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Generalized solution of the first boundary value problem for parabolic Monge-Ampère equation. (English) Zbl 0990.35034
In 1993 R. Wang and G. Wang [J. Partial Differ. Equations 6, 273-254 (1993; Zbl 0811.35053)] introduced a measure theoretic notion of generalized solution for the problem $-u_t\det D^2_xu = f \quad\text{in}\quad Q=\Omega\times(0,T], \qquad u=\varphi \quad\text{on}\quad \partial_pQ, (*)$ where $$\Omega$$ is a bounded convex domain in $${\mathbb R}^n$$ and $$\partial_pQ$$ denotes the parabolic boundary of $$Q$$. They also proved the existence and uniqueness of such solutions. Similar results were also obtained by J. Spiliotis [Nonlinear Stud. 4, 233-255 (1997; Zbl 0883.35058)]. The notion of generalized solution is defined for functions $$u\in C(\overline Q)$$ that are convex with respect to $$x$$ and nonincreasing with respect to $$t$$, and is based on an observation of K. Tso [Commun. Partial Differ. Equations 10, 543-553 (1985; Zbl 0581.35027)] that $$-u_t\det D^2_xu$$ is the Jacobian of a Legendre type transformation. This permits the development of a theory that parallels the well-known Aleksandrov theory of generalized solutions of elliptic Monge-Ampère equations.
Here the authors prove the existence and uniqueness of generalized solutions of $$(*)$$ under somewhat different hypotheses than required in previous work. Specifically, the assumptions are that $$f\in L^\infty(Q)$$ is nonnegative, $$\varphi\in C(\partial_pQ)$$, $$\varphi(\cdot,0)$$ is convex on $$\overline\Omega$$, $$\varphi(x_0,\cdot)\in C^\alpha([0,T])$$ for all $$x_0\in\partial\Omega$$, and finally, there is a strict generalized supersolution $$u_\varphi\in C(\overline Q)$$ of $$(*)$$.
The assumptions on $$f$$ and $$\varphi$$ are weaker than required previously; however the existence of $$u_\varphi$$ was not assumed in previous work. The authors point out that this condition can be omitted, and that a full exposition of the existence of a generalized supersolution of $$(*)$$ will be given in future work.

##### MSC:
 35D05 Existence of generalized solutions of PDE (MSC2000) 35K55 Nonlinear parabolic equations 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations