Elliptic Partial Differential Equations of Second Order
01 Jan 1997-pp 213-267
TL;DR: In this paper, a class of partial differential equations that generalize and are represented by Laplace's equation was studied. And the authors used the notation D i u, D ij u for partial derivatives with respect to x i and x i, x j and the summation convention on repeated indices.
Abstract: We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations of second order. A linear partial differential operator L defined by
$$ Lu{\text{: = }}{a_{ij}}\left( x \right){D_{ij}}u + {b_i}\left( x \right){D_i}u + c\left( x \right)u $$
is elliptic on Ω ⊂ ℝ n if the symmetric matrix [a ij ] is positive definite for each x ∈ Ω. We have used the notation D i u, D ij u for partial derivatives with respect to x i and x i , x j and the summation convention on repeated indices is used. A nonlinear operator Q,
$$ Q\left( u \right): = {a_{ij}}\left( {x,u,Du} \right){D_{ij}}u + b\left( {x,u,Du} \right) $$
[D u = (D 1 u, ..., D n u)], is elliptic on a subset of ℝ n × ℝ × ℝ n ] if [a ij (x, u, p)] is positive definite for all (x, u, p) in this set. Operators of this form are called quasilinear. In all of our examples the domain of the coefficients of the operator Q will be Ω × ℝ × ℝ n for Ω a domain in ℝ n . The function u will be in C 2(Ω) unless explicitly stated otherwise.
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TL;DR: In this article, the authors deal with the fractional Sobolev spaces W s;p and analyze the relations among some of their possible denitions and their role in the trace theory.
Abstract: This paper deals with the fractional Sobolev spaces W s;p . We analyze the relations among some of their possible denitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.
3,555 citations
TL;DR: In this article, the authors studied the problem of finding the optimal regularity result for the contact set of a function ϕ and s ∈ (0, 1) when ϕ is C 1,s or smoother, and showed that the solution u is in the space c 1,α for every α < s.
Abstract: Given a function ϕ and s ∈ (0, 1), we will study the solutions of the following obstacle problem: • u ≥ ϕ in R n , • (−� ) s u ≥ 0i nR n , • (−� ) s u(x) = 0 for those x such that u(x )>ϕ (x), • lim|x|→+∞ u(x) = 0. We show that when ϕ is C 1,s or smoother, the solution u is in the space C 1,α for every α< s. In the case where the contact set {u = ϕ} is convex, we prove the optimal regularity result u ∈ C 1,s . When ϕ is only C 1,β for a β< s, we prove that our solution u is C 1,α for every α< β. c � 2006 Wiley Periodicals, Inc.
1,259 citations
Cites background from "Elliptic Partial Differential Equat..."
...We will show how (−4)σ interacts with Cα norms, and a characterization of its supersolutions....
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Book•
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01 Jan 2000
TL;DR: In this article, the Poincare and Sobolev inequalities, pointwise estimates, and pointwise classifications of Soboleve classes are discussed. But they do not cover the necessary conditions for Poincarse inequalities.
Abstract: Introduction What are Poincare and Sobolev inequalities? Poincare inequalities, pointwise estimates, and Sobolev classes Examples and necessary conditions Sobolev type inequalities by means of Riesz potentials Trudinger inequality A version of the Sobolev embedding theorem on spheres Rellich-Kondrachov Sobolev classes in John domains Poincare inequality: examples Carnot-Caratheodory spaces Graphs Applications to PDE and nonlinear potential theory Appendix References.
1,093 citations
Book•
21 Oct 2015
TL;DR: In this paper, the primal and dual problems of one-dimensional problems are considered. But they do not consider the dual problems in L^1 and L^infinity theory.
Abstract: Preface.- Primal and Dual Problems.- One-Dimensional Issues.- L^1 and L^infinity Theory.- Minimal Flows.- Wasserstein Spaces.- Numerical Methods.- Functionals over Probabilities.- Gradient Flows.- Exercises.- References.- Index.
