ON G-COMPACTNESS OF THE BELTRAMI OPERATORS

摘要

Quasiconformal mappings continue to play an important role in the modern theory of partial differential equations (PDEs). Here we shall focus on elliptic equations in two variables for which quasiconformal analysis suites very well. There are a number of central themes about convergence of differential operators. While quasiconformal mappings have been widely acknowledged in this context there are some ideas still unexplored. Here, using the normal family arguments we shall develop a fairly general method of constructing the G-limits of some differential operators. The very definition of a G-convergence is concerned with the second order elliptic equations. Nowadays this concept evolves much further to include general PDEs. Let us briefly discuss such a general framework. Roughly speaking, G-compactness is based on analysis of the solutions to the limit equation. In many situations, it has very little to do with convergence of the coefficients.