Singularities of Solutions of Second-Order Quasilinear Equations

Singularities of Solutions of Second-Order Quasilinear Equations PDF Author: Laurent Veron
Publisher: CRC Press
ISBN: 9780582035393
Category : Mathematics
Languages : en
Pages : 396

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Book Description
This text examines the singularity problem for solutions of elliptic and parabolic quasilinear equations of second order.

Singularities of Solutions of Second-Order Quasilinear Equations

Singularities of Solutions of Second-Order Quasilinear Equations PDF Author: Laurent Veron
Publisher: CRC Press
ISBN: 9780582035393
Category : Mathematics
Languages : en
Pages : 396

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Book Description
This text examines the singularity problem for solutions of elliptic and parabolic quasilinear equations of second order.

Singular Solutions of Nonlinear Elliptic and Parabolic Equations

Singular Solutions of Nonlinear Elliptic and Parabolic Equations PDF Author: Alexander A. Kovalevsky
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110332248
Category : Mathematics
Languages : en
Pages : 447

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Book Description
This 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

Linear and Quasi-linear Equations of Parabolic Type

Linear and Quasi-linear Equations of Parabolic Type PDF Author: Olʹga A. Ladyženskaja
Publisher: American Mathematical Soc.
ISBN: 9780821815731
Category : Mathematics
Languages : en
Pages : 74

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Book Description
Equations of parabolic type are encountered in many areas of mathematics and mathematical physics, and those encountered most frequently are linear and quasi-linear parabolic equations of the second order. In this volume, boundary value problems for such equations are studied from two points of view: solvability, unique or otherwise, and the effect of smoothness properties of the functions entering the initial and boundary conditions on the smoothness of the solutions.

Existence of Solutions of Quasilinear Elliptic Equations on Manifolds with Conic Points

Existence of Solutions of Quasilinear Elliptic Equations on Manifolds with Conic Points PDF Author:
Publisher:
ISBN:
Category :
Languages : en
Pages : 202

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Book Description
Existence and regularity of solutions of quasilinear elliptic equations in nonsmooth domains have been interesting topics in the development of partial differential equations. The existence of finite-energy solutions of higher-order equations, also those with degenerations and singularities, can be shown by theories of monotone operators and topological methods. There are few results about singular solutions of second-order equations involving the p-Laplacian with the Dirac distribution on the right-hand side. So far the existence of singular solutions of higher-order equations with a presc...

Nonlinear Second Order Elliptic Equations Involving Measures

Nonlinear Second Order Elliptic Equations Involving Measures PDF Author: Moshe Marcus
Publisher: Walter de Gruyter
ISBN: 3110305313
Category : Mathematics
Languages : en
Pages : 264

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Book Description
In the last 40 years semi-linear elliptic equations became a central subject of study in the theory of nonlinear partial differential equations. On the one hand, the interest in this area is of a theoretical nature, due to its deep relations to other branches of mathematics, especially linear and nonlinear harmonic analysis, dynamical systems, differential geometry and probability. On the other hand, this study is of interest because of its applications. Equations of this type come up in various areas such as problems of physics and astrophysics, curvature problems in Riemannian geometry, logistic problems related for instance to population models and, most importantly, the study of branching processes and superdiffusions in the theory of probability. The aim of this book is to present a comprehensive study of boundary value problems for linear and semi-linear second order elliptic equations with measure data. We are particularly interested in semi-linear equations with absorption. The interactions between the diffusion operator and the absorption term give rise to a large class of nonlinear phenomena in the study of which singularities and boundary trace play a central role. This book is accessible to graduate students and researchers with a background in real analysis and partial differential equations.

