Some Qualitative Properties of Solutions to Second Order Elliptic and Parabolic Equations
Author: Yu Yuan (Mathematician)
Publisher:
Published: 1998
Total Pages: 158
ISBN-13:
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Second Order Equations of Elliptic and Parabolic Type
Author: Evgeniĭ Mikhaĭlovich Landis
Publisher: American Mathematical Soc.
Published: 1998
Total Pages: 203
ISBN-13: 9780821808573
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.
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: 531
ISBN-13: 3110390086
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
A Priori Estimate and Some Properties of Solutions of Elliptic and Parabolic Equations
Author: S. N. Kruzhkov
Publisher:
Published: 1968
Total Pages: 62
ISBN-13:
DOWNLOAD EBOOKIn this paper we establish a priori estimates and costudy certain qualitative properties of generalized solutions of second-order elliptic and parabolic equations. The first fundamental results in this direction for equations with many independent variables were established in the papers by de Giorgi and Nash. A refined new approach to the consideration of similar problems and, in particular, a new proof of the de Giorgio theorem were given by Moser. In the present paper the method proposed by Moser is expanded.
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.
Qualitative Properties of Second Order Parabolic Equations
Author: Juraj Hűska
Publisher:
Published: 2006
Total Pages: 144
ISBN-13:
DOWNLOAD EBOOKSobolev 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.
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
Some Qualitative Properties of Fully Nonlinear Elliptic and Parabolic Equations
Author: Congming Li
Publisher:
Published: 1989
Total Pages: 164
ISBN-13:
DOWNLOAD EBOOKElliptic and Parabolic Problems
Author: C Bandle
Publisher: CRC Press
Published: 2020-11-26
Total Pages: 272
ISBN-13: 1000115275
DOWNLOAD EBOOKThis Research Note presents some recent advances in various important domains of partial differential equations and applied mathematics including equations and systems of elliptic and parabolic type and various applications in physics, mechanics and engineering. These topics are now part of various areas of science and have experienced tremendous development during the last decades. -------------------------------------
