We consider the problem of constructing quasi-conformal mappings between surfaces by solving Beltrami equations. This is of great importance for shape registration. In the physical world, most surface deformations can be rigorously modeled as quasi-conformal maps. The local deformation is characterized by a complex-value function, Beltrami coefficient, which describes the deviation from conform...

We present some recent sharp estimates for the Hölder exponent of solutions of linear second order elliptic equations in divergence form with measurable coefficients. We apply such results to planar Beltrami equations, and we exhibit a mapping of the “angular stretching” type for which our estimates are attained.

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.

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...

The eigenvalue problem of the Laplace-Beltrami operators on curved surfaces plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve this operator. In this note we shall combine the local tangential lifting (LTL) method with the configuration equation to develop a new effective and convergent algori...

The physical situation which has initiated this research is that of a dielectric particle with electric charges on its surface, placed in electric field. Here, the diffusion equation of the charges is coupled with the Maxwell equations. There is an analytical solution of this system of equation [1] which involves some functional calculus with operators, in particular with Laplace-Beltrami opera...

The convergence problem of the Laplace-Beltrami operators plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve the operator. In this note we present a new effective and convergent algorithm to compute discrete Laplace-Beltrami operators acting on functions over surfaces. We prove a convergence t...