Book Details :
LanguageEnglish
Pages311
FormatPDF
Size3.9 MB

# Methods on Nonlinear Elliptic Equations

### The Author of Methods on Nonlinear Elliptic Equations

Wenxiong Chen and Congming Li are the editors of Methods on Nonlinear Elliptic Equations PDF Book

• Introduction to Sobolev Spaces
• Existence of Weak Solutions
• Regularity of Solutions
• Preliminary Analysis of Riemannian Manifold
• Prescribing Gaussian Curvature on Compact 2-Manifolds
• The Yamabe Problem and Prescribing Scalar Curvature on S", for n. > 3
• Maximum Principles
• Methods of Moving Planes and Moving Spheres

## Preface to Methods on Nonlinear Elliptic Equations PDF

In this book, we present basic concepts as well as real research examples to young researchers interested in the field of non-linear analysis of partial differential equations (PDEs); in particular, the text focuses on the analysis of semi-linear elliptic PDEs.

We hope that graduate students consider our text good reading material and that professors consider it a handy textbook for use in a topics course on non-linear analysis.

After necessary preparations for basic knowledge have been made, a series of typical methods in the non-linear analysis will be introduced, some of which are well known while others are relatively new.

We will first illustrate these ideas and techniques using simple examples; we will then lead readers to the research front and explain how these methods can be applied to solve practical problems through careful analysis of a series of recent research articles.

Roughly speaking, in applying these commonly used methods, there are usually two aspects: i) A general scheme (more or less universal) and, ii) Key points for each individual problem.

By understanding these research examples, readers should then be able to apply these general schemes to solve their own research problems by discovering their own key points.

In Chapter 1, we introduce basic concepts of Sobolev spaces and some commonly used inequalities. These are the major spaces in which we will seek weak solutions to PDEs.

Chapter 2 shows how to find weak solutions for some typical linear and semi-linear PDEs by using functional analysis methods, mainly, the calculus of variations and critical point theories, including the well-known Mountain Pass Lemma. In Chapter 3, we establish W2, P a priori estimates, and regularity.

We prove that, in most cases, weak solutions are actually differentiable and hence are classical solutions. Our approach here is quite different from the traditional approach.

We will also present two Regularity Lifting Theorems. The first Theorem uses an operator which is contracting in both spaces.

It is a simple method to boost the regularity of solutions and has been used extensively in various forms in the authors' previous works.

The essence of the approach is well known in the analysis community, while the version here contains some new developments.

The second Theorem employs an operator which is contracting in one space and shrinking in the other space. It is a brand new idea, and as we believe, it will find broad applications in a variety of nonlinear problems.

We will use examples to show how these Theorems can be applied to systems of PDEs and integral equations including a fully nonlinear system of Wolff type.

Chapter 4 is a preparation for chapters 5 and 6. We introduce Riemannian manifolds, curvatures, covariant derivatives, and Sobolev embedding on manifolds.

Chapter 5 deals with semi-linear elliptic equations arising from prescribing Gaussian curvature on both positively and negatively curved manifolds.

We show the existence of weak solutions in both subcritical and critical cases via variational approaches. We also introduce the method of lower and upper solutions.

Chapter 6 focuses on the well-known Yamabe problem and its generalization, prescribing scalar curvature on S' for n > 3. The latter is in the critical case where the corresponding variational functional is not compact at any level sets.

To recover the compactness, we construct a max-mini variational scheme. The outline is clearly presented; however, the detailed proofs are rather complex, and beginners are welcome to skip these proofs.

Chapter 7 is devoted to the study of various Maximum Principles, in particular, the ones based on comparisons.

Besides classical ones, we also introduce a version of the Maximum Principle at infinity and a Maximum Principle for integral equations that basically depends on the absolute continuity of a Lebesgue integral.

It is a preparation for the Method of Moving Planes.

In Chapter 8, we introduce the Method of Moving Planes and its variant the Method of Moving Spheres-and apply them to obtain the symmetry, monotonicity, a priori estimates, and even non-existence of solutions.

We also introduce an integral form of the Method of Moving Planes, a relatively new idea that is quite different from the traditional ones for PDEs.

Instead of using local properties of a PDE, global norms of solutions for integral equations will be exploited.