Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations
| Author | : Nikolaĭ Vladimirovich Krylov |
| Publisher | : |
| Total Pages | : 458 |
| Release | : 2018 |
| Genre | : MATHEMATICS |
| ISBN | : 9781470448530 |
Sobolev And Viscosity Solutions For Fully Nonlinear Elliptic And Parabolic Equations
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Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations
| Author | : N. V. Krylov |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 458 |
| Release | : 2018-09-07 |
| Genre | : Mathematics |
| ISBN | : 1470447401 |
This 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.
Nonlinear Elliptic and Parabolic Equations of the Second Order
| Author | : N.V. Krylov |
| Publisher | : Springer |
| Total Pages | : 0 |
| Release | : 2001-11-30 |
| Genre | : Mathematics |
| ISBN | : 9781402003349 |
Approach your problems from the It isn't that they can't see the right end and begin with the solution. It is that they can't see answers. Then one day, perhaps the problem. you will find the final question. G.K. Chesterton. The Scandal of 'The Hermit Clad in Crane Father Brown 'The Point of a Pin'. Feathers' in R. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of mono graphs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theor.etical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces.
Comparison Principles for General Potential Theories and PDEs
| Author | : Marco Cirant |
| Publisher | : Princeton University Press |
| Total Pages | : 224 |
| Release | : 2023-10-03 |
| Genre | : Mathematics |
| ISBN | : 069124362X |
An examination of the symbiotic and productive relationship between fully nonlinear partial differential equations and generalized potential theories In recent years, there has evolved a symbiotic and productive relationship between fully nonlinear partial differential equations and generalized potential theories. This book examines important aspects of this story. One main purpose is to prove comparison principles for nonlinear potential theories in Euclidian spaces straightforwardly from duality and monotonicity under the weakest possible notion of ellipticity. The book also shows how to deduce comparison principles for nonlinear differential operators, by marrying these two points of view, under the correspondence principle. The authors explain that comparison principles are fundamental in both contexts, since they imply uniqueness for the Dirichlet problem. When combined with appropriate boundary geometries, yielding suitable barrier functions, they also give existence by Perron’s method. There are many opportunities for cross-fertilization and synergy. In potential theory, one is given a constraint set of 2-jets that determines its subharmonic functions. The constraint set also determines a family of compatible differential operators. Because there are many such operators, potential theory strengthens and simplifies the operator theory. Conversely, the set of operators associated with the constraint can influence the potential theory.
Linear and Quasilinear Parabolic Systems: Sobolev Space Theory
| Author | : David Hoff |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 226 |
| Release | : 2020-11-18 |
| Genre | : Education |
| ISBN | : 1470461617 |
This monograph presents a systematic theory of weak solutions in Hilbert-Sobolev spaces of initial-boundary value problems for parabolic systems of partial differential equations with general essential and natural boundary conditions and minimal hypotheses on coefficients. Applications to quasilinear systems are given, including local existence for large data, global existence near an attractor, the Leray and Hopf theorems for the Navier-Stokes equations and results concerning invariant regions. Supplementary material is provided, including a self-contained treatment of the calculus of Sobolev functions on the boundaries of Lipschitz domains and a thorough discussion of measurability considerations for elements of Bochner-Sobolev spaces. This book will be particularly useful both for researchers requiring accessible and broadly applicable formulations of standard results as well as for students preparing for research in applied analysis. Readers should be familiar with the basic facts of measure theory and functional analysis, including weak derivatives and Sobolev spaces. Prior work in partial differential equations is helpful but not required.
Maximal Function Methods for Sobolev Spaces
| Author | : Juha Kinnunen |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 354 |
| Release | : 2021-08-02 |
| Genre | : Education |
| ISBN | : 1470465752 |
This book discusses advances in maximal function methods related to Poincaré and Sobolev inequalities, pointwise estimates and approximation for Sobolev functions, Hardy's inequalities, and partial differential equations. Capacities are needed for fine properties of Sobolev functions and characterization of Sobolev spaces with zero boundary values. The authors consider several uniform quantitative conditions that are self-improving, such as Hardy's inequalities, capacity density conditions, and reverse Hölder inequalities. They also study Muckenhoupt weight properties of distance functions and combine these with weighted norm inequalities; notions of dimension are then used to characterize density conditions and to give sufficient and necessary conditions for Hardy's inequalities. At the end of the book, the theory of weak solutions to the p p-Laplace equation and the use of maximal function techniques is this context are discussed. The book is directed to researchers and graduate students interested in applications of geometric and harmonic analysis in Sobolev spaces and partial differential equations.
