Mid Sweden University

In this note we prove a product property for the pluricomplex energy, and then give some applications.

It has previously been shown that a surface current density J on a closed surface § of class C1 in R3 induces a static magnetic ¯eld BJ in R3 n §, which has some discontinuity along §. In this note, we represent BJ by use of jump theorem for harmonic forms in the case where § is of class C!.We then apply this result to prove the existence of a surface current density J, which induces the nonzero magnetic ¯eld BJ such that BJ ´ 0 inside (or outside) of the domain bounded by § in R3. This has previously been called the equilibrium magnetic ¯eld for §.

We show that if a decreasing sequence of subharmonic functions converges to a function in W-loc(1,2) then the convergence is in W-loc(1,2) .

We study the convergence of sequences of Monge-Ampère measures (dd c u j ) n where (u j ) is a given sequence of plurisubharmonic functions. Our main theorem is about approximation by multipole pluricomplex Green functions.

We consider differences of plurisubharmonic functions in the energy classF as a linear space, and equip this space with a norm, depending on the generalized complex Monge-Ampère operator, turning the linear space into a Banach space δF. Fundamental topological questions for this space is studied, and we prove that δF is not separable. Moreover we investigate the dual space. The study is concluded with comparison between δF and the space of delta-plurisubharmonic functions, with norm depending on the total variation of the Laplace mass, studied by the first author in an earlier paper

Boundary values of plurisubharmonic functions and a general Dirichlet problem for the Complex Monge-Amp`ere operator. Research reports, No 1, 2005. Mid Sweden Univ.