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Convex-monotone functions and generalized solution of parabolic Monge–Ampère equation. (English) Zbl 1014.35042
The authors extend and improve their previous results [J. Partial Differ. Equations 14, 149-162 (2001; Zbl 0990.35034)] on the existence of generalized solutions of the first initial-boundary value problem for the parabolic Monge-Ampère equation \begin{aligned} -u_t \det D^2u & = f(x,t) \quad\text{in}\quad Q=\Omega\times(0,T],\\ u & = \varphi(x,t) \quad\text{on}\quad\partial_p Q.\end{aligned}\tag{*} The paper contains two main results. The first is the Hölder continuity in $$t$$ of $$u$$ if $$\varphi(t,x_0)$$ is Hölder continuous in $$t$$ for each $$x_0\in \partial\Omega$$. The second is a geometric characterization of the convex-monotone solution $$U$$ of (*) with $$f\equiv 0$$ as follows: $U(x,t) = \sup \{ l(x): l \text{ is affine and } l(x)\leq\varphi(x,0) \text{ in }\Omega,$ $l(x) \leq \varphi(x,t) \text{ on } \partial\Omega \}, \qquad (x,t)\in \bar Q.$ Using these results the authors establish the existence of generalized solutions of (*) under somewhat weaker assumptions than required in previous work.
Reviewer: John Urbas (Bonn)

##### MSC:
 35K55 Nonlinear parabolic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000)
##### Keywords:
Hölder continuity in $$t$$
Full Text:
##### References:
 [1] Aleksandrov, A.D., Dirichlet’s problem for the equation det||Zij||=φ, Vestn. leningr. un-ta. ser. matematika, mekhanika, astronomiya, 1, 1, 5-24, (1958) · Zbl 0114.30202 [2] I. Ya. Bakel’man, The Dirichlet problem for the elliptic n-dimensional Monge-Ampère equations and related problems in the theory of quasilinear equations, Proceedings of Seminar on Monge-Ampère Equations and Related Topics, Firenze 1980, Instituto Nazionale di Alta Matematica, Roma, 1982, pp. 1-78. [3] Chen, L., Existence and uniqueness of generalized solution for parabolic monge – ampère equation, Acta. scien. natur. univer. jilin., 4, 1-9, (2000), (in Chinese) · Zbl 0988.35511 [4] Chen, L.; Guanglie, Wang; Songzhe, Lian, Generalized solution of the first boundary value problem for parabolic type monge – ampère equation, J. partial differential equations, 14, 2, 149-162, (2001) · Zbl 0990.35034 [5] L. Chen, Parabolic type Monge-Ampère equation with zero initial-boundary value, submitted for publication. [6] Cheng, S.Y.; Yau, S.T., On the regularity of the monge – ampère equation, Comm. pure appl. math., 30, 41-68, (1977) · Zbl 0347.35019 [7] Kaising, Tso, On an aleksandrov – bakel’man type maximum principle for second order parabolic equation, Comm. partial differential equations, 10, 543-553, (1985) · Zbl 0581.35027 [8] Pogorelov, A.V., The Minkowski multidimensional problem, (1978), Wiley New York [9] Rauch, J.; Taylor, B.A., The Dirichlet problem for the multidimensional monge – ampère equation, Rocky mount. J. math., 7, 2, 345-364, (1977) · Zbl 0367.35025 [10] Wang, Rouhuai; Wang, Guanglie, The geometric measure theoretical characterization of viscosity solutions to parabolic monge – ampère equation, J. partial differential equations, 3, 237-254, (1993) · Zbl 0811.35053
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