By Kari Astala

This publication explores the newest advancements within the conception of planar quasiconformal mappings with a specific concentrate on the interactions with partial differential equations and nonlinear research. It offers an intensive and sleek method of the classical idea and offers vital and compelling functions throughout a spectrum of arithmetic: dynamical structures, singular critical operators, inverse difficulties, the geometry of mappings, and the calculus of adaptations. It additionally offers an account of modern advances in harmonic research and their purposes within the geometric thought of mappings.

The publication explains that the lifestyles, regularity, and singular set buildings for second-order divergence-type equations--the most crucial classification of PDEs in applications--are decided via the maths underpinning the geometry, constitution, and size of fractal units; moduli areas of Riemann surfaces; and conformal dynamical structures. those issues are inextricably associated through the idea of quasiconformal mappings. extra, the interaction among them permits the authors to increase classical effects to extra normal settings for wider applicability, offering new and sometimes optimum solutions to questions of life, regularity, and geometric houses of recommendations to nonlinear platforms in either elliptic and degenerate elliptic settings.

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**Extra info for Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane**

**Example text**

1]. Any map ˆ →C ˆ has a well-deﬁned degree independent of a chosen point, so we simply g:C write deg(g) for such maps. 8. DEGREE AND JACOBIAN 33 These two notions are equivalent for suﬃciently regular maps and are more generally proven to be equivalent via approximation. The topological deﬁnition has the virtue of clearly deﬁning a homotopy invariant notion, while the analytic deﬁnition is computable. We recall the basic facts concerning degree. 1. Let f : Ω → C be continuous, p ∈ C \ f (∂Ω) and f −1 (p) compact.

Later we shall use this metric of D, together with the uniformization theorem, to deﬁne the hyperbolic metric of an arbitrary planar domain (other than the full plane or the punctured plane). This is quite a technical and deep fact from complex analysis, in fact, one of the CHAPTER 2. CONFORMAL GEOMETRY 16 most important results and themes of 19th century mathematics, and we shall not oﬀer proofs here. Mostly, we shall need only the hyperbolic metric on the disk and the triply punctured sphere.

14. B0 For locally univalent mappings (so with the further assumption f (z) = 0, z ∈ D), the similarly deﬁned constant is denoted B∞ , and it is known that B∞ > 12 . The exact determination of B0 remains a famously diﬃcult open problem. 4719 . .. 10. 1. An elementary proof using classical complex analysis can be found in [6]. 15. If f is holomorphic in D, if it is continuous in D and if f ∂D is homotopic to the identity relative to C \ {0}, then f (z0 ) = 0 at precisely one point z0 ∈ D. 10 Distortion by Conformal Mapping From a historical perspective, the notion of a quasiconformal mapping has its roots in the study of the geometric properties of conformal mappings.