Ebook: The Beltrami Equation: A Geometric Approach

This book is devoted to the Beltrami equations that play a significant role in Geometry, Analysis and Physics and, in particular, in the study of quasiconformal mappings and their generalizations, Riemann surfaces, Kleinian groups, Teichmuller spaces, Clifford analysis, meromorphic functions, low dimensional topology, holomorphic motions, complex dynamics, potential theory, electrostatics, magnetostatics, hydrodynamics and magneto-hydrodynamics.

The purpose of this book is to present the recent developments in the theory of Beltrami equations; especially those concerning degenerate and alternating Beltrami equations. The authors study a wide circle of problems like convergence, existence, uniqueness, representation, removal of singularities, local distortion estimates and boundary behavior of solutions to the Beltrami equations. The monograph contains a number of new types of criteria in the given problems, particularly new integral conditions for the existence of regular solutions to the Beltrami equations that turned out to be not only sufficient but also necessary.

The most important feature of this book concerns the unified geometric approach based on the modulus method that is effectively applied to solving the mentioned problems. Moreover, it is characteristic for the book application of many new concepts as strong ring solutions, tangent dilatations, weakly flat and strongly accessible boundaries, functions of finite mean oscillations and new integral conditions that make possible to realize a more deep and refined analysis of problems related to the Beltrami equations. Mastering and using these new tools also gives essential advantages for the reader in the research of modern problems in many other domains. Every mathematics graduate library should have a copy of this book.​

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This volume emphasises studies related toclassical Stefan problems. The term "Stefan problem" isgenerally used for heat transfer problems with phase-changes suchas from the liquid to the solid. Stefan problems have somecharacteristics that are typical of them, but certain problemsarising in fields such as mathematical physics and engineeringalso exhibit characteristics similar to them. The term``classical" distinguishes the formulation of these problems fromtheir weak formulation, in which the solution need not possessclassical derivatives. Under suitable assumptions, a weak solutioncould be as good as a classical solution. In hyperbolic Stefanproblems, the characteristic features of Stefan problems arepresent but unlike in Stefan problems, discontinuous solutions areallowed because of the hyperbolic nature of the heat equation. Thenumerical solutions of inverse Stefan problems, and the analysis ofdirect Stefan problems are so integrated that it is difficult todiscuss one without referring to the other. So no strict line ofdemarcation can be identified between a classical Stefan problemand other similar problems. On the other hand, including everyrelated problem in the domain of classical Stefan problem wouldrequire several volumes for their description. A suitablecompromise has to be made.The basic concepts, modelling, and analysis of the classicalStefan problems have been extensively investigated and there seemsto be a need to report the results at one place. This bookattempts to answer that need. Within the framework of theclassical Stefan problem with the emphasis on the basic concepts,modelling and analysis, it tries to include some weaksolutions and analytical and numerical solutions also. The mainconsiderations behind this are the continuity and the clarity ofexposition. For example, the description of some phase-fieldmodels in Chapter 4 arose out of this need for a smooth transitionbetween topics. In the mathematical formulation of Stefanproblems, the curvature effects and the kinetic condition areincorporated with the help of the modified Gibbs-Thomson relation.On the basis of some thermodynamical and metallurgicalconsiderations, the modified Gibbs-Thomson relation can bederived, as has been done in the text, but the rigorousmathematical justification comes from the fact that this relationcan be obtained by taking appropriate limits of phase-fieldmodels. Because of the unacceptability of some phase-field modelsdue their so-called thermodynamical inconsistency, some consistentmodels have also been described. This completes the discussion ofphase-field models in the present context.Making this volume self-contained would require reporting andderiving several results from tensor analysis, differentialgeometry, non-equilibrium thermodynamics, physics and functionalanalysis. The text is enriched with appropriatereferences so as not to enlarge the scope of the book. The proofsof propositions and theorems are often lengthy and different fromone another. Presenting them in a condensed way may not be of muchhelp to the reader. Therefore only the main features of proofsand a few results have been presented to suggest the essentialflavour of the theme of investigation. However at each place,appropriate references have been cited so that inquisitivereaders can follow them on their own.Each chapter begins with basic concepts, objectives and thedirections in which the subject matter has grown. This is followedby reviews - in some cases quite detailed - of published works. In awork of this type, the author has to make a suitable compromisebetween length restrictions and understandability Introduction -- Preliminaries -- The Classical Beltrami Equation [µ][infinity symbol]<1 -- The Degenerate Case -- BMO- and FMO-Quasiconformal Mappings -- Ring Q-Homeomorphisms at Boundary Points -- Strong Ring Solutions of Beltrami Equations -- On the Dirichlet Problem for Beltrami Equations -- On the Beltrami Equations with Two Characteristics -- Alternating Beltrami Equation