1,015 citations
Cites background or methods from "Elliptic Partial Differential Equat..."
...To do that, note that we have ||ψn||C2,α ≤ C||ρ − ρn||C0,α from standard elliptic regularity theory (see Schauder Estimates in Chapter 6 in [185])....
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...The proof is based on a priori bounds derived from both elliptic regularity theory (see [185], for instance) and Caffarelli’s regularity for Monge-Ampère (see Section 1....
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Book•
26 Oct 2006
TL;DR: The Porous Medium Equation (PME) as discussed by the authors is one of the classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood.
Abstract: The Heat Equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. In this monograph, aimed at research students and academics in mathematics and engineering, as well as engineering specialists, Professor Vazquez provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises for the reader.
978 citations
Cites methods from "Elliptic Partial Differential Equat..."
...We will use notations that are rather standard in PDE texts, like Evans [229], GilbargTrudinger [261] or equivalent, which we assume known to the reader....
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07 Jan 2013
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
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
18,443 citations
Book•
01 Jan 1980
TL;DR: In this paper, the SIAM edition Preface Glossary of notations Introduction Part I. Variational Inequalities in Rn Part II. Free Boundary Problems Governed by Elliptic Equations and Systems Part VII. A One Phase Stefan Problem Bibliography Index.
Abstract: Preface to the SIAM edition Preface Glossary of notations Introduction Part I. Variational Inequalities in Rn Part II. Variational Inequalities in Hilbert Space Part III. Variational Inequalities for Monotone Operators Part IV. Problems of Regularity Part V. Free Boundary Problems and the Coincidence Set of the Solution Part VI. Free Boundary Problems Governed by Elliptic Equations and Systems Part VII. Applications of Variational Inequalities Part VIII. A One Phase Stefan Problem Bibliography Index.
4,107 citations
Book•
01 Jan 1977
TL;DR: In this article, a priori estimation of the gradient of the Bernstein problem is given. But the gradient is not a priorimate of the radius of the singular set, and it is not known whether the gradient can be estimated by direct methods.
Abstract: I: Parametric Minimal Surfaces.- 1. Functions of Bounded Variation and Caccioppoli Sets.- 2. Traces of BV Functions.- 3. The Reduced Boundary.- 4. Regularity of the Reduced Boundary.- 5. Some Inequalities.- 6. Approximation of Minimal Sets (I).- 7. Approximation of Minimal Sets (II).- 8. Regularity of Minimal Surfaces.- 9. Minimal Cones.- 10. The First and Second Variation of the Area.- 11. The Dimension of the Singular Set.- II: Non-Parametric Minimal Surfaces.- 12. Classical Solutions of the Minimal Surface Equation.- 13. The a priori Estimate of the Gradient.- 14. Direct Methods.- 15. Boundary Regularity.- 16. A Further Extension of the Notion of Non-Parametric Minimal Surface.- 17. The Bernstein Problem.- Appendix A.- Appendix B.- Appendix C.
2,479 citations
Book•
16 Dec 1985
TL;DR: In this paper, the authors propose a method of Gauss characterisation of the Energies of a narrow tube, and apply it to the problem of estimating the diameter of a small tube.
