An Introduction to Maximum Principles and Symmetry in Elliptic Problems (Cambridge Tracts in Mathematics, Series Number 128)

This book presents the basic theory of the symmetry of solutions to second-order elliptic partial differential equations by means of the maximum principle. It proceeds from elementary facts about the linear case to recent results about positive solutions of nonlinear elliptic equations. Gidas, Ni and Nirenberg, building on the work of Alexandrov and Serrin, have shown that the shape of the set on which such elliptic equations are solved has a strong effect on the form of positive solutions. In particular, if the equation and its boundary condition allow spherically symmetric solutions, then, remarkably, all positive solutions are spherically symmetric. These recent and important results are presented with minimal prerequisites, in a style suited to graduate students. Two long appendices give a leisurely account of basic facts about the Laplace and Poisson equations, and there is an abundance of exercises, with detailed hints, some of which contain new results. Contents......Page 6 Preface......Page 8 0 Some Notation, Terminology and Basic Calculus......Page 12 1.1 A glimpse of objectives......Page 28 1.2 What are maximum principles?......Page 30 1.3 On reflection in hyperplanes......Page 35 1.4 What is symmetry?......Page 38 1.5 Exercises......Page 43 2.1 Linear elliptic operators of order two......Page 50 2.2 The weak maximum principle......Page 52 2.3 The boundary-point lemma and the strong maximum principle......Page 61 2.4 A maximum principle for thin sets S2......Page 67 2.5 Steps towards Phragmen-Lindelof theory......Page 72 2.6 Comparison functions of Siegel type......Page 83 2.7 Some Phragmen-Lindelof theory for subharmonic functions......Page 88 2.8 Exercises......Page 95 3.1 The simplest case......Page 98 3.2 A discontinuous non-linearity f......Page 104 3.3 Exercises......Page 112 4.1 Statement of the main result......Page 117 4.2 Four lemmas about reflection of v......Page 122 4.3 Proof of Theorem 4.2 and a corollary......Page 131 4.4 Application to some Newtonian potentials......Page 133 4.5 Exercises......Page 144 5.1 Prospectus......Page 152 5.2 On the geometry of caps and reflected caps......Page 153 5.3 Monotonicity in fl......Page 164 5.4 A little topology......Page 170 5.5 Exercises......Page 173 A.1 Point sources in R3......Page 178 A.2 The Newtonian potential: first steps......Page 185 A.3 Continuity of the force field Vu......Page 205 A.4 Multipoles and the far field......Page 210 A.5 Second derivatives of u at points in G......Page 214 A.6 Exercises......Page 224 B.1 Real-analytic functions......Page 232 B.2 Smoothness and mean-value properties of harmonic functions......Page 235 B.3 The Kelvin transformation......Page 243 B.4 On the Dirichlet and Neumann problems......Page 246 B.5 The solution of the Dirichlet problem for a ball......Page 259 B.6 Exercises......Page 273 Appendix C. Construction of the Primary Function of Siegel Type......Page 281 D.1 A first divergence theorem......Page 290 D.2 Extension to some sets with edges and vertices......Page 296 D.3 Interior approximations to the boundary 00......Page 304 D.4 Exercises......Page 311 E.1 Preliminaries......Page 316 E.2 Bluntness and ellipticity under co-ordinate transformations......Page 320 E.3 Two stages of the edge-point lemma......Page 322 Notes on Sources......Page 335 References......Page 343 Index......Page 348 This Is The First Book To Present The Basic Theory Of The Symmetry Of Solutions To Second-order Elliptic Partial Differential Equations By Means Of The Maximum Principle. It Proceeds From Elementary Facts About The Linear Case To Recent Results About Positive Solutions Of Non-linear Elliptic Equations. Gidas, Ni And Nirenberg, Building On Work Of Alexandrov And Of Serrin, Have Shown That The Shape Of The Set On Which Such Elliptic Equations Are Solved Has A Strong Effect On The Form Of Positive Solutions. In Particular, If The Equation And Its Boundary Condition Allow Spherically Symmetric Solutions, Then, Remarkably, All Positive Solutions Are Spherically Symmetric.--jacket. Some Notation, Terminology And Basic Calculus -- 1. Introduction -- 2. Some Maximum Principles For Elliptic Equations -- 3. Symmetry For A Non-linear Poisson Equation In A Symmetric Set [omega] -- 4. Symmetry For The Non-linear Poisson Equation In R[superscript N] -- 5. Monotonicity Of Positive Solutions In A Bounded Set [omega] -- App. A. On The Newtonian Potential -- App. B. Rudimentary Facts About Harmonic Functions And The Poisson Equation -- App. C. Construction Of The Primary Function Of Siegel Type -- App. D. On The Divergence Theorem And Related Matters -- App. E. The Edge-point Lemma. L.e. Fraenkel. Includes Bibliographical References (p. 332-336) And Index. Originally published in 2000, this was the first book to present the basic theory of the symmetry of solutions to second-order elliptic partial differential equations by means of the maximum principle. It proceeds from elementary facts about the linear case to recent results about positive solutions of non-linear elliptic equations. Gidas, Ni and Nirenberg, building on work of Alexandrov and of Serrin, have shown that the shape of the set on which such elliptic equations are solved has a strong effect on the form of positive solutions. In particular, if the equation and its boundary condition allow spherically symmetric solutions, then, remarkably, all positive solutions are spherically symmetric. Results are presented with minimal prerequisites in a style suited to graduate students. Two long and leisurely appendices give basic facts about the Laplace and Poisson equations. There is a plentiful supply of exercises, with detailed hints. The author of this 2000 volume proceeds from elementary facts about the linear case to recent results about positive solutions of non-linear elliptic equations, with minimal prerequisites in a style suited to graduate students. There is a plentiful supply of exercises, with detailed hints, some of which contain new results. Preamble Let denote, now and henceforth, an open, non-empty subset of the real, N-dimensional, Euclidean space RN, and let denote its closure.