On Dirichlet's principle and problem (Q2890398)

Let \({\mathcal E}_1\) be the Cegrell class of plurisubharmonic functions \(u\) that are limits of decreasing sequences of test plurisubharmonic functions \(u_j\) with uniformly bounded energies NEWLINE\[NEWLINEe_1(u_j)=\int_\Omega |u_j|(dd^cu_j)^n,NEWLINE\]NEWLINE where \(\Omega\) is a bounded, hyperconvex domain in \({\mathbb C}^n\). In [\textit{U. Cegrell}, Acta Math. 180, No. 2, 187--217 (1998; Zbl 0926.32042)], the Monge-Ampère measures \(\mu=(dd^cu)^n\) of such functions \(u\) were characterized as those satisfying NEWLINE\[NEWLINE\int_\Omega|\phi|\,d\mu\leq \text{const}\cdot e_1(\phi)^{1/n+1}NEWLINE\]NEWLINE (Theorem A). Using this, in [\textit{L. Persson}, Ark. Mat. 37, No. 2, 345--356 (1999; Zbl 1045.34056)] solutions \(u\) to \((dd^cu)^n=\mu\) were described as minimizers for the functional NEWLINE\[NEWLINEw\mapsto {\mathcal J}_\mu(w)={1\over n+1}e_1(w)-\|w\|_{L_1(\Omega)}NEWLINE\]NEWLINE (Theorem B).NEWLINENEWLINEIn the present paper, a variational approach to these problems in the spirit of [\textit{R. J. Berman, S. Boucksom, V. Guedj} and \textit{A. Zeriahi}, ``A variational approach to complex Monge-Ampère equations '', Preprint, \url{arXiv:0907.4490}] is treated, which gives a new proof of Theorem B, independent of Theorem A. Moreover, the authors then use Theorem B in order to prove Theorem A.