Weak solutions to degenerate complex Monge–Ampère flows I
Résumé
The authors apply the method of viscosity solutions to the parabolic version of the complex Monge-Amp\`ere equation in a strictly pseudoconvex domain $\Omega$ that is, $$ \exp (\partial _t \varphi _t +F(t,z, \varphi _t ))\mu (z) - (dd^c \varphi _t )^n =0 \ \ \text{ in } [0,T]\times \Omega , $$ with $F$ continuous and nondecreasing in the last variable, $\mu$ a bounded continuous nonnegative volume form, and $\varphi _t =\varphi (t, \cdot )$ the unknown function. First, they review the results of Carandall, Ishii, Lions and others in classical PDE viscosity theory. Then, the following results are shown. Any bounded subsolution is plurisubharmonic on space slices. The parabolic comparison principle is valid. Under certain assumptions on the initial data (satisfied when $\mu >0$) the solution of the Cauchy-Dirichlet problem exists for all time, and it is an upper envelope of all subsolutions. Finally, with the same assumptions and $F$ independent of time, $\varphi _t $ converges to the solution of the related elliptic equation as $t\to\infty$.