Nonlinear Elliptic and Parabolic Equations of the Second Order
Author: N.V. Krylov
Publisher: Springer
Published: 1987-06-30
Total Pages: 480
ISBN-13: 9027722897
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Singular Solutions of Nonlinear Elliptic and Parabolic Equations
Author: Alexander A. Kovalevsky
Publisher: Walter de Gruyter GmbH & Co KG
Published: 2016-03-21
Total Pages: 448
ISBN-13: 3110332248
DOWNLOAD EBOOKThis monograph looks at several trends in the investigation of singular solutions of nonlinear elliptic and parabolic equations. It discusses results on the existence and properties of weak and entropy solutions for elliptic second-order equations and some classes of fourth-order equations with L1-data and questions on the removability of singularities of solutions to elliptic and parabolic second-order equations in divergence form. It looks at localized and nonlocalized singularly peaking boundary regimes for different classes of quasilinear parabolic second- and high-order equations in divergence form. The book will be useful for researchers and post-graduate students that specialize in the field of the theory of partial differential equations and nonlinear analysis. Contents: Foreword Part I: Nonlinear elliptic equations with L^1-data Nonlinear elliptic equations of the second order with L^1-data Nonlinear equations of the fourth order with strengthened coercivity and L^1-data Part II: Removability of singularities of the solutions of quasilinear elliptic and parabolic equations of the second order Removability of singularities of the solutions of quasilinear elliptic equations Removability of singularities of the solutions of quasilinear parabolic equations Quasilinear elliptic equations with coefficients from the Kato class Part III: Boundary regimes with peaking for quasilinear parabolic equations Energy methods for the investigation of localized regimes with peaking for parabolic second-order equations Method of functional inequalities in peaking regimes for parabolic equations of higher orders Nonlocalized regimes with singular peaking Appendix: Formulations and proofs of the auxiliary results Bibliography
Nonlinear Elliptic Equations of the Second Order
Author: Qing Han
Publisher: American Mathematical Soc.
Published: 2016-04-15
Total Pages: 378
ISBN-13: 1470426072
DOWNLOAD EBOOKNonlinear elliptic differential equations are a diverse subject with important applications to the physical and social sciences and engineering. They also arise naturally in geometry. In particular, much of the progress in the area in the twentieth century was driven by geometric applications, from the Bernstein problem to the existence of Kähler–Einstein metrics. This book, designed as a textbook, provides a detailed discussion of the Dirichlet problems for quasilinear and fully nonlinear elliptic differential equations of the second order with an emphasis on mean curvature equations and on Monge–Ampère equations. It gives a user-friendly introduction to the theory of nonlinear elliptic equations with special attention given to basic results and the most important techniques. Rather than presenting the topics in their full generality, the book aims at providing self-contained, clear, and “elementary” proofs for results in important special cases. This book will serve as a valuable resource for graduate students or anyone interested in this subject.
Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations
Author: N. V. Krylov
Publisher: American Mathematical Soc.
Published: 2018-09-07
Total Pages: 458
ISBN-13: 1470447401
DOWNLOAD EBOOKThis book concentrates on first boundary-value problems for fully nonlinear second-order uniformly elliptic and parabolic equations with discontinuous coefficients. We look for solutions in Sobolev classes, local or global, or for viscosity solutions. Most of the auxiliary results, such as Aleksandrov's elliptic and parabolic estimates, the Krylov–Safonov and the Evans–Krylov theorems, are taken from old sources, and the main results were obtained in the last few years. Presentation of these results is based on a generalization of the Fefferman–Stein theorem, on Fang-Hua Lin's like estimates, and on the so-called “ersatz” existence theorems, saying that one can slightly modify “any” equation and get a “cut-off” equation that has solutions with bounded derivatives. These theorems allow us to prove the solvability in Sobolev classes for equations that are quite far from the ones which are convex or concave with respect to the Hessians of the unknown functions. In studying viscosity solutions, these theorems also allow us to deal with classical approximating solutions, thus avoiding sometimes heavy constructions from the usual theory of viscosity solutions.
