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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

Gregorio Díaz 1, and  Jesús Ildefonso Díaz 2,

1. 

Instituto Matemático Interdisciplinar and Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de las Ciencias 3, 28040-Madrid, Spain
2. 

Instituto de Matemática Interdiciplinar, Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de las Ciencias, 3, 28040 Madrid

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.
Keywords: Monge--Ampère equation, Gauss curvatures surfaces, free boundary problem..
Mathematics Subject Classification: Primary: 35J96, 35K96, 35R35; Secondary: 53C3.
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 & 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)  Google Scholar

[2]

L. Álvarez, On the behavior of the free boundary of some nonhomogeneous elliptic problems, Appl. Anal., 36 (1990), 131-144. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. M. Ampère, Mémoire contenant l'application de la théorie, J. l'École Polytechnique, 1820. Google Scholar

[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.  Google Scholar

[7]

B. Brandolini and J. I. Díaz, work, in progress., ().   Google Scholar

[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.  Google Scholar

[9]

H. Brezis and L. Nirenberg, Removable singularities for nonlinear elliptic equations, Topol. Methods Nonlinear Anal., 9 (1997), 201-219.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

L. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problemas, American Mathematical Society, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[18]

G. Díaz, The Liouville Theorem on Hessian Equations Perturbed with a Superlinear Zeroth Order Term,, work in progress ., ().   Google Scholar

[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., ().   Google Scholar

[20]

G. Díaz and J. I. Díaz, Parabolic MongeAmpre equations giving rise to a free boundary: the worn stone model,, to appear., ().   Google Scholar

[21]

J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Vol. 1 Elliptic Equations, Res. Notes Math, 106. Pitman, 1985.  Google Scholar

[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. Google Scholar

[23]

W. J. Firey, Shapes of worn stones, Mathematika, 21 (1974), 1-11. doi: 10.1112/S0025579300005714.  Google Scholar

[24]

W. Gangbo and R. J. Mccann, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161. doi: 10.1007/BF02392620.  Google Scholar

[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.  Google Scholar

[26]

E. Goursat, Leçons sur l'Integration des Équations aux Derivées Partielles du Second Order à Deux Variables Indepéndantes, Herman, Paris, 1896. Google Scholar

[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.  Google Scholar

[28]

C. E. Gutiérrez, The Monge-Ampère Equation, Birkhauser, Boston, MA, 2001. doi: 10.1007/978-1-4612-0195-3.  Google Scholar

[29]

R. Hamilton, Worn stones with at sides; in a tribute to Ilya Bakelman, Discourses Math. Appl., 3 (1993), 69-78.  Google Scholar

[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.  Google Scholar

[31]

G. Monge, Sur le Calcul Intégral Des Équations Aux Differences Partielles, Mémoires de l'Académie des Sciences, 1784. Google Scholar

[32]

L. Nirenberg, Monge-Ampère Equations and Some Associated Problems in Geometry, in Proccedings of the International Congress of Mathematics, Vancouver, 1974. Google Scholar

[33]

P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.  Google Scholar

[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.  Google Scholar

[35]

N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Ration. Mech. Anal., 111 (1990), 153-179. doi: 10.1007/BF00375406.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

C. Villani, Optimal Transport: Old and New, Springer Verlag (Grundlehren der mathematischen Wissenschaften), 2008. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[39]

J. L. Vázquez, A strong Maximum Principle for some quasilinear elliptic equations, Appl Math Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.  Google Scholar

show all references

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)  Google Scholar

[2]

L. Álvarez, On the behavior of the free boundary of some nonhomogeneous elliptic problems, Appl. Anal., 36 (1990), 131-144. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. M. Ampère, Mémoire contenant l'application de la théorie, J. l'École Polytechnique, 1820. Google Scholar

[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.  Google Scholar

[7]

B. Brandolini and J. I. Díaz, work, in progress., ().   Google Scholar

[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.  Google Scholar

[9]

H. Brezis and L. Nirenberg, Removable singularities for nonlinear elliptic equations, Topol. Methods Nonlinear Anal., 9 (1997), 201-219.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

L. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problemas, American Mathematical Society, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[18]

G. Díaz, The Liouville Theorem on Hessian Equations Perturbed with a Superlinear Zeroth Order Term,, work in progress ., ().   Google Scholar

[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., ().   Google Scholar

[20]

G. Díaz and J. I. Díaz, Parabolic MongeAmpre equations giving rise to a free boundary: the worn stone model,, to appear., ().   Google Scholar

[21]

J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Vol. 1 Elliptic Equations, Res. Notes Math, 106. Pitman, 1985.  Google Scholar

[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. Google Scholar

[23]

W. J. Firey, Shapes of worn stones, Mathematika, 21 (1974), 1-11. doi: 10.1112/S0025579300005714.  Google Scholar

[24]

W. Gangbo and R. J. Mccann, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161. doi: 10.1007/BF02392620.  Google Scholar

[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.  Google Scholar

[26]

E. Goursat, Leçons sur l'Integration des Équations aux Derivées Partielles du Second Order à Deux Variables Indepéndantes, Herman, Paris, 1896. Google Scholar

[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.  Google Scholar

[28]

C. E. Gutiérrez, The Monge-Ampère Equation, Birkhauser, Boston, MA, 2001. doi: 10.1007/978-1-4612-0195-3.  Google Scholar

[29]

R. Hamilton, Worn stones with at sides; in a tribute to Ilya Bakelman, Discourses Math. Appl., 3 (1993), 69-78.  Google Scholar

[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.  Google Scholar

[31]

G. Monge, Sur le Calcul Intégral Des Équations Aux Differences Partielles, Mémoires de l'Académie des Sciences, 1784. Google Scholar

[32]

L. Nirenberg, Monge-Ampère Equations and Some Associated Problems in Geometry, in Proccedings of the International Congress of Mathematics, Vancouver, 1974. Google Scholar

[33]

P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007.  Google Scholar

[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.  Google Scholar

[35]

N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Ration. Mech. Anal., 111 (1990), 153-179. doi: 10.1007/BF00375406.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

C. Villani, Optimal Transport: Old and New, Springer Verlag (Grundlehren der mathematischen Wissenschaften), 2008. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[39]

J. L. Vázquez, A strong Maximum Principle for some quasilinear elliptic equations, Appl Math Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.  Google Scholar

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