Potential Theory for Fully Nonlinear Elliptic Equations PDF Download

Author: Denis A. Labutin
Publisher:
ISBN:
Category : Differential equations, Elliptic
Languages : en
Pages : 122

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Author: Luis A. Caffarelli
Publisher: American Mathematical Soc.
ISBN: 0821804375
Category : Mathematics
Languages : en
Pages : 114

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Book Description
The goal of the book is to extend classical regularity theorems for solutions of linear elliptic partial differential equations to the context of fully nonlinear elliptic equations. This class of equations often arises in control theory, optimization, and other applications. The authors give a detailed presentation of all the necessary techniques. Instead of treating these techniques in their greatest generality, they outline the key ideas and prove the results needed for developing the subsequent theory. Topics discussed in the book include the theory of viscosity solutions for nonlinear equations, the Alexandroff estimate and Krylov-Safonov Harnack-type inequality for viscosity solutions, uniqueness theory for viscosity solutions, Evans and Krylov regularity theory for convex fully nonlinear equations, and regularity theory for fully nonlinear equations with variable coefficients.

Author: Juha Heinonen
Publisher: Courier Dover Publications
ISBN: 0486830462
Category : Mathematics
Languages : en
Pages : 417

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Book Description
A self-contained treatment appropriate for advanced undergraduates and graduate students, this text offers a detailed development of the necessary background for its survey of the nonlinear potential theory of superharmonic functions. 1993 edition.

Author: Adisak Seesanea
Publisher:
ISBN:
Category :
Languages : en
Pages : 0

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Author: Juha Heinonen
Publisher: Courier Dover Publications
ISBN: 048682425X
Category : Mathematics
Languages : en
Pages : 417

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Book Description
A self-contained treatment appropriate for advanced undergraduate and graduate students, this volume offers a detailed development of the necessary background for its survey of the nonlinear potential theory of superharmonic functions. Starting with the theory of weighted Sobolev spaces, the text advances to the theory of weighted variational capacity. Succeeding chapters investigate solutions and supersolutions of equations, with emphasis on refined Sobolev spaces, variational integrals, and harmonic functions. Chapter 7 defines superharmonic functions via the comparison principle, and chapters 8 through 14 form the core of the nonlinear potential theory of superharmonic functions. Topics include balayage; Perron's method, barriers, and resolutivity; polar sets; harmonic measure; fine topology; harmonic morphisms; and quasiregular mappings. The book concludes with explorations of axiomatic nonlinear potential theory and helpful appendixes.

Author: J. Kral
Publisher: Springer Science & Business Media
ISBN: 1461344255
Category : Mathematics
Languages : en
Pages : 138

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Book Description
Preface.- Gottfried Anger: Direct and inverse problems in potential theory.- Viorel Barbu: Regularity results for sane differential equations associated with maximal monotone operators in Hilbert spaces.- Haim Brezis: Classes d'interpolation associées à un opérateur monotone et applications.- Siegfried Dnümmel: On inverse problems for k-dimensional potentials.- Jozef Ka?ur: Application of Rothe's method to nonlinear parabolic boundary value problems.- Josef Král: Potentials and removability of singularities.- Vladimir Lovicar: Theorem of Fréchet and asymptotically almost periodid solutions of.

Author: Denis A. Labutin
Publisher:
ISBN:
Category : Approximation theory
Languages : en
Pages : 46

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Author: Nikolai Nadirashvili
Publisher: American Mathematical Soc.
ISBN: 1470417103
Category : Mathematics
Languages : en
Pages : 250

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Book Description
This book presents applications of noncommutative and nonassociative algebras to constructing unusual (nonclassical and singular) solutions to fully nonlinear elliptic partial differential equations of second order. The methods described in the book are used to solve a longstanding problem of the existence of truly weak, nonsmooth viscosity solutions. Moreover, the authors provide an almost complete description of homogeneous solutions to fully nonlinear elliptic equations. It is shown that even in the very restricted setting of "Hessian equations", depending only on the eigenvalues of the Hessian, these equations admit homogeneous solutions of all orders compatible with known regularity for viscosity solutions provided the space dimension is five or larger. To the contrary, in dimension four or less the situation is completely different, and our results suggest strongly that there are no nonclassical homogeneous solutions at all in dimensions three and four. Thus this book gives a complete list of dimensions where nonclassical homogeneous solutions to fully nonlinear uniformly elliptic equations do exist; this should be compared with the situation of, say, ten years ago when the very existence of nonclassical viscosity solutions was not known.

Author: Anders Björn
Publisher: European Mathematical Society
ISBN: 9783037190999
Category : Harmonic functions
Languages : en
Pages : 422

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Book Description
The $p$-Laplace equation is the main prototype for nonlinear elliptic problems and forms a basis for various applications, such as injection moulding of plastics, nonlinear elasticity theory, and image processing. Its solutions, called p-harmonic functions, have been studied in various contexts since the 1960s, first on Euclidean spaces and later on Riemannian manifolds, graphs, and Heisenberg groups. Nonlinear potential theory of p-harmonic functions on metric spaces has been developing since the 1990s and generalizes and unites these earlier theories. This monograph gives a unified treatment of the subject and covers most of the available results in the field, so far scattered over a large number of research papers. The aim is to serve both as an introduction to the area for interested readers and as a reference text for active researchers. The presentation is rather self contained, but it is assumed that readers know measure theory and functional analysis. The first half of the book deals with Sobolev type spaces, so-called Newtonian spaces, based on upper gradients on general metric spaces. In the second half, these spaces are used to study p-harmonic functions on metric spaces, and a nonlinear potential theory is developed under some additional, but natural, assumptions on the underlying metric space. Each chapter contains historical notes with relevant references, and an extensive index is provided at the end of the book.

Author: Marco Cirant
Publisher: Princeton University Press
ISBN: 0691243646
Category : Mathematics
Languages : en
Pages : 225

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Book Description
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.