied in the literature.
Citation: Gregorio Díaz, Jesús Ildefonso Díaz. On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1447-1468. doi: 10.3934/dcds.2015.35.1447
References:
 [1] A. D. Aleksandrov, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Uzen. Zap. Leningrad. Gos. Univ., 6 (1939), 3-35. (Russian) [2] L. Álvarez, On the behavior of the free boundary of some nonhomogeneous elliptic problems, Appl. Anal., 36 (1990), 131-144. [3] L. Álvarez and J. I. Díaz, On the retention of the interfaces in some elliptic and parabolic nonlinear problems, Discrete Contin. Dyn. Syst., 25 (2009), 1-17. doi: 10.3934/dcds.2009.25.1. [4] L. Ambrosio, Lecture Notes on Optimal Transport Problems, Mathematical Aspects of Evolving Interfaces, Springer Verlag, Berlin, Lecture Notes in Mathematics, 1812, (2003), 1-52. doi: 10.1007/978-3-540-39189-0_1. [5] A. M. Ampère, Mémoire contenant l'application de la théorie, J. l'École Polytechnique, 1820. [6] G. Barles and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Comm. in P.D.E., 26 (2001), 2323-2337. doi: 10.1081/PDE-100107824. [7] B. Brandolini and J. I. Díaz, work, in progress., (). [8] B. Brandolini and C. Trombetti, Comparison results for Hessian equations via symmetrization, J. Eur. Math. Soc. (JEMS), 9 (2007), 561-575. doi: 10.4171/JEMS/88. [9] H. Brezis and L. Nirenberg, Removable singularities for nonlinear elliptic equations, Topol. Methods Nonlinear Anal., 9 (1997), 201-219. [10] L. Caffarelli, Some regularity properties of solutions of the Monge-Ampère equation, Comm. Pure Appl. Math., 44 (1991), 965-969. doi: 10.1002/cpa.3160440809. [11] L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second-order elliptic equations. V. The Dirichlet problem for Weingarten hypersurfaces, Comm. Pure Appl. Math., 41 (1988), 47-70. doi: 10.1002/cpa.3160410105. [12] L. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problemas, American Mathematical Society, 2005. [13] 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: 10.1090/S0273-0979-1992-00266-5. [14] M. G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298. doi: 10.2307/2373376. [15] P. Daskalopoulos and K. Lee, Fully degenerate Monge-Ampère equations, J. Differential. Equations, 253 (2012), 1665-1691. doi: 10.1016/j.jde.2012.06.006. [16] G. Díaz, Some properties of second order of degenerate second order P.D.E. in non-divergence form, Appl. Anal., 20 (1985), 309-336. doi: 10.1080/00036818508839576. [17] G. Díaz, The Influence of the Geometry in the Large Solution of Hessian Equations Perturbed with a Superlinear Zeroth Order Term,, work in progress., (). [18] G. Díaz, The Liouville Theorem on Hessian Equations Perturbed with a Superlinear Zeroth Order Term,, work in progress ., (). [19] G. Díaz and J. I. Díaz, On some free boundary problems for stationary fully nonlinear equations involving Hessian functions: application to optimal multi-antennas,, to appear., (). [20] G. Díaz and J. I. Díaz, Parabolic MongeAmpre equations giving rise to a free boundary: the worn stone model,, to appear., (). [21] J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Vol. 1 Elliptic Equations, Res. Notes Math, 106. Pitman, 1985. [22] J. I. Díaz, T. Mingazzini and A. M. Ramos, On an optimal control problem involving the location of a free boundary, Proceedings of the XII Congreso de Ecuaciones Diferenciales y Aplicaciones/Congreso de Matemática Aplicada (Palma de Mallorca), Spain, September, (2011), 5-9. [23] W. J. Firey, Shapes of worn stones, Mathematika, 21 (1974), 1-11. doi: 10.1112/S0025579300005714. [24] W. Gangbo and R. J. Mccann, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161. doi: 10.1007/BF02392620. [25] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [26] E. Goursat, Leçons sur l'Integration des Équations aux Derivées Partielles du Second Order à Deux Variables Indepéndantes, Herman, Paris, 1896. [27] P. Guan, N. S. Trudinger and X. Wang, On the Dirichlet problem for degenerate Monge-Ampère equations, Acta Math., 182 (1999), 87-104. doi: 10.1007/BF02392824. [28] C. E. Gutiérrez, The Monge-Ampère Equation, Birkhauser, Boston, MA, 2001. doi: 10.1007/978-1-4612-0195-3. [29] R. Hamilton, Worn stones with at sides; in a tribute to Ilya Bakelman, Discourses Math. Appl., 3 (1993), 69-78. [30] P.