no. 5
A Calabi theorem for solutions to the parabolic Monge–Ampère equation with periodic data
Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 5, pp. 1143-1173.

Nous classifions toutes les solutions à

−utdet⁡D2u=f(x) in R−n+1,
où f∈Cα(Rn) est une fonction périodique positive en x. Plus précisément, si u est une solution paraboliquement convexe de l'équation ci-dessus, alors u est la somme d'un polynôme quadratique convexe en x, une fonction périodique en x et une fonction linéaire de t. Cela peut être considéré comme une généralisation du travail de Gutiérrez et Huang en 1998. Et le long de la ligne d'approche dans cet article, nous pouvons traiter d'autres équations paraboliques Monge–Ampère.

We classify all solutions to

−utdet⁡D2u=f(x) in R−n+1,
where f∈Cα(Rn) is a positive periodic function in x. More precisely, if u is a parabolically convex solution to above equation, then u is the sum of a convex quadratic polynomial in x, a periodic function in x and a linear function of t. It can be viewed as a generalization of the work of Gutiérrez and Huang in 1998. And along the line of approach in this paper, we can treat other parabolic Monge–Ampère equations.

DOI : 10.1016/j.anihpc.2017.09.007
Mots clés : Parabolic Monge–Ampère equation, Jörgens–Calabi–Pogorelov type theorem, Periodic data, Local maximum principle, Linearized parabolic Monge–Ampère equation 
@article{AIHPC_2018__35_5_1143_0,
     author = {Zhang, Wei and Bao, Jiguang},
     title = {A {Calabi} theorem for solutions to the parabolic {Monge{\textendash}Amp\`ere} equation with periodic data},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1143--1173},
     publisher = {Elsevier},
     volume = {35},
     number = {5},
     year = {2018},
     doi = {10.1016/j.anihpc.2017.09.007},
     mrnumber = {3813961},
     zbl = {1439.35310},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2017.09.007/}
}
TY  - JOUR
AU  - Zhang, Wei
AU  - Bao, Jiguang
TI  - A Calabi theorem for solutions to the parabolic Monge–Ampère equation with periodic data
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2018
SP  - 1143
EP  - 1173
VL  - 35
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2017.09.007/
DO  - 10.1016/j.anihpc.2017.09.007
LA  - en
ID  - AIHPC_2018__35_5_1143_0
ER  - 
%0 Journal Article
%A Zhang, Wei
%A Bao, Jiguang
%T A Calabi theorem for solutions to the parabolic Monge–Ampère equation with periodic data
%J Annales de l'I.H.P. Analyse non linéaire
%D 2018
%P 1143-1173
%V 35
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2017.09.007/
%R 10.1016/j.anihpc.2017.09.007
%G en
%F AIHPC_2018__35_5_1143_0
Zhang, Wei; Bao, Jiguang. A Calabi theorem for solutions to the parabolic Monge–Ampère equation with periodic data. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 5, pp. 1143-1173. doi : 10.1016/j.anihpc.2017.09.007. http://www.numdam.org/articles/10.1016/j.anihpc.2017.09.007/

[1] Xiong, J.; Bao, J. On Jörgens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge–Ampère equations, J. Differ. Equ., Volume 250 (2011), pp. 367–385 | DOI | MR | Zbl

[2] Caffarelli, L. A localization property of viscosity solutions to the Monge–Ampère equation and their strict convexity, Ann. Math., Volume 131 (1990), pp. 129–134 | DOI | MR | Zbl

[3] Caffarelli, L. Topics in PDEs: The Monge–Ampère Equation, Courant Institute, New York University, 1995 (Graduate course)

[4] Caffarelli, L.; Gutiérrez, C.E. Properties of the solutions of the linearized Monge–Ampère equation, Am. J. Math., Volume 119 (1997), pp. 423–465 | MR | Zbl

[5] Caffarelli, L.; Li, Y.Y. An extension to a theorem of Jorgens, Calabi, and Pogorelov, Commun. Pure Appl. Math., Volume 56 (2003), pp. 549–583 | DOI | MR | Zbl

[6] Caffarelli, L.; Li, Y.Y. A Liouville theorem for solutions of the Monge–Ampère equation with periodic data, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 21 (2004), pp. 97–120 | DOI | Numdam | MR | Zbl

[7] Calabi, E. Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Mich. Math. J., Volume 5 (1958), pp. 105–126 | DOI | MR | Zbl

