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Elliptic boundary value problem

About: Elliptic boundary value problem is a(n) research topic. Over the lifetime, 5423 publication(s) have been published within this topic receiving 118122 citation(s). more


Open accessBook
07 Jan 2013-
Abstract: Chapter 1. Introduction Part I: Linear Equations Chapter 2. Laplace's Equation 2.1 The Mean Value Inequalities 2.2 Maximum and Minimum Principle 2.3 The Harnack Inequality 2.4 Green's Representation 2.5 The Poisson Integral 2.6 Convergence Theorems 2.7 Interior Estimates of Derivatives 2.8 The Dirichlet Problem the Method of Subharmonic Functions 2.9 Capacity Problems Chapter 3. The Classical Maximum Principle 3.1 The Weak Maximum Principle 3.2 The Strong Maximum Principle 3.3 Apriori Bounds 3.4 Gradient Estimates for Poisson's Equation 3.5 A Harnack Inequality 3.6 Operators in Divergence Form Notes Problems Chapter 4. Poisson's Equation and Newtonian Potential 4.1 Holder Continuity 4.2 The Dirichlet Problem for Poisson's Equation 4.3 Holder Estimates for the Second Derivatives 4.4 Estimates at the Boundary 4.5 Holder Estimates for the First Derivatives Notes Problems Chapter 5. Banach and Hilbert Spaces 5.1 The Contraction Mapping 5.2 The Method of Cintinuity 5.3 The Fredholm Alternative 5.4 Dual Spaces and Adjoints 5.5 Hilbert Spaces 5.6 The Projection Theorem 5.7 The Riesz Representation Theorem 5.8 The Lax-Milgram Theorem 5.9 The Fredholm Alternative in Hilbert Spaces 5.10 Weak Compactness Notes Problems Chapter 6. Classical Solutions the Schauder Approach 6.1 The Schauder Interior Estimates 6.2 Boundary and Global Estimates 6.3 The Dirichlet Problem 6.4 Interior and Boundary Regularity 6.5 An Alternative Approach 6.6 Non-Uniformly Elliptic Equations 6.7 Other Boundary Conditions the Obliue Derivative Problem 6.8 Appendix 1: Interpolation Inequalities 6.9 Appendix 2: Extension Lemmas Notes Problems Chapter 7. Sobolev Spaces 7.1 L^p spaces 7.2 Regularization and Approximation by Smooth Functions 7.3 Weak Derivatives 7.4 The Chain Rule 7.5 The W^(k,p) Spaces 7.6 DensityTheorems 7.7 Imbedding Theorems 7.8 Potential Estimates and Imbedding Theorems 7.9 The Morrey and John-Nirenberg Estimes 7.10 Compactness Results 7.11 Difference Quotients 7.12 Extension and Interpolation Notes Problems Chapter 8 Generalized Solutions and Regularity 8.1 The Weak Maximum Principle 8.2 Solvability of the Dirichlet Problem 8.3 Diferentiability of Weak Solutions 8.4 Global Regularity 8.5 Global Boundedness of Weak Solutions 8.6 Local Properties of Weak Solutions 8.7 The Strong Maximum Principle 8.8 The Harnack Inequality 8.9 Holder Continuity 8.10 Local Estimates at the Boundary 8.11 Holder Estimates for the First Derivatives 8.12 The Eigenvalue Problem Notes Problems Chapter 9. Strong Solutions 9.1 Maximum Princiles for Strong Solutions 9.2 L^p Estimates: Preliminary Analysis 9.3 The Marcinkiewicz Interpolation Theorem 9.4 The Calderon-Zygmund Inequality 9.5 L^p Estimates 9.6 The Dirichlet Problem 9.7 A Local Maximum Principle 9.8 Holder and Harnack Estimates 9.9 Local Estimates at the Boundary Notes Problems Part II: Quasilinear Equations Chapter 10. Maximum and Comparison Principles 10.1 The Comparison Principle 10.2 Maximum Principles 10.3 A Counterexample 10.4 Comparison Principles for Divergence Form Operators 10.5 Maximum Principles for Divergence Form Operators Notes Problems Chapter 11. Topological Fixed Point Theorems and Their Application 11.1 The Schauder Fixes Point Theorem 11.2 The Leray-Schauder Theorem: a Special Case 11.3 An Application 11.4 The Leray-Schauder Fixed Point Theorem 11.5 Variational Problems Notes Chapter 12. Equations in Two Variables 12.1 Quasiconformal Mappings 12.2 holder Gradient Estimates for Linear Equations 12.3 The Dirichlet Problem for Uniformly Elliptic Equations 12.4 Non-Uniformly Elliptic Equations Notes Problems Chapter 13. Holder Estimates for more

Topics: Dirichlet problem (66%), Schauder estimates (65%), Elliptic boundary value problem (63%) more

17,825 Citations

Open accessBook
01 Apr 1985-
Abstract: Foreword Preface 1. Sobolev spaces 2. Regular second-order elliptic boundary value problems 3. Second-order elliptic boundary value problems in convex domains 4. Second-order boundary value problems in polygons 5. More singular solutions 6. Results in spaces of Holder functions 7. A model fourth-order problem 8. Miscellaneous Bibliography Index. more

4,904 Citations

Open accessJournal ArticleDOI: 10.1080/03605300600987306
Luis A. Caffarelli1, Luis Silvestre2Institutions (2)
Abstract: The operator square root of the Laplacian (−△) 1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems. more

Topics: Elliptic boundary value problem (63%), Vector Laplacian (62%), Laplacian matrix (62%) more

2,374 Citations

Open accessBook
01 Jan 1987-
Abstract: Part 1 Hypergeometric Functions and Elliptic Integrals: Some Basic Topics In Analytical Dynamics The Problem of Two Bodies Two-Body Orbits and the Initial-Value Problem Solving Kepler's Equation Two-Body Orbital Boundary Value Problem Solving Lambert's Problem Appendices Part 2 Non-Keplerian Motion: Patched-Conic Orbits and Perturbation Methods Variation of Parameters Two-Body Orbital Transfer Numerical Integration of Differential Equations The Celestial Position Fix Space Navigation Appendices more

Topics: Lambert's problem (60%), Elliptic boundary value problem (58%), Free boundary problem (57%) more

1,997 Citations

Open accessBook
Shmuel Agmon1Institutions (1)
01 Jan 1965-
Abstract: This book, which is a new edition of a book originally published in 1965, presents an introduction to the theory of higher-order elliptic boundary value problems. The book contains a detailed study of basic problems of the theory, such as the problem of existence and regularity of solutions of higher-order elliptic boundary value problems. It also contains a study of spectral properties of operators associated with elliptic boundary value problems. Weyl's law on the asymptotic distribution of eigenvalues is studied in great generality. more

1,508 Citations

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Topic's top 5 most impactful authors

Ravi P. Agarwal

15 papers, 206 citations

Q-Heung Choi

13 papers, 26 citations

Serguei A. Nazarov

13 papers, 271 citations

Allaberen Ashyralyev

11 papers, 177 citations

Vladimir Maz'ya

10 papers, 693 citations

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