-L. Lions, Sur les equations de Monge-Ampère I, II, Manuscripta Math., 41 (1983), 1-44; Arch. Rational Mech. Anal., 89 (1985), 93-122. doi: 10.1007/BF00282327. [31] G. Monge, Sur le Calcul Intégral Des Équations Aux Differences Partielles, Mémoires de l'Académie des Sciences, 1784. [32] L. Nirenberg, Monge-Ampère Equations and Some Associated Problems in Geometry, in Proccedings of the International Congress of Mathematics, Vancouver, 1974. [33] P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007. [34] G. Talenti, Some estimates of solutions to Monge-Ampère type equations in dimension two, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 8 (1981), 183-230. [35] N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Ration. Mech. Anal., 111 (1990), 153-179. doi: 10.1007/BF00375406. [36] N. S. Trudinger and X.-J. Wang, The Monge-Ampère equation and its geometric applications, in Handbook of Geometric Analysis, Vol. I, International Press, (2008), 467-524. [37] J. Urbas, On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations, Indiana Univ. Math. J., 39 (1990), 355-382. doi: 10.1512/iumj.1990.39.39020. [38] C. Villani, Optimal Transport: Old and New, Springer Verlag (Grundlehren der mathematischen Wissenschaften), 2008. doi: 10.1007/978-3-540-71050-9. [39] J. L. Vázquez, A strong Maximum Principle for some quasilinear elliptic equations, Appl Math Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.

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References:
 [1] A. D. Aleksandrov, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Uzen. Zap. Leningrad. Gos. Univ., 6 (1939), 3-35. (Russian) [2] L. Álvarez, On the behavior of the free boundary of some nonhomogeneous elliptic problems, Appl. Anal., 36 (1990), 131-144. [3] L. Álvarez and J. I. Díaz, On the retention of the interfaces in some elliptic and parabolic nonlinear problems, Discrete Contin. Dyn. Syst., 25 (2009), 1-17. doi: 10.3934/dcds.2009.25.1. [4] L. Ambrosio, Lecture Notes on Optimal Transport Problems, Mathematical Aspects of Evolving Interfaces, Springer Verlag, Berlin, Lecture Notes in Mathematics, 1812, (2003), 1-52. doi: 10.1007/978-3-540-39189-0_1. [5] A. M. Ampère, Mémoire contenant l'application de la théorie, J. l'École Polytechnique, 1820. [6] G. Barles and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Comm. in P.D.E., 26 (2001), 2323-2337. doi: 10.1081/PDE-100107824. [7] B. Brandolini and J. I. Díaz, work, in progress., (). [8] B. Brandolini and C. Trombetti, Comparison results for Hessian equations via symmetrization, J. Eur. Math. Soc. (JEMS), 9 (2007), 561-575. doi: 10.4171/JEMS/88. [9] H. Brezis and L. Nirenberg, Removable singularities for nonlinear elliptic equations, Topol. Methods Nonlinear Anal., 9 (1997), 201-219. [10] L. Caffarelli, Some regularity properties of solutions of the Monge-Ampère equation, Comm. Pure Appl. Math., 44 (1991), 965-969. doi: 10.1002/cpa.3160440809. [11] L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second-order elliptic equations. V. The Dirichlet problem for Weingarten hypersurfaces, Comm. Pure Appl. Math., 41 (1988), 47-70. doi: 10.1002/cpa.3160410105. [12] L. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problemas, American Mathematical Society, 2005. [13] 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: 10.1090/S0273-0979-1992-00266-5. [14] M. G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298. doi: 10.2307/2373376. [15] P. Daskalopoulos and K. Lee, Fully degenerate Monge-Ampère equations, J. Differential. Equations, 253 (2012), 1665-1691. doi: 10.1016/j.jde.2012.06.006. [16] G. Díaz, Some properties of second order of degenerate second order P.D.E. in non-divergence form, Appl. Anal., 20 (1985), 309-336. doi: 10.1080/00036818508839576. [17] G. Díaz, The Influence of the Geometry in the Large Solution of Hessian Equations Perturbed with a Superlinear Zeroth Order Term,, work in progress., (). [18] G. Díaz, The Liouville Theorem on Hessian Equations Perturbed with a Superlinear Zeroth Order Term,, work in progress ., (). [19] G. Díaz and J. I. Díaz, On some free boundary problems for stationary fully nonlinear equations involving Hessian functions: application to optimal multi-antennas,, to appear., (). [20] G. Díaz and J. I. Díaz, Parabolic MongeAmpre equations giving rise to a free boundary: the worn stone model,, to appear., (). [21] J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Vol. 1 Elliptic Equations, Res. Notes Math, 106. Pitman, 1985. [22] J. I. Díaz, T. Mingazzini and A. M. Ramos, On an optimal control problem involving the location of a free boundary, Proceedings of the XII Congreso de Ecuaciones Diferenciales y Aplicaciones/Congreso de Matemática Aplicada (Palma de Mallorca), Spain, September, (2011), 5-9. [23] W. J. Firey, Shapes of worn stones, Mathematika, 21 (1974), 1-11. doi: 10.1112/S0025579300005714. [24] W. Gangbo and R. J. Mccann, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161. doi: 10.1007/BF02392620. [25] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [26] E. Goursat, Leçons sur l'Integration des Équations aux Derivées Partielles du Second Order à Deux Variables Indepéndantes, Herman, Paris, 1896. [27] P. Guan, N. S. Trudinger and X. Wang, On the Dirichlet problem for degenerate Monge-Ampère equations, Acta Math., 182 (1999), 87-104. doi: 10.1007/BF02392824. [28] C. E. Gutiérrez, The Monge-Ampère Equation, Birkhauser, Boston, MA, 2001. doi: 10.1007/978-1-4612-0195-3. [29] R. Hamilton, Worn stones with at sides; in a tribute to Ilya Bakelman, Discourses Math. Appl., 3 (1993), 69-78. [30] P.-L. Lions, Sur les equations de Monge-Ampère I, II, Manuscripta Math., 41 (1983), 1-44; Arch. Rational Mech. Anal., 89 (1985), 93-122. doi: 10.1007/BF00282327. [31] G. Monge, Sur le Calcul Intégral Des Équations Aux Differences Partielles, Mémoires de l'Académie des Sciences, 1784. [32] L. Nirenberg, Monge-Ampère Equations and Some Associated Problems in Geometry, in Proccedings of the International Congress of Mathematics, Vancouver, 1974. [33] P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007. [34] G. Talenti, Some estimates of solutions to Monge-Ampère type equations in dimension two, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 8 (1981), 183-230. [35] N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Ration. Mech. Anal., 111 (1990), 153-179. doi: 10.1007/BF00375406. [36] N. S. Trudinger and X.-J. Wang, The Monge-Ampère equation and its geometric applications, in Handbook of Geometric Analysis, Vol. I, International Press, (2008), 467-524. [37] J. Urbas, On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations, Indiana Univ. Math. J., 39 (1990), 355-382. doi: 10.1512/iumj.1990.39.39020. [38] C. Villani, Optimal Transport: Old and New, Springer Verlag (Grundlehren der mathematischen Wissenschaften), 2008. doi: 10.1007/978-3-540-71050-9. [39] J. L. Vázquez, A strong Maximum Principle for some quasilinear elliptic equations, Appl Math Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.
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