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.
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"The originality of this book mainly consists in new proofs and new extensions of quite well known results concerning maximum principles and symmetry properties related to semilinear elliptic equations." Mathematical Reviews
Book Description
This is the first book to present the basic theory of the symmetry of solutions to second-order elliptic equations by means of the maximum principle. It proceeds from elementary facts about the linear case to recent results, 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 a plentiful supply of exercises, with detailed hints, some of which contain new results.
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Front Cover | Front Flap | Table of Contents | First Pages | Index | Back Flap | Back Cover | Surprise Me!
Inside This Book
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First Sentence:
Preamble Let denote, now and henceforth, an open, non-empty subset of the real, N-dimensional, Euclidean space RN, and let denote its closure. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
subsolution relative, admissible asymptotic behaviour, generalized subsolution, weak maximum principle, largest open interval, distributional solution, strong maximum principle, subharmonic functions, uniform ellipticity, bounded open subset, symmetric relative, smoothing kernel, comparison function, thin sets, invariance group, divergence theorem, generalized solution Key Phrases - Capitalized Phrases (CAPs): (learn more)
Proof Let, Exercises Exercise New!
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Front Cover | Front Flap | Table of Contents | First Pages | Index | Back Flap | Back Cover | Surprise Me!
Preamble Let denote, now and henceforth, an open, non-empty subset of the real, N-dimensional, Euclidean space RN, and let denote its closure. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
subsolution relative, admissible asymptotic behaviour, generalized subsolution, weak maximum principle, largest open interval, distributional solution, strong maximum principle, subharmonic functions, uniform ellipticity, bounded open subset, symmetric relative, smoothing kernel, comparison function, thin sets, invariance group, divergence theorem, generalized solution Key Phrases - Capitalized Phrases (CAPs): (learn more)
Proof Let, Exercises Exercise New!
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Front Cover | Front Flap | Table of Contents | First Pages | Index | Back Flap | Back Cover | Surprise Me!
Citations (learn more)
This book cites 2
books:
2
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cite this book:
- Let Newton Be! by John Fauvel in Back Matter
- Differential Equations (2nd Edition) by John Polking in Back Matter
- Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type (Oxford Lecture Series in Mathematics and Its Applications) by Juan Luis Vï¿1/2zquez in Back Matter
- Order Structure And Topological Methods in Nonlinear Partial Differential Equations Maximum... (Partial Differential Equations and Application) by Yihong Du in Back Matter
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