A variational approach to complex Hessian equations in Cn

Abstract

Let Ω be an m-hyperconvex domain of C n and β be the standard Kähler form in C n . We introduce finite energy classes of m-subharmonic functions of Cegrell type, E p m (Ω), p > 0 and F m (Ω). Using a variational method we show that the degenerate complex Hessian equation (dd c ϕ) m ∧ β n−m = μ has a unique solution in E 1 m (Ω) if and only if every function in E 1 m (Ω) is integrable with respect to μ. If μ has finite total mass and does not charge m-polar sets, then the equation has a unique solution in F m (Ω).

Poincaré- and Sobolev- Type Inequalities for Complex m-Hessian Equations

“…Next, we shall recall the function classes that are of our interest. As said in the introduction we shall use the following notations: In [32,33], it was proved that for u ∈ E p,m (Ω) the complex Hessian operator, H m (u), is well-defined, and…”

“…This example will be used in our proof of the Sobolev type inequality (Theorem 5.4). In this special case Theorem 3.2 was proved by Cegrell, see [18], and Lu [32,33]. Inspired by Ψ 1 , we define for 1 ≤ l ≤ n the following:…”

“…( [6,18,33,34]). For μ ∈ δM p,m let now u + , u − ∈ E p,m (Ω) be the unique msubharmonic functions such that…”

Modulus of continuity of solutions to complex Hessian equations

“…We first prove the statement for p = q = 1. Thanks to the Cauchy-Schwarz inequality (see [Lu13c]), we have…”

On a Family of Quasimetric Spaces in Generalized Potential Theory

“…where H m (u) = (dd c u) m ∧ β n−m is the complex Hessian operator. (ii) E p,m ( ) if, there exists a decreasing sequence, {u j }, u j ∈ E 0,m ( ), that converges pointwise to u on , as j tends to ∞, and Proof See, e.g., Lu [29,30], and Nguy ễn [33]. For the case when m = n, see [3,15,17,35].…”