Weak Solutions to the Complex m-Hessian Equation on Open Subsets of $${{\mathbb {C}}}^{n}$$
Tóm tắt
In this paper, we prove the existence of weak solutions to the complex m-Hessian equations in the class
$${\mathcal {D}}_{m}(\Omega )$$
on an open subset
$$\Omega $$
of
$${\mathbb {C}}^n$$
. In the end of the paper we give an example shows that in the unit ball
$${\mathbb {B}}^{2}(0,1)\subset {\mathbb {C}}^{2}$$
the complex Monge-Ampère equation
$$(dd^{c} .)^{2}=\mu $$
is solvable but the complex Hessian equation
$$H_{1}(.)=\mu $$
has not any weak solutions where
$$\mu $$
is a nonnegative Radon measure on
$${\mathbb {B}}^{2}(0,1)$$
.
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