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Nonlinear Elliptic Equations of the Second Order

Qing Han
Publisher: 
American Mathematical Society
Publication Date: 
2016
Number of Pages: 
368
Format: 
Hardcover
Series: 
Graduate Studies in Mathematics 171
Price: 
89.00
ISBN: 
9781470426071
Category: 
Textbook
[Reviewed by
Jason M. Graham
, on
07/20/2016
]

Let \(\Omega \subset \mathbb{R}^{n}\) be a domain. Typically, \(\Omega\) is taken to be open, connected, and having a smooth boundary, \(\partial \Omega\). Consider the three following forms of second-order partial differential equations (PDEs): \[ \begin{align} \sum_{i,j=1}^{n}a_{ij}(x)\partial_{ij}u + \sum_{i=1}^{n}b_{i}(x)\partial_{i}u + c(x)u &= f(x), \tag{1} \\ \sum_{i,j=1}^{n}a_{ij}(x,u,\nabla u)\partial_{ij}u &= f(x,u,\nabla u), \tag{2}\\ F(x,u,\nabla u, \nabla^{2} u) &= 0.\tag{3} \end{align} \] The equations (1), (2), (3) are linear, quasi-linear, and fully nonlinear, respectively. Additionally, with certain simple conditions placed on the coefficients matrices \(a_{ij}(x)\), \(a_{ij}(x,u,\nabla u)\) and the function \(F\), we get so-called second-order elliptic PDEs . Some representative examples of each case are Poisson’s equation (linear) \[ \begin{align} -\sum_{i=1}^{n}\partial_{ii}u &= f(x), \end{align}\] the minimal surface equation (quasi-linear) \[\begin{align} \nabla \cdot \left(\frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} \right) &= 0, \end{align}\] and the Monge-Ampére equation (fully nonlinear) \[\begin{align} \text{det}(\nabla^{2} u) &= f(x). \end{align}\] We note that both the minimal surface and the Monge-Ampére equations arise from differential geometry.

There is a well-developed and by now classical theory for the existence, uniqueness and continuity/differentiability properties, i.e. the regularity, of solutions to second-order linear elliptic PDEs. Nonlinear Elliptic Equations of the Second Order by Qing Han presents the more recently developed theory for second-order quasi-linear and fully nonlinear, collectively simply termed nonlinear, elliptic equations. Specifically, Han covers the theory of solutions to Dirichlet boundary value problems for second-order nonlinear elliptic PDEs in which the solution \(u\) to one of (2) or (3) is a prescribed function on the boundary \(\partial \Omega\). As is the case for linear equations, the elliptic structure is fundamental in determining the properties of solutions to the Dirichlet problem for (2) or (3).

Nonlinear Elliptic Equations of the Second Order is an advanced text that assumes that the reader has prior exposure to the analysis of solutions of PDEs at the level of Evans’s Partial Differential Equations. Nevertheless, the text begins with a detailed review of essential theory for linear equations of the form (1). This overview of the linear theory simultaneously reminds the reader of results to be used later in the text, and sets the stage for the approach that is taken to study solutions of the nonlinear equations. Namely the derivation and use of a priori estimates for the solutions of equations. It is here that the ellipticity comes into play since it is the elliptic structure of the equations that forces solutions to satisfy a maximum principle, Harnack inequality, etc.

This book is well-written and the theory is developed very carefully by the author. It is clear that Professor Han had the student in mind when writing this book. A strong point for the text is the excellent job that the author does in motivating difficult material. Each chapter begins with an overview. Also, there is a good amount of discussion between results that provides perspective on the material. These two features aid the reader by helping to keep the big picture in view among a mass of detail. It is also interesting to see the general theory in action applied to specific equations such as mean curvature and Monge-Ampére. This text should prove to be a valuable reference for students and researchers interested in the modern theory of elliptic PDE and its relation to differential geometry.

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Jason M. Graham is Assistant Professor of Mathematics at the University of Scranton.

See the table of contents in the publisher's webpage.