Analysis of Monge–Ampère Equations
| Author | : Nam Q. Le |
| Publisher | : American Mathematical Society |
| Total Pages | : 599 |
| Release | : 2024-03-07 |
| Genre | : Mathematics |
| ISBN | : 1470474204 |
This book presents a systematic analysis of the Monge–Ampère equation, the linearized Monge–Ampère equation, and their applications, with emphasis on both interior and boundary theories. Starting from scratch, it gives an extensive survey of fundamental results, essential techniques, and intriguing phenomena in the solvability, geometry, and regularity of Monge–Ampère equations. It describes in depth diverse applications arising in geometry, fluid mechanics, meteorology, economics, and the calculus of variations. The modern treatment of boundary behaviors of solutions to Monge–Ampère equations, a very important topic of the theory, is thoroughly discussed. The book synthesizes many important recent advances, including Savin's boundary localization theorem, spectral theory, and interior and boundary regularity in Sobolev and Hölder spaces with optimal assumptions. It highlights geometric aspects of the theory and connections with adjacent research areas. This self-contained book provides the necessary background and techniques in convex geometry, real analysis, and partial differential equations, presents detailed proofs of all theorems, explains subtle constructions, and includes well over a hundred exercises. It can serve as an accessible text for graduate students as well as researchers interested in this subject.
Computational Management
| Author | : Srikanta Patnaik |
| Publisher | : Springer Nature |
| Total Pages | : 682 |
| Release | : 2021-05-29 |
| Genre | : Technology & Engineering |
| ISBN | : 303072929X |
This book offers a timely review of cutting-edge applications of computational intelligence to business management and financial analysis. It covers a wide range of intelligent and optimization techniques, reporting in detail on their application to real-world problems relating to portfolio management and demand forecasting, decision making, knowledge acquisition, and supply chain scheduling and management.
An Introduction To Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L∞
| Author | : Nikos Katzourakis |
| Publisher | : Springer |
| Total Pages | : 125 |
| Release | : 2014-11-26 |
| Genre | : Mathematics |
| ISBN | : 3319128299 |
The purpose of this book is to give a quick and elementary, yet rigorous, presentation of the rudiments of the so-called theory of Viscosity Solutions which applies to fully nonlinear 1st and 2nd order Partial Differential Equations (PDE). For such equations, particularly for 2nd order ones, solutions generally are non-smooth and standard approaches in order to define a "weak solution" do not apply: classical, strong almost everywhere, weak, measure-valued and distributional solutions either do not exist or may not even be defined. The main reason for the latter failure is that, the standard idea of using "integration-by-parts" in order to pass derivatives to smooth test functions by duality, is not available for non-divergence structure PDE.
Nonlinear Dirac Equation: Spectral Stability of Solitary Waves
| Author | : Nabile Boussaïd |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 306 |
| Release | : 2019-11-21 |
| Genre | : Education |
| ISBN | : 1470443953 |
This monograph gives a comprehensive treatment of spectral (linear) stability of weakly relativistic solitary waves in the nonlinear Dirac equation. It turns out that the instability is not an intrinsic property of the Dirac equation that is only resolved in the framework of the second quantization with the Dirac sea hypothesis. Whereas general results about the Dirac-Maxwell and similar equations are not yet available, we can consider the Dirac equation with scalar self-interaction, the model first introduced in 1938. In this book we show that in particular cases solitary waves in this model may be spectrally stable (no linear instability). This result is the first step towards proving asymptotic stability of solitary waves. The book presents the necessary overview of the functional analysis, spectral theory, and the existence and linear stability of solitary waves of the nonlinear Schrödinger equation. It also presents the necessary tools such as the limiting absorption principle and the Carleman estimates in the form applicable to the Dirac operator, and proves the general form of the Dirac-Pauli theorem. All of these results are used to prove the spectral stability of weakly relativistic solitary wave solutions of the nonlinear Dirac equation.
