Beltrami Equations on Rossi Spheres

On Planar Beltrami Equations and Hölder Regularity

We provide estimates for the Hölder exponent of solutions to the Beltrami equation ∂f = μ∂f + ν∂f , where the Beltrami coefficients μ, ν satisfy ‖|μ|+ |ν|‖∞ < 1 and =(ν) = 0. Our estimates depend on the arguments of the Beltrami coefficients as well as on their moduli. Furthermore, we exhibit a class of mappings of the “angular stretching” type, on which our estimates are actually attained.

On Beltrami equations and Hölder regularity

We estimate the Hölder exponent α of solutions to the Beltrami equation ∂f = μ∂f , where the Beltrami coefficient satisfies ‖μ‖∞ < 1. Our estimate improves the classical estimate α ≥ ‖Kμ‖ , where Kμ = (1 + |μ|)/(1 − |μ|), and it is sharp, in the sense that it is actually attained in a class of mappings which generalize the radial stretchings. Some other properties of such mappings are also prov...

Solving Beltrami Equations by Circle Packing

We use Andreev-Thurston's theorem on the existence of circle packings to construct approximating solutions to the Beltrami equations on Riemann surfaces. The convergence of the approximating solutions on compact subsets will be shown. This gives a constructive proof of the existence theorem for Beltrami equations.

Beltrami Equations with Coefficient in the Sobolev Space

Abstract We study the removable singularities for solutions to the Beltrami equation ∂f = μ∂f , where μ is a bounded function, ‖μ‖∞ ≤ K−1 K+1 < 1, and such that μ ∈ W 1,p for some p ≤ 2. Our results are based on an extended version of the well known Weyl’s lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly s...

Boundary Integral Equations for the Laplace–Beltrami Operator