This book deals with the tangential oblique derivative problem for second order linear and non-linear elliptic and parabolic operators. In a large survey a lot of the most interesting results obtained during the last 30 years are proposed. Historically, the problem was stated first by Poincar? when studying the tides, but the same problem arises in the theory of Brownian motion, too. The main difficulties in investigating this problem are due to the fact that at the points of tangency between the vector field, representing the boundary operator, and the boundary of the domain the Lopatinskii condition is failed and boundary value problems with infinite dimensional kernel or cokernel can appear. By using subelliptic type estimates for pseudodifferential operators in Sobolev and H?lder spaces many interesting results have been proved for linear problems during the last 30 years. The authors propose for the first time an investigation of the degenerate oblique derivative problem for semilinear elliptic and parabolic operators. To do this, they use subelliptic estimates (Egorov, H?rmander, Tr?ves, Winzell, Guan, Sawyer) and the Leray-Schauder fixed point principle. In this way theorems on existence, uniqueness and regularity of the classical solutions in H?lder classes are derived. In a lot of cases considered the coefficients are not infinitely smooth, and the set of degeneration of the problem is a rather massive one, i.e., it is not obliged to be a submanifold of the boundary and can have positive measure.

The aim of our book is the investigation of the behavior of strong and weak solutions to the regular oblique derivative problems for second order elliptic equations, linear and quasi-linear, in the neighborhood of the boundary singularities. The main goal is to establish the precise exponent of the solution decrease rate and under the best possible conditions. The question on the behavior of solutions of elliptic boundary value problems near boundary singularities is of great importance for its many applications, e.g., in hydrodynamics, aerodynamics, fracture mechanics, in the geodesy etc. Only few works are devoted to the regular oblique derivative problems for second order elliptic equations in non-smooth domains. All results are given with complete proofs. The monograph will be of interest to graduate students and specialists in elliptic boundary value problems and their applications.

This book gives an up-to-date exposition on the theory of oblique derivative problems for elliptic equations. The modern analysis of shock reflection was made possible by the theory of oblique derivative problems developed by the author. Such problems also arise in many other physical situations such as the shape of a capillary surface and problems of optimal transportation. The author begins the book with basic results for linear oblique derivative problems and work through the theory for quasilinear and nonlinear problems. The final chapter discusses some of the applications. In addition, notes to each chapter give a history of the topics in that chapter and suggestions for further reading.

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This book gives an up-to-date exposition on the theory of oblique derivative problems for elliptic equations. The modern analysis of shock reflection was made possible by the theory of oblique derivative problems developed by the author. Such problems also arise in many other physical situations such as the shape of a capillary surface and problems of optimal transportation. The author begins the book with basic results for linear oblique derivative problems and work through the theory for quasilinear and nonlinear problems. The final chapter discusses some of the applications. In addition, notes to each chapter give a history of the topics in that chapter and suggestions for further reading.

Boundary problems constitute an essential field of common mathematical interest, they lie in the center of research activities both in analysis and geometry. This book encompasses material from both disciplines, and focuses on their interactions which are particularly apparent in this field. Moreover, the survey style of the contributions makes the topics accessible to a broad audience with a background in analysis or geometry, and enables the reader to get a quick overview.

The Oblique Derivative Problem (ODP), introduced and first studied by Henry Poincaré, is one of the classical problems not only in the theory of Partial Differential Equations but also in Mathematical Physics. This is the first monograph, written by one of the leading scientists in this area, which is completely devoted to the ODP. All main results in this field are described with full proofs based on modern techniques. The book contains a lot of results that have been unknown to a wide audience till now. A special chapter containing extensive material from geometry, functional analysis and differential equations, which is used in the proofs, makes the book self–contained to a large extent. A short Appendix containig open problems will stimulate the reader to further research in this area.

In the recent half-century, many mathematicians have investigated various problems on several equations of mixed type and obtained interesting results, with important applications to gas dynamics. However, the Tricomi problem of general mixed type equations of second order with parabolic degeneracy has not been completely solved, particularly the Tricomi and Frankl problems for general Chaplygin equation in multiply connected domains posed by L Bers, and the existence, regularity of solutions of the above problems for mixed equations with non-smooth degenerate curve in several domains posed by J M Rassias. The method revealed in this book is unlike any other, in which the hyperbolic number and hyperbolic complex function in hyperbolic domains, and the complex number and complex function in elliptic domains are used. The corresponding problems for first order complex equations with singular coefficients are first discussed, and then the problems for second order complex equations are considered, where we pose the new partial derivative notations and complex analytic methods such that the forms of the above first order complex equations in hyperbolic and elliptic domains are wholly identical. In the meantime, the estimates of solutions for the above problems are obtained, hence many open problems including the above TricomiOCo Bers and TricomiOCoFranklOCoRassias problems can be solved. Sample Chapter(s). Chapter 1: Elliptic Complex Equations of First Order (247 KB). Contents: Elliptic Complex Equations of First Order; Elliptic Complex Equations of Second Order; Hyperbolic Complex Equations of First and Second Orders; First Order Complex Equations of Mixed Type; Second Order Linear Equations of Mixed Type; Second Order Quasilinear Equations of Mixed Type. Readership: Graduate students and academics in analysis, differential equations and applied mathematics.

Authored by well-known researchers, this book presents its material as accessible surveys, giving readers access to comprehensive coverage of results scattered throughout the literature. A unique source of information for graduate students and researchers in mathematics and theoretical physics, and engineers interested in the subject.