Non-Linear Analysis and Application to Partial Differential Equations

Bali, Saransh and Manna, Bhakti Bhusan (2018) Non-Linear Analysis and Application to Partial Differential Equations. Masters thesis, Indian Institute of Technology, Hyderabad.

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Abstract

In the past couple of decades, when Linear Functional Analysis was quite widely and completely established, the interest of mathematicians towards Non-linear Analysis has increased a lot. At one hand the treatment of various classical problems has been uni�ed, on the other, theories speci�cally non-linear one of great signi�cance and applicability have come out. In Non-linear Functional Analysis we study the properties of (continuous) mappings between the normed linear spaces and we describe the methods for solving non-linear equaions involving such mappings. For �nding the solution of non-linear equations there are primarily two major approaches which are known as topological methods and variational methods. Topological methods are derived from �xed point theorems and one of the important tools used in this direction are the Topological Degree and Morse Theory. Variational methods describe the solutions as critical points of a suitable functional and study ways of locating them. Moreover, there is an important fact to be noted. The fact is that the problems that are often considered to be di�cult, once they are framed in an appropriate functional setting, may be faced and solved quite easily. Here in this project, we provide an introduction to the basic aspect of Non-Linear Analysis mainly those which are based on di�erential calculus in Banach spaces. We have expressed the results here in geometric style in such a way that they are often a transposition of in�nite dimensions of events, which are intutive in R2 or R3. A particular nature of Non-Linear Analysis is that its theory has direct applications, especially to those related to di�erntial equations, where the power of non-linear methods is expressed in a more striking way like Degree Theory, Bifurcation Theory and Morse Theory.

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