Function Spaces and Weak Formulation of Partial Differential Equations

Abstract

Functional analysis plays an increasing role in the applied sciences and mathematics. It is an abstract branch of mathematics that is developed from classical analysis. Proper functional analytic setting is important for the study of initial and boundary value problems. It is also important for the construction of effective numerical schemes. With the discovery of distributions, the role of functional analysis in partial differential equations has become more important. Numerical methods like finite element and finite volume methods depend on the results of functional analysis both for the construction and error analysis of the schemes. The accuracy of several approximations to partial differential equations very much depends on the smoothness of the analytical solution to the equation under consideration. So, smoothness of data becomes very important in the analysis. For this motive, we will consider some classes of functions with some specific differentiability and integrability properties, called function spaces. Chapter 1 starts with an introduction to function spaces with examples and we define the different norms which are used in analysis of partial differential equations. In addition to this, we have included the different inequalities which will be used in forthcoming chapters. In Chapter 2, we discuss the theory of distributions and also discussed the need of distributions. Several properties like differentiability, measure as distribution and functions as distributions have discussed. The concept of weak derivatives and convolution of distributions have included. Finally, in Chapter 3, we study the Sobolev spaces and norm defined on theses spaces. Then we study some extension theorems, imbedding theorems, compactness theorems, and trace theory. Then motivation for weak solution and its existence and uniqueness through LaxMilgram Theorem. Some test examples have been studied for weak formulation from elliptic partial differential equations.