Viscosity solutions to degenerate complex monge‐ampère equations

By Philippe Eyssidieux, Vincent Guedj, Ahmed Zeriahi. 2010

106 citations recorded by the source corpus.

Degenerate complex Monge‐Ampère equations on compact Kähler manifolds have recently been studied intensively using tools from pluripotential theory. We develop an alternative approach based on the concept of viscosity solutions and systematically compare viscosity concepts with pluripotential theoretic ones.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1e370cc188eff778ecce6815b13c6d9ddd412200

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Hessian measures on m-polar sets and applications to the complex Hessian equations (2017, methodology): “A viscosity approach to this equation has been developed in [11] which is a generalization of results in [12,13].”
On the $C^{2,α}$-regularity of the complex Monge–Ampère equation (2012, background): “referee of this paper, it suffices to assume that u is a weak solution to the equation (viscosity solution or pluripotential solution, see [9, 17] for the definition of viscosity solution to the complex Monge–Ampère) and ‖Δu‖L∞ < Λ.”
Plurisubharmonic envelopes and supersolutions (2017, methodology): “For each β>0, let ϕβ be the unique function in PSH(D) ∩C0(D¯) such that (3.7) (ddcϕβ)n= eβ(ϕβ−u)fdV in D with boundary values ui.e. ϕβ= uin ∂D[BT76]. Using the local viscosity comparison principle [EGZ11], we deduce that ϕβ≤ uin D¯ for any β>0. Arguing as in Step 2 of the proof of Theorem 3.5,”
The Kähler–Ricci flow through singularities (2009, background): “Its generalization was independently carried out in [11,50] for the case when [ω] is rational, big and semi-ample and in [13,15] for the general case when [ω] is big and semi-positive.”