Study on Elliptic Partial Differential Equations

Abstract

The thesis entitled with Study on Elliptic Partial Differential Equations is a serious and a keen study towards the ambit of most influential and practical scenario of Mathematics i.e. Partial Differential Equations. Before in-sighting towards PDE, this thesis includes two chapters dedicated to partial differential equations. Chapter 1 discusses the classification of partial differential equations providing its detailed classification and definitions (see 1.1.1) and also, insight classification of partial differential equations is also included in B. Chapter 1 is an introductory step towards chapter 2 which provides the basic knowledge of symbols and notations that are immensely used in chapter 2. Chapter 1 keeps its concerns with Quasi-Linear Partial Differential Equations and Semi-Linear Partial Differential Equations. Chapter 2 has a serious agenda of this thesis. Moving further in chapter 2, the governing definition of elliptic partial differential equations is provided (see 2.1.1) which is all the way important in the course of thesis. The section 2.1 provides the outline of all those aspects which have to analysed in section 2.2 which is, thus very important. Chapter 2 begins with the Reisz- Representation Theorem. This theorem establishes an important connection between a Hilbert space and its (continuous) dual space. Moving on, this thesis presents the Bilinear Forms in Elliptic Partial Differential Equations. The second last topic is Poinc˜re Inequality which helps in estimating the norm of a function in terms of a norm of its derivative. The Lax-Milgram Theorem incorporates as more general form of Reisz-Representation Theorem, as it applies to bilinear forms that are not necessarily symmetric.