Local And Global Aspects Of Quasilinear Degenerate Elliptic Equations: Quasilinear Elliptic Singular Problems

Local And Global Aspects Of Quasilinear Degenerate Elliptic Equations: Quasilinear Elliptic Singular Problems PDF Author: Laurent Veron
Publisher: World Scientific
ISBN: 9814730343
Category : Mathematics
Languages : en
Pages : 474

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Book Description
This book is devoted to the study of elliptic second-order degenerate quasilinear equations, the model of which is the p-Laplacian, with or without dominant lower order reaction term. Emphasis is put on three aspects:

The Boundary Behavior of Solutions of Quasilinear Second Order Elliptic Partial Differential Equations Arising from Geometrical Problems

The Boundary Behavior of Solutions of Quasilinear Second Order Elliptic Partial Differential Equations Arising from Geometrical Problems PDF Author: Chi-Ping Lau
Publisher:
ISBN:
Category :
Languages : en
Pages : 224

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Book Description


Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains

Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains PDF Author: Michail Borsuk
Publisher: Elsevier
ISBN: 0080461735
Category : Mathematics
Languages : en
Pages : 538

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Book Description
The book contains a systematic treatment of the qualitative theory of elliptic boundary value problems for linear and quasilinear second order equations in non-smooth domains. The authors concentrate on the following fundamental results: sharp estimates for strong and weak solutions, solvability of the boundary value problems, regularity assertions for solutions near singular points. Key features: * New the Hardy – Friedrichs – Wirtinger type inequalities as well as new integral inequalities related to the Cauchy problem for a differential equation.* Precise exponents of the solution decreasing rate near boundary singular points and best possible conditions for this.* The question about the influence of the coefficients smoothness on the regularity of solutions.* New existence theorems for the Dirichlet problem for linear and quasilinear equations in domains with conical points.* The precise power modulus of continuity at singular boundary point for solutions of the Dirichlet, mixed and the Robin problems.* The behaviour of weak solutions near conical point for the Dirichlet problem for m – Laplacian.* The behaviour of weak solutions near a boundary edge for the Dirichlet and mixed problem for elliptic quasilinear equations with triple degeneration. * Precise exponents of the solution decreasing rate near boundary singular points and best possible conditions for this.* The question about the influence of the coefficients smoothness on the regularity of solutions.* New existence theorems for the Dirichlet problem for linear and quasilinear equations in domains with conical points.* The precise power modulus of continuity at singular boundary point for solutions of the Dirichlet, mixed and the Robin problems.* The behaviour of weak solutions near conical point for the Dirichlet problem for m - Laplacian.* The behaviour of weak solutions near a boundary edge for the Dirichlet and mixed problem for elliptic quasilinear equations with triple degeneration.

Propagation and Interaction of Singularities in Nonlinear Hyperbolic Problems

Propagation and Interaction of Singularities in Nonlinear Hyperbolic Problems PDF Author: Michael Beals
Publisher: Springer Science & Business Media
ISBN: 1461245540
Category : Mathematics
Languages : en
Pages : 153

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Book Description
This book developed from a series of lectures I gave at the Symposium on Nonlinear Microlocal Analysis held at Nanjing University in October. 1988. Its purpose is to give an overview of the use of microlocal analysis and commutators in the study of solutions to nonlinear wave equations. The weak singularities in the solutions to such equations behave up to a certain extent like those present in the linear case: they propagate along the null bicharacteristics of the operator. On the other hand. examples exhibiting singularities not present in the linear case can also be constructed. I have tried to present a crossection of both the regularity results and the singular examples. for problems on the interior of a domain and on domains with boundary. The main emphasis is on the case of more than one space dimen sion. since that case is treated in great detail in the paper of Rauch-Reed 159]. The results presented here have for the most part appeared elsewhere. and are the work of many authors. but a few new examples and proofs are given. I have attempted to indicate the essential ideas behind the arguments. so that only some of the results are proved in full detail. It is hoped that the central notions of the more technical proofs appearing in research papers will be illuminated by these simpler cases.

Handbook of Differential Equations: Stationary Partial Differential Equations

Handbook of Differential Equations: Stationary Partial Differential Equations PDF Author: Michel Chipot
Publisher: Elsevier
ISBN: 0080560598
Category : Mathematics
Languages : en
Pages : 618

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Book Description
This handbook is the sixth and last volume in the series devoted to stationary partial differential equations. The topics covered by this volume include in particular domain perturbations for boundary value problems, singular solutions of semilinear elliptic problems, positive solutions to elliptic equations on unbounded domains, symmetry of solutions, stationary compressible Navier-Stokes equation, Lotka-Volterra systems with cross-diffusion, and fixed point theory for elliptic boundary value problems. * Collection of self-contained, state-of-the-art surveys * Written by well-known experts in the field * Informs and updates on all the latest developments