Abstract: 1 Introduction.- 1.1. Mean Curvature.- 1.2. Laplace's Equation.- 1.3. Angle of Contact.- 1.4. The Method of Gauss Characterization of the Energies.- 1.5. Variational Considerations.- 1.6. The Equation and the Boundary Condition.- 1.7. Divergence Structure.- 1.8. The Problem as a Geometrical One.- 1.9. The Capillary Tube.- 1.10. Dimensional Considerations.- Notes to Chapter 1.- 2 The Symmetric Capillary Tube.- 2.1. Historical and General.- 2.2. The Narrow Tube Center Height.- 2.3. The Narrow Tube Outer Height.- 2.4. The Narrow Tube Estimates Throughout the Trajectory.- 2.5. Height Estimates for Tubes of General Size.- 2.6. Meniscus Height Narrow Tubes.- 2.7. Meniscus Height General Case.- 2.8. Comparisons with Earlier Theories.- Notes to Chapter 2.- 3 The Symmetric Sessile Drop.- 3.1. The Correspondence Principle.- 3.2. Continuation Properties.- 3.3. Uniqueness and Existence.- 3.4. The Envelope.- 3.5. Comparison Theorems.- 3.6. Geometry of the Sessile Drop Small Drops.- 3.7. Geometry of the Sessile Drop Larger Drops.- Notes to Chapter 3.- 4 The Pendent Liquid Drop.- 4.1. Mise en Scene.- 4.2. Local Existence.- 4.3. Uniqueness.- 4.4. Global Behavior General Remarks.- 4.5. Small |u0|.- 4.6. Appearance of Vertical Points.- 4.7. Behavior for Large |u0|.- 4.8. Global Behavior.- 4.9. Maximum Vertical Diameter.- 4.10. Maximum Diameter.- 4.11. Maximum Volume.- 4.12. Asymptotic Properties.- 4.13. The Singular Solution.- 4.14. Isolated Character of Global Solutions.- 4.15. Stability.- Notes to Chapter 4.- 5 Asymmetric Case Comparison Principles and Applications.- 5.1. The General Comparison Principle.- 5.2. Applications.- 5.3. Domain Dependence.- 5.4. A Counterexample.- 5.5. Convexity.- Notes to Chapter 5.- 6 Capillary Surfaces Without Gravity.- 6.1. General Remarks.- 6.2. A Necessary Condition.- 6.3. Sufficiency Conditions.- 6.4. Sufficiency Conditions II.- 6.5. A Subsidiary Extremal Problem.- 6.6. Minimizing Sequences.- 6.7. The Limit Configuration.- 6.8. The First Variation.- 6.9. The Second Variation.- 6.10. Solution of the Jacobi Equation.- 6.11. Convex Domains.- 6.12. Continuous and Discontinuous Disappearance.- 6.13. An Example.- 6.14. Another Example.- 6.15. Remarks on the Extremals.- 6.16. Example 1.- 6.17. Example 2.- 6.18. Example 3.- 6.19. The Trapezoid.- 6.20. Tail Domains A Counterexample.- 6.21. Convexity.- 6.22. A Counterexample.- 6.23. Transition to Zero Gravity.- Notes to Chapter 6.- 7 Existence Theorems.- 7.1. Choice of Venue.- 7.2. Variational Solutions.- 7.3. Generalized Solutions.- 7.4. Construction of a Generalized Solution.- 7.5. Proof of Boundedness.- 7.6. Uniqueness.- 7.7. The Variational Condition Limiting Case.- 7.8. A Necessary and Sufficient Condition.- 7.9. A Limiting Configuration.- 7.10. The Case > 0>1.- 7.11. Application: A General Gradient Bound.- Notes to Chapter 7.- 8 The Capillary Contact Angle.- 8.1. Everyday Experience.- 8.2. The Hypothesis.- 8.3. The Horizontal Plane Preliminary Remarks.- 8.4. Necessity for ?.- 8.5. Proof that ? is Monotone.- 8.6. Geometrically Imposed Stability Bounds.- 8.7. A Further Kind of Instability.- 8.8. The Inclined Plane Preliminary Remarks.- 8.9. Integral Relations, and Impossibility of Constant Contact Angle.- 8.10. The Zero-Gravity Solution.- 8.11. Postulated Form for ?.- 8.12. Formal Analytical Solution.- 8.13. The Expansion Leading Terms.- 8.14. Computer Calculations.- 8.15. Discussion.- 8.16. Further Discussion.- Notes to Chapter 8.- 9 Identities and Isoperimetric Relations.
763 citations