Nonlinear Elliptic Partial Differential Equations
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Nonlinear Elliptic Partial Differential Equations
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Author : Hervé Le Dret
language : en
Publisher: Springer
Release Date : 2018-05-25
Nonlinear Elliptic Partial Differential Equations written by Hervé Le Dret and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-05-25 with Mathematics categories.
This textbook presents the essential parts of the modern theory of nonlinear partial differential equations, including the calculus of variations. After a short review of results in real and functional analysis, the author introduces the main mathematical techniques for solving both semilinear and quasilinear elliptic PDEs, and the associated boundary value problems. Key topics include infinite dimensional fixed point methods, the Galerkin method, the maximum principle, elliptic regularity, and the calculus of variations. Aimed at graduate students and researchers, this textbook contains numerous examples and exercises and provides several comments and suggestions for further study.
Qualitative Analysis Of Nonlinear Elliptic Partial Differential Equations
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Author : Vicenţiu Rǎdulescu
language : en
Publisher: Hindawi Publishing Corporation
Release Date : 2008
Qualitative Analysis Of Nonlinear Elliptic Partial Differential Equations written by Vicenţiu Rǎdulescu and has been published by Hindawi Publishing Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Differential equations, Elliptic categories.
This book provides a comprehensive introduction to the mathematical theory of nonlinear problems described by elliptic partial differential equations. These equations can be seen as nonlinear versions of the classical Laplace equation, and they appear as mathematical models in different branches of physics, chemistry, biology, genetics, and engineering and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on the calculus of variations and functional analysis. Concentrating on single-valued or multivalued elliptic equations with nonlinearities of various types, the aim of this volume is to obtain sharp existence or nonexistence results, as well as decay rates for general classes of solutions. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including bifurcation, stability, asymptotic analysis, and optimal regularity of solutions. The method of presentation should appeal to readers with different backgrounds in functional analysis and nonlinear partial differential equations. All chapters include detailed heuristic arguments providing thorough motivation of the study developed later on in the text, in relationship with concrete processes arising in applied sciences. A systematic description of the most relevant singular phenomena described in this volume includes existence (or nonexistence) of solutions, unicity or multiplicity properties, bifurcation and asymptotic analysis, and optimal regularity. The book includes an extensive bibliography and a rich index, thus allowing for quick orientation among the vast collection of literature on the mathematical theory of nonlinear phenomena described by elliptic partial differential equations.
Nonlinear Elliptic Partial Differential Equations
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Author : J. P. Gossez
language : en
Publisher: American Mathematical Soc.
Release Date : 2011
Nonlinear Elliptic Partial Differential Equations written by J. P. Gossez and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Mathematics categories.
This volume contains papers on semi-linear and quasi-linear elliptic equations from the workshop on Nonlinear Elliptic Partial Differential Equations, in honor of Jean-Pierre Gossez's 65th birthday, held September 2-4, 2009 at the Universite Libre de Bruxelles, Belgium. The workshop reflected Gossez's contributions in nonlinear elliptic PDEs and provided an opening to new directions in this very active research area. Presentations covered recent progress in Gossez's favorite topics, namely various problems related to the $p$-Laplacian operator, the antimaximum principle, the Fucik Spectrum, and other related subjects. This volume will be of principle interest to researchers in nonlinear analysis, especially in partial differential equations of elliptic type.
Boundary Value Problems For Nonlinear Elliptic Equations In Divergence Form
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Author : Abubakar Mwasa
language : en
Publisher: Linköping University Electronic Press
Release Date : 2021-02-23
Boundary Value Problems For Nonlinear Elliptic Equations In Divergence Form written by Abubakar Mwasa and has been published by Linköping University Electronic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-02-23 with categories.
The thesis consists of three papers focussing on the study of nonlinear elliptic partial differential equations in a nonempty open subset Ω of the n-dimensional Euclidean space Rn. We study the existence and uniqueness of the solutions, as well as their behaviour near the boundary of Ω. The behaviour of the solutions at infinity is also discussed when Ω is unbounded. In Paper A, we consider a mixed boundary value problem for the p-Laplace equation ∆pu := div(|∇u| p−2∇u) = 0 in an open infinite circular half-cylinder with prescribed Dirichlet boundary data on a part of the boundary and zero Neumann boundary data on the rest. By a suitable transformation of the independent variables, this mixed problem is transformed into a Dirichlet problem for a degenerate (weighted) elliptic equation on a bounded set. By analysing the transformed problem in weighted Sobolev spaces, it is possible to obtain the existence of continuous weak solutions to the mixed problem, both for Sobolev and for continuous data on the Dirichlet part of the boundary. A characterisation of the boundary regularity of the point at infinity is obtained in terms of a new variational capacity adapted to the cylinder. In Paper B, we study Perron solutions to the Dirichlet problem for the degenerate quasilinear elliptic equation div A(x, ∇u) = 0 in a bounded open subset of Rn. The vector-valued function A satisfies the standard ellipticity assumptions with a parameter 1 < p < ∞ and a p-admissible weight w. For general boundary data, the Perron method produces a lower and an upper solution, and if they coincide then the boundary data are called resolutive. We show that arbitrary perturbations on sets of weighted p-capacity zero of continuous (and quasicontinuous Sobolev) boundary data f are resolutive, and that the Perron solutions for f and such perturbations coincide. As a consequence, it is also proved that the Perron solution with continuous boundary data is the unique bounded continuous weak solution that takes the required boundary data outside a set of weighted p-capacity zero. Some results in Paper C are a generalisation of those in Paper A, extended to quasilinear elliptic equations of the form div A(x, ∇u) = 0. Here, results from Paper B are used to prove the existence and uniqueness of continuous weak solutions to the mixed boundary value problem for continuous Dirichlet data. Regularity of the boundary point at infinity for the equation div A(x, ∇u) = 0 is characterised by a Wiener type criterion. We show that sets of Sobolev p-capacity zero are removable for the solutions and also discuss the behaviour of the solutions at ∞. In particular, a certain trichotomy is proved, similar to the Phragmén–Lindelöf principle.