Lectures on Elliptic and Parabolic Equations in Holder Spaces
Author: Nikolaĭ Vladimirovich Krylov
Publisher: American Mathematical Soc.
Published: 1996
Total Pages: 178
ISBN-13: 082180569X
DOWNLOAD EBOOKThese lectures concentrate on fundamentals of the modern theory of linear elliptic and parabolic equations in H older spaces. Krylov shows that this theory - including some issues of the theory of nonlinear equations - is based on some general and extremely powerful ideas and some simple computations. The main object of study is the first boundary-value problems for elliptic and parabolic equations, with some guidelines concerning other boundary-value problems such as the Neumann or oblique derivative problems or problems involving higher-order elliptic operators acting on the boundary. Numerical approximations are also discussed. This book, containing 200 exercises, aims to provide a good understanding of what kind of results are available and what kinds of techniques are used to obtain them.
Second Order Equations of Elliptic and Parabolic Type
Author: E. M. Landis
Publisher: American Mathematical Soc.
Published: 1997-12-02
Total Pages: 224
ISBN-13: 9780821897812
DOWNLOAD EBOOKMost books on elliptic and parabolic equations emphasize existence and uniqueness of solutions. By contrast, this book focuses on the qualitative properties of solutions. In addition to the discussion of classical results for equations with smooth coefficients (Schauder estimates and the solvability of the Dirichlet problem for elliptic equations; the Dirichlet problem for the heat equation), the book describes properties of solutions to second order elliptic and parabolic equations with measurable coefficients near the boundary and at infinity. The book presents a fine elementary introduction to the theory of elliptic and parabolic equations of second order. The precise and clear exposition is suitable for graduate students as well as for research mathematicians who want to get acquainted with this area of the theory of partial differential equations.
Elliptic and Parabolic Equations with Discontinuous Coefficients
Author: Antonino Maugeri
Publisher: Wiley-VCH
Published: 2000-12-13
Total Pages: 266
ISBN-13:
DOWNLOAD EBOOKThis book unifies the different approaches in studying elliptic and parabolic partial differential equations with discontinuous coefficients. To the enlarging market of researchers in applied sciences, mathematics and physics, it gives concrete answers to questions suggested by non-linear models. Providing an up-to date survey on the results concerning elliptic and parabolic operators on a high level, the authors serve the reader in doing further research. Being themselves active researchers in the field, the authors describe both on the level of good examples and precise analysis, the crucial role played by such requirements on the coefficients as the Cordes condition, Campanato's nearness condition, and vanishing mean oscillation condition. They present the newest results on the basic boundary value problems for operators with VMO coefficients and non-linear operators with discontinuous coefficients and state a lot of open problems in the field.
Second Order Parabolic Differential Equations
Author: Gary M. Lieberman
Publisher: World Scientific
Published: 1996
Total Pages: 472
ISBN-13: 9789810228835
DOWNLOAD EBOOKIntroduction. Maximum principles. Introduction to the theory of weak solutions. Hölder estimates. Existence, uniqueness, and regularity of solutions. Further theory of weak solutions. Strong solutions. Fixed point theorems and their applications. Comparison and maximum principles. Boundary gradient estimates. Global and local gradient bounds. Hölder gradient estimates and existence theorems. The oblique derivative problem for quasilinear parabolic equations. Fully nonlinear equations. Introduction. Monge-Ampère and Hessian equations.
Nonlinear Elliptic and Parabolic Problems
Author: Michel Chipot
Publisher: Springer Science & Business Media
Published: 2006-02-09
Total Pages: 531
ISBN-13: 3764373857
DOWNLOAD EBOOKCelebrates the work of the renowned mathematician Herbert Amann, who had a significant and decisive influence in shaping Nonlinear Analysis. Containing 32 contributions, this volume covers a range of nonlinear elliptic and parabolic equations, with applications to natural sciences and engineering.
Quasilinear Degenerate and Nonuniformly Elliptic and Parabolic Equations of Second Order
Author: A. V. Ivanov
Publisher: American Mathematical Soc.
Published: 1984
Total Pages: 306
ISBN-13: 9780821830802
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