Weak Solutions to the Complex Hessian Type Equations for Arbitrary Measures
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Complex Analysis and Operator Theory
Weak Solutions to the Complex Hessian Type Equations for Arbitrary Measures Hichame Amal1 · Saïd Asserda2 · Ayoub El Gasmi2 Received: 17 February 2020 / Accepted: 1 October 2020 © Springer Nature Switzerland AG 2020
Abstract We study the complex equations of Hessian type (dd c u)m ∧ β n−m = F(u, .)dμ, where μ is a positive Borel measure defined on an m-hyperconvex domain of Cn , m is an integer such that 1 ≤ m ≤ n and β := dd c |z|2 is the standard kähler form in Cn . We show that, under some regularity conditions on the density F, if this equation admits a (weak) subsolution in , then it admits a (weak) solution with a prescribed least maximal m-subharmonic majorant in . Keywords Complex Hessian equations · m-subharmonic functions · Dirichlet problem · m-hyperconvex domain · Maximal m-subharmonic function Mathematics Subject Classification 32U05 · 32U15 · 32U40 · 32W50
Communicated by Ronen Peretz.
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Ayoub El Gasmi [email protected] Hichame Amal [email protected] Saïd Asserda [email protected]
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Department of mathematics, Laboratory LaREAMI, Regional Centre of trades of education and training, Kenitra, Morocco
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Faculty of sciences, Department of mathematics, Ibn tofail university, Kenitra, Morocco 0123456789().: V,-vol
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1 Introduction Studying the complex Hessian equations has attracted the attention of many researchers in this last decarde. This is due to its importance in Pluripotential theory, Complex Analysis and Kähler Geometry. They appear in many geometric problems, including the J -flow [24] and quaternionic geometry [1]. They are also important from PDE’s point of view as they can be considered as non linear PDE’s of second order which interpolate between (linear) complex Poisson equations (m = 1) and (non linear) complex Monge-Ampère equations (m = n). Let be an m-hyperconvex domain of Cn and μ be a nonnegative Borel measure defined on . We consider the following complex equations of Hessian type Hm (u) := (dd c u)m ∧ β n−m = F(u, .)dμ,
(1)
where F : R × → [0, +∞) is a measurable function and u is an m-subharmonic function belonging to the domain of definition of the Hessian operator Hm . We mention polynomial of that when u is smooth, then Hm (u) is the m th elementary symmetric the eigenvalues of the complex Hessian matrix ∂ 2 u/∂z j ∂ z¯ k . In the degenerate case, Eq. (1) should be understood in the sense of currents. The case m = n, which corresponds to the complex equation of Monge–Ampère type (2) (dd c u)n = F(u, .)dμ, was first treated by E. Bedford and A. Taylor in [2], who constructed a weak solution on bounded strictly pseudoconvex domains in C2 . (see also [7]). When the measure μ is finite, vanishing on pluripolar sets, and F is non-negative, continuous in the first variable, upper bounded by a function from L 1 (dμ), Cegrell and Kołodjiez proved in [5] the existence of a solution to (2) in the class of Cegrell F a (, f ). A generalization of the corresponding result in [5] was ma
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