Contributions To Nonlinear Elliptic Equations And Systems
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Author : Alexandre N. Carvalho
language : en
Publisher: Birkhäuser
Release Date : 2015-11-14
Contributions To Nonlinear Elliptic Equations And Systems written by Alexandre N. Carvalho and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-11-14 with Mathematics categories.
This volume of contributions pays tribute to the life and work of Djairo Guedes de Figueiredo on the occasion of his 80th birthday. The articles it contains were born out of the ICMC Summer Meeting on Differential Equations – 2014 Chapter, also dedicated to de Figueiredo and held at the Universidade de São Paulo at São Carlos, Brazil from February 3-7, 2014. The contributing authors represent a group of international experts in the field and discuss recent trends and new directions in nonlinear elliptic partial differential equations and systems. Djairo Guedes de Figueiredo has had a very active scientific career, publishing 29 monographs and over one hundred research articles. His influence on Brazilian mathematics has made him one of the pillars of the subject in that country. He had a major impact on the development of analysis, especially in its application to nonlinear elliptic partial differential equations and systems throughout the entire world. The articles collected here pay tribute to him and his legacy and are intended for graduate students and researchers in mathematics and related areas who are interested in nonlinear elliptic partial differential equations and systems.
Elliptic Partial Differential Equations
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Author : Lucio Boccardo
language : en
Publisher: Walter de Gruyter
Release Date : 2013-10-29
Elliptic Partial Differential Equations written by Lucio Boccardo and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-10-29 with Mathematics categories.
Elliptic partial differential equations is one of the main and most active areas in mathematics. This book is devoted to the study of linear and nonlinear elliptic problems in divergence form, with the aim of providing classical results, as well as more recent developments about distributional solutions. For this reason this monograph is addressed to master's students, PhD students and anyone who wants to begin research in this mathematical field.
Numerical Solution Of Nonlinear Elliptic Partial
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Author : P. Concus
language : en
Publisher:
Release Date : 1976
Numerical Solution Of Nonlinear Elliptic Partial written by P. Concus and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1976 with categories.
Fully Nonlinear Elliptic Equations
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Author : Luis A. Caffarelli
language : en
Publisher: American Mathematical Soc.
Release Date : 1995
Fully Nonlinear Elliptic Equations written by Luis A. Caffarelli and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995 with Mathematics categories.
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.
Numerical Methods For Nonlinear Elliptic Partial Differential Equations
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Author : Tiago Miguel Saldanha Salvador
language : en
Publisher:
Release Date : 2017
Numerical Methods For Nonlinear Elliptic Partial Differential Equations written by Tiago Miguel Saldanha Salvador and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with categories.
"The goal of this thesis is to widen the class of provably convergent schemes for elliptic partial differential equations (PDEs) and improve their accuracy. We accomplish this by building on the theory of Barles and Souganidis, and its extension by Froese and Oberman to build monotone and filtered schemes.The first problem considered is the widely studied class of first order Hamilton-Jacobi (HJ) equations. The goal is to construct provably convergent accurate schemes, together with an efficient solver, by making use of the large number of discretizations and solvers already available. To this end, we build filtered schemes, whose main idea is to blend a stable monotone convergent scheme with an accurate scheme while retaining the advantages of both: stability and convergence of the former, and higher accuracy of the latter. Indeed, we are able to build schemes which are second, third, and fourth order accurate in one dimension, as well as schemes that are second order accurate in two dimensions. Moreover, the schemes are explicit, allowing us to use the easy-to-implement fast sweeping method. Using different accurate schemes (e.g. from standard centred differences, higher order upwinding and ENO interpolation), the accuracy of the filtered schemes is validated with computational results for the eikonal equation, as well as other HJ equations (both in one and two dimensions).The second problem considered is the 2-Hessian equation, a fully nonlinear PDE related to the intrinsic curvature for three-dimensional manifolds. The goal is to build numerical methods to compute its solution on a bounded domain given prescribed boundary data. We propose two distinct methods. The first is provably convergent to the unique viscosity solution. The second has higher accuracy and converges in practice, but lacks a formal proof of convergence. The PDE is elliptic on a restricted set of functions: a convexity-type constraint is needed for the ellipticity of the PDE operator, which poses additional difficulties when building the numerical methods. Solutions with both discretizations are obtained using Newton's method. Computational results are presented on a number of exact solutions which range in regularity from smooth to non-differentiable, and in shape from convex to non-convex.The third and last problem is to build a provably convergent scheme for the nonlinear PDE that governs the motion of level sets by affine curvature. It is closely related to mean curvature but exhibits instabilities not found in the former. These instabilities and the lack of regularity of the affine curvature operator posed unexpected and additional difficulties in building monotone schemes. A standard finite difference scheme is proposed and an example that illustrates its nonlinear instability is given. We build provably convergent monotone finite difference schemes. Numerical experiments demonstrate the accuracy and stability of the discretization, as well as the fact that our approximate solutions capture the affine invariance and morphological properties of the evolution." --
Nonlinear Elliptic Boundary Value Problems And Their Applications
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Author : H Begehr
language : en
Publisher: CRC Press
Release Date : 1996-05-15
Nonlinear Elliptic Boundary Value Problems And Their Applications written by H Begehr and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996-05-15 with Mathematics categories.