an:01891254
Zbl 1014.35042
Chen, Li; Wang, Guanglie; Lian, Songzhe
Convex-monotone functions and generalized solution of parabolic Monge--Amp??re equation
EN
J. Differ. Equations 186, No. 2, 558-571 (2002).
00091146
2002
j
35K55 35D05 35D10
H??lder continuity in \(t\)
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-Amp??re 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 H??lder continuity in \(t\) of \(u\) if \(\varphi(t,x_0)\) is H??lder 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.
John Urbas (Bonn)
Zbl 0990.35034