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Nonlocal refuge model with a partial control
On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions
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 |
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] | |
[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. |
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) |
[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] | |
[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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