[8] Cheng, S.Y.; Yau, S.T. Complete affine hypersurfaces I. The completeness of affine metrics, Commun. Pure Appl. Math., Volume 39 (1986), pp. 839–866 | DOI | MR | Zbl

[9] Firey, W.J. Shapes of worn stones, Mathematika, Volume 21 (1974), pp. 1–11 | DOI | MR | Zbl

[10] Gutiérrez, C.E. The Monge–Ampère Equation, Birkhäuser, Boston, 2001 | DOI | MR | Zbl

[11] Gutiérrez, C.E.; Huang, Q. A generalization of a theorem by Calabi to the parabolic Monge–Ampère equation, Indiana Univ. Math. J., Volume 47 (1998) no. 4, pp. 1459–1480 | DOI | MR | Zbl

[12] Gutiérrez, C.E.; Huang, Q. Geometric properties of the sections of solutions to the Monge–Ampère equation, Trans. Am. Math. Soc., Volume 352 (2000) no. 9, pp. 4381–4396 | DOI | MR | Zbl

[13] Gutiérrez, C.E.; Huang, Q. W2,p estimates for the parabolic Monge–Ampère equation, Arch. Ration. Mech. Anal., Volume 159 (2001), pp. 137–177 | MR | Zbl

[14] Huang, Q. Harnack inequality for the linearized parabolic Monge–Ampère equation, Trans. Am. Math. Soc., Volume 351 (1999), pp. 2025–2054 | DOI | MR | Zbl

[15] Jörgens, K. Über die Lösungen der Differentialgleichung rt−s2=1 , Math. Ann., Volume 127 (1954), pp. 130–134 | DOI | MR | Zbl

[16] Jost, J.; Xin, Y.L. Some aspects of the global geometry of entire space-like submanifolds. Dedicated to Shiing-Shen Chern on his 90th birthday, Results Math., Volume 40 (2001), pp. 233–245 | DOI | MR | Zbl

[17] Krylov, N.V. Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation, Sib. Math. J., Volume 17 (1976), pp. 226–236 | DOI | MR | Zbl

[18] Krylov, N.V. On controllable diffusion processes with unbounded coefficients, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 45 (1981), pp. 734–759 (Russian) | MR | Zbl

[19] Krylov, N.V. On the control of a diffusion process until the moment of the first exit from the domain, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 45 (1981), pp. 1029–1048 (in Russian) | MR | Zbl

[20] Krylov, N.V. Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 46 (1982), pp. 487–523 (in Russian) | MR | Zbl

[21] Krylov, N.V.; Safonov, M.V. A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR, Izv., Volume 16 (1981), pp. 151–164 | DOI | MR | Zbl

[22] Li, Y.Y. Some existence results of fully nonlinear elliptic equations of Monge–Ampère type, Commun. Pure Appl. Math., Volume 43 (1990), pp. 233–271 | MR | Zbl

[23] Pogorelov, A.V. On the improper affine hyperspheres, Geom. Dedic., Volume 1 (1972), pp. 33–46 | DOI | MR | Zbl

[24] Teixeira, E.; Zhang, L. Global Monge–Ampère equations with asymptotically periodic data, Indiana Univ. Math. J., Volume 65 (2016), pp. 399–422 | DOI | MR | Zbl

[25] Tso, K. On an Aleksandrov–Bakelman type maximum principle for second-order parabolic equations, Commun. Partial Differ. Equ., Volume 10 (1985), pp. 543–553 | MR | Zbl

[26] Tso, K. Deforming a hypersurface by its Gauss–Kronecker curvature, Commun. Pure Appl. Math., Volume 38 (1985), pp. 867–882 | MR | Zbl

[27] Wang, L. On the regularity theory of fully nonlinear parabolic equations I, Commun. Pure Appl. Math., Volume 45 (1992), pp. 27–76 | MR | Zbl

[28] Wang, R.; Wang, G. On existence, uniqueness and regularity of viscosity solutions for the first initial boundary value problems to parabolic Monge–Ampère equation, Northeast. Math. J., Volume 8 (1992) no. 4, pp. 417–446 | MR | Zbl

[29] W. Zhang, J. Bao, B. Wang, An extension of Jörgens–Calabi–Pogorelov theorem to parabolic Monge–Ampère equation, to appear in Calc. Var. Partial Differ. Equ. | MR

[30] W. Zhang, Regularity results for solutions to parabolic Monge–Ampère equation with VMO ψ data, submitted for publication.

Cité par Sources :