Generalized solution of the first boundary value problem for parabolic Monge-Ampère equation.

*(English)*Zbl 0990.35034In 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.

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.

Reviewer: John Urbas (Canberra)