Isolated Singularities in Partial Differential Inequalities 1st Edition by Marius Ghergu, Steven Taliaferro ISBN 1107138388 978-1107138384
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Product details:
ISBN 10: 1107138388
ISBN 13: 978-1107138384
Author: Marius Ghergu, Steven D. Taliaferro
In this monograph, the authors present some powerful methods for dealing with singularities in elliptic and parabolic partial differential inequalities. Here, the authors take the unique approach of investigating differential inequalities rather than equations, the reason being that the simplest way to study an equation is often to study a corresponding inequality; for example, using sub and superharmonic functions to study harmonic functions. Another unusual feature of the present book is that it is based on integral representation formulae and nonlinear potentials, which have not been widely investigated so far. This approach can also be used to tackle higher order differential equations. The book will appeal to graduate students interested in analysis, researchers in pure and applied mathematics, and engineers who work with partial differential equations. Readers will require only a basic knowledge of functional analysis, measure theory and Sobolev spaces.
Table of contents:
1 Representation formulae for singular solutions of polyharmonic and parabolic inequalities
1.1 Introduction
1.2 Harmonic inequalities in the punctured ball
1.3 Polyharmonic inequalities in the punctured ball
1.4 Parabolic inequalities
1.5 Comments
2 Isolated singularities of nonlinear second-order elliptic inequalities
2.1 Introduction
2.2 Case n ≥ 3
2.3 Case n = 2
2.4 Comments
3 More on isolated singularities for semilinear elliptic inequalities
3.1 Introduction
3.2 Local behavior of solutions
3.3 Global existence of solutions
3.4 Case p 1-2
3.5 Comments
4 Elliptic inequalities for the Laplace operator with Hardy potential
4.1 Introduction
4.2 Some comparison principles for the Laplace operator with L¹ potential
4.3 Elliptic inequalities with Hardy terms in cone-like domains
4.4 Elliptic inequalities with Hardy terms in smooth domains with boundary singularities
4.5 Elliptic inequalities with Hardy terms and higher dimensional singularity
4.6 Comments
5 Singular solutions for second-order nondivergence type elliptic inequalities
5.1 Introduction
5.2 Effective dimension and other important quantities
5.3 An Emden-Fowler equation
5.4 First result. Estimates on the critical exponent
5.5 Second result. The critical case
5.6 Examples
5.7 Comments
6 Isolated singularities of polyharmonic inequalities
6.1 Introduction
6.2 Polyharmonic inequalities in punctured balls
6.3 Polyharmonic inequalities in exterior domains
6.4 Comments
7 Nonlinear biharmonic inequalities
7.1 Introduction
7.2 Main results
7.3 Preliminary results
7.4 Case n ≥ 3
7.5 Case n = 2
7.6 Comments
8 Initial blow-up for nonlinear parabolic inequalities
8.1 Introduction
8.2 Optimal exponents for a priori bounds
8.3 Parabolic inequalities with boundary conditions
8.4 Comments
9 Semilinear elliptic systems of differential inequalities
9.1 Introduction
9.2 Case n = 2
9.3 Case n ≥ 3
9.4 Comments
10 Isolated singularities for nonlocal elliptic systems
10.1 Introduction: What is a nonlocal equation?
10.2 A nonlocal elliptic system with convolution terms
10.3 Some integral estimates
10.4 Case min{a, b} ≤ 2
10.5 Some auxiliary results for integral systems
10.6 Case a, b (2, h)
10.7 Comments
11 Isolated singularities for systems of parabolic inequalities
11.1 Introduction
11.2 Preliminary lemmas
11.3 The main results
11.4 Comments
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Tags: Marius Ghergu, Steven Taliaferro, Isolated Singularities, Partial Differential Inequalities