Maximal Monotone Operators on Hilbert Spaces

Maximal Monotone Operators on Hilbert Spaces.

ABSTRACT 

Let H be a real Hilbert space and A: D(A) ⊂ H → H be an unbounded, linear, self-adjoint, and maximal monotone operator. The aim of this thesis is to solve u 0 (t) + Au(t) = 0, when A is linear but not bounded. The classical theory of differential linear systems cannot be applied here because the exponential formula exp(tA) does not make sense, since A is not continuous.

Here we assume A is maximal monotone on a real Hilbert space, then we use the Yosida approximation to solve. Also, we provide many results on regularity of solutions. To illustrate the basic theory of the thesis, we propose to solve the heat equation in L 2 (Ω). In order to do that, we use many important properties from Sobolev spaces, Green’s formula and Lax-Milgram’s theorem. 

TABLE OF CONTENTS

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i
Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .v
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vi
1 Hilbert Spaces and Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Lp Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Maximal Monotone Operators on Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . .8
2.1 Examples of maximal monotone operators . . . . . . . . . . . . . . . 11
2.2 Yosida Approximation of a maximal monotone operator . . . . . . . . 14
2.3 Self adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

INTRODUCTION  

This work exhibits existence and uniqueness results for a differential equation of first order with initial condition, governed by a maximal monotone operator on a Hilbert space. Firstly, we show the main proofs in maximal monotone operators needed for the existence and uniqueness theorem, the Hille-Yosida’s Theorem on Hilbert spaces and its proof are also presented. One very efficient way to describe some natural phenomena is through partial differential equation.

This field of mathematics constantly explains and solves real world problems emanating from different fields of study. The concept of maximal monotonicity results is a very rich theory that has been developed in this field of mathematics since the early 1960s. Maximal monotone operators are considered due to the existence and uniqueness Theorem presented. Definition 0.1. Let A: D(A) ⊂ H → H be a linear operator.

We define graph of A by Gr(A) = {(x, y) ∈ H × H: Ax = y}. Definition 0.2. Let A: D(A) ⊂ H → H be an operator. A is said to be closed if for all sequence (xn) ∈ D(A) such that xn → x in H and Axn → y in H. Then: (a) x ∈ D(A), (b) y = Ax. vi The aim of this work is to solve the following differential equation (P)    u 0 + Au = 0 on [0,∞) u(0) = u0 where H is a real Hilbert space, u : [0,∞) → H is the unknown function, A : D(A) ⊂ H → H is maximal monotone, u0 ∈ H is a given initial state. If A is linear and continuous, then existence and uniqueness is established by Theorem 2.7.

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StudentsandScholarship Team.