An essentially unique homeomorphic solution to the Beltrami equation with measurable coefficients was found in 1930s by Morrey. The most well-known proof from 1960s uses theory of Calderón–Zygmund and singular integral operators $$L^p(\mathbb {C})$$ . We will present an alternative method solve using Hodge star operator standard elliptic PDE theory. also discuss a different prove regularity sol...

We present a class of asymptotic-preserving (AP) schemes for the nonhomogeneous Fokker–Planck–Landau (nFPL) equation. Filbet and Jin [16] designed a class of AP schemes for the classical Boltzmann equation, by penalization with the BGK operator, so they become efficient in the fluid dynamic regime. We generalize their idea to the nFPL equation, with a different penalization operator, the Fokker...