A variational approach to complex Hessian equations in C-n

Journal article, 2015

Let Omega be an m-hyperconvex domain of C-n and beta be the standard lathier form in C-n. We introduce finite energy classes of ni,subharmonic functions of Cegrell type, epsilon(p)(m)(Omega), p > 0 and F-m(Omega). Using a variational method we show that the degenerate complex Hessian equation (ddc phi)(m) A beta(n-m) = mu has a unique solution in epsilon(1)(m)(Omega) if and only if every function in a epsilon(1)(m) is integrable with respect to mu. If has finite total mass and does not charge m-polar sets, then the equation has a unique solution in F-m (Omega). (C) 2015 Elsevier Inc. All rights reserved.