Pure and Applied Mathematics

Characteristic Inequalities in Banach Spaces and Applications

2013-05-14T00:00:00Z

Characteristic Inequalities in Banach Spaces and Applications Abdulrashid, Ismail The contribution of this project falls within the general area of nonlinear functional analysis and applications. We focus on an important topic within this area: Inequalities in Banach spaces and applications. As is well known, among all infinite dimensional Banach spaces, Hilbert spaces generally have simple geometric structures. This makes problems posed in them easier to handle, this is as a result of the existence of inner product, the proximity map, and the two characteristic identities. Refer to the main thesis for more detailed Abstract

Differential Forms and Applications

2011-12-06T00:00:00Z

Differential Forms and Applications Uchechukwu, Michael, Opara This project deals mainly with Differential Forms on smooth Riemannian manifolds and their applications through the properties of their classical Differential and Integral Operators. The calculus of Differential Forms provides a simple and flexible alternative to vector calculus. It is not dependent on any coordinate system, simplifies or condenses variational principles, offers a more comprehensive means of evaluating multivariate integrals, and is crucial in the analysis of the variation of differentiable functions on smooth manifolds. Differential Forms have numerous applications within (and beyond) Differential Geometry and Mathematical Physics. Needless to mention, Differential Forms constitute the ingredients (test functions) of the Theory of k-current which is analogous to Distribution Theory, and so they offer diverse potential tools for research.

Evolution Equations and Applications

2011-12-06T00:00:00Z

Evolution Equations and Applications Ndambomve, Patrice This project concerns Evolution Equations in Banach spaces and lies at the interface between Functional Analysis, Dynamical Systems, Modeling Theory and Natural Sciences. Evolution Equations include Partial Di erential Equations (PDEs) with time t as one of the independent variables and arise from many elds of Mathematics as well as Physics, Mechanics and Material Sciences (e.g., Systems of Conservation Law from Dynamics, Navier-Stokes and Eu- ler equations from Fluid Mechanics, Di usion equations from Heat transfer and Natural Sciences, Klein-Gordon and Schrödinger equations from Quantum Mechanics, Cahn-Hilliard equations and Porous media equations from Material Sciences, Evolution equations with memory from Pharma- cokinetics). In this project, we present the fundamental theory of abstract Evolution Equations by using the semigroup approach (which arises naturally from well-posed Cauchy problems: Theorem 2.2.6) and Fixed-point methods. More precisely, rst we review the basic notions of Functional Analysis and Di erential Analysis, secondly we study the theory of semigroups of bounded linear operators, and thirdly we consider Linear Evolution Equations (with emphasis on the di erence between the nite dimensional and the in nite dimensional case, that is due to domain restrictions) and moreover we give existence results (in appropriate sense) for Semilinear Evolution Equations of the form du = Au + f (t, u), dt t>0 ; u(0) = u0 where A is a linear operator that generates a C0 -semigroup and f satis es certain conditions. As applications we start with the evolution equation ∂t u + ∂x u = 0 in R and then after we show the existence of solutions to some Homogeneous Heat Equations, classical Wave equations, nonlinear Heat Equation, and to some nonlinear Wave equation.

Sobolev Spaces and Linear Elliptic Partial Differential Equations

2011-12-06T00:00:00Z

Sobolev Spaces and Linear Elliptic Partial Differential Equations Iyiola, Olaniyi, Samuel The cardinal goal to the study of theory of Partial Differential Equations (PDEs) is to insure or find out properties of solutions of PDE that are not directly at- tainable by direct analytical means. Certain function spaces have certain known properties for which solutions of PDEs can be classified. As a result, this work critically looked into some function spaces and their properties. We consider extensively, Lp − spaces, distribution theory and sobolev spaces. The emphasis is made on sobolev spaces, which permit a modern approach to the study of differential equations, defined as W m,p (Ω) = {u ∈ Lp (Ω), | Dα u ∈ Lp (Ω), for all α ∈ Nn : |α| ≤ m}. (1) it is provided with the norm: u W m,p (Ω) Dα u = (2) Lp (Ω) . |α|≤m for 1 ≤ p < ∞, we have 1 p  u W m,p (Ω) Dα u = p  Lp (Ω) . (3) |α|≤m Finally,we also have u W m,p (Ω) = max{ Dα u Lp (Ω) : |α| ≤ m} (4) This is the key to the entire work, even though others spaces are ingredients. The study is based on variational formulation of some Boundary Value Problems using some known thoerem (Lax-Milgram Theorem) to ascertain the existence and uniqueness of solution to such linear PDEs. v We do not consider all types of PDEs (Parabolic, Hyperbolic and Elliptic).