The dirichlet problem for a complex Monge-Amp�re equation

Counterexamples to regularity for the complex Monge-Amp�re equation

“…By [4] or [13], u is unique, so it follows that u(z, w) = u(-z, w), which contradicts the Proposition. …”

“…The best regularity obtained in [4] occurs when G=IB" is the unit ball, ~o~C2(#G) and fl/"~C2(G): in this case u~CI,I(IB"). Presumably the solution is C 1,1 for more general domains.…”

“…It has been shown in [4] that there exists a unique generalized solution for the Dirichlet problem u plurisubharmonic (02u~ det \OziO2J] =f>0 on G…”

The range of the complex Monge-Ampere operator II

“…The Lebesgue measure will be denoted throughout by λ and the Kähler form in C n by β := dd c |z| 2 = (i/2)dz j ∧ dz j . For locally bounded psh functions u k , the wedge product dd c u 1 ∧ ··· ∧ dd c u N is a well defined positive current (see [BT1]). In particular, it is continuous with respect to monotone sequences of functions appearing in it.…”

On the fundamental solution for the real Monge-Ampère operator

“…However, as shown by Bedford and Taylor [2] (see also [11]), M c u can be de¢ned as a nonnegative Borel measure for all continuous psh functions in such a way that M c u j converges weakly to M c u if u j converges uniformly to u. This determines M c u uniquely since every continuous psh function can be locally uniformly approximated by smooth psh functions.…”