Abstrac t . We consider several results, each o f which uses some type o f " L 2' ' es t imate to provide information about harmonic measure on planar domains. The first gives an a.e. characterization of tangent points o f a curve in terms of a certain geometric square function. Our next result is an LP est imate relating the derivative of a conformal mapping to its Schwarzian derivative. One c...

The Harmonic Mapping Problem asks when there exists a harmonic homeomorphism between two given domains. It arises in the theory of minimal surfaces and in calculus of variations, specifically in hyperelasticity theory. We investigate this problem for doubly connected domains in the plane, where it already presents considerable challenge and leads to several interesting open questions.

We generalize Martio’s paper [14]. Indeed the problem studied in this paper is under which conditions on a homeomorphism f between the unit circle S1 := {z : |z| = 1} and a fix convex Jordan curve γ the harmonic extension of f is a quasiconformal mapping. In addition, we give some results for some classes of harmonic diffeomorphisms. Further, we give some results concerning harmonic quasiconfor...

A continuous function f(x+ iy) = u(x, y) + iv(x, y) defined in a domain D ⊂ C (Complex plane) is harmonic in D if u and v are real harmonic in D. Clunie and Shiel-Small [3] showed that in a simply connected domain such functions can be written in the form f = h+g, where both h and g are analytic. We call h the analytic part and g, the co-analytic part of f . Let w(z) = g ′(z) h′(z) be the dilat...

sectionHarmonic Volumetric Parameterization using MFS After the decomposition of a given object M , we get a set of star shapes {Mi}, each region being guarded by a point gi. Then we can parameterize each subregion onto a solid sphere. A key property that we will show shortly is that such a harmonic map is guaranteed to be bijective. The harmonic map can be computed using the method of fundamen...