Parabolic Quasilinear Equations Minimizing Linear Growth Functionals

1 Total Variation Based Image Restoration.- 1.1 Introduction.- 1.2 Equivalence between Constrained and Unconstrained Restoration.- 1.3 The Partial Differential Equation Satisfied by the Minimum of (1.17).- 1.4 Algorithm and Numerical Experiments.- 1.5 Review of Numerical Methods.- 2 The Neumann Problem for the Total Variation Flow.- 2.1 Introduction.- 2.2 Strong Solutions in L2(?).- 2.3 The Semigroup Solution in L1(?).- 2.4 Existence and Uniqueness of Weak Solutions.- 2.5 An LN-L? Regularizing Effect.- 2.6 Asymptotic Behaviour of Solutions.- 2.7 Regularity of the Level Lines.- 3 The Total Variation Flow in ?N.- 3.1 Initial Conditions in L2(?N).- 3.2 The Notion of Entropy Solution.- 3.3 Uniqueness in Ló(?N).- 3.4 Existence in Lloc1.- 3.5 Initial Conditions in L2(?N).- 3.6 Time Regularity.- 3.7 An LN-L? Regularizing Effect.- 3.8 Measure Initial Conditions.- 4 Asymptotic Behaviour and Qualitative Properties of Solutions.- 4.1 Radially Symmetric Explicit Solutions.- 4.2 Some Qualitative Properties.- 4.3 Asymptotic Behaviour.- 4.4 Evolution of Sets in ?2: The Connected Case.- 4.5 Evolution of Sets in ?2: The Nonconnected Case.- 4.6 Some Examples.- 4.7 Explicit Solutions for the Denoising Problem.- 5 The Dirichlet Problem for the Total Variation Flow.- 5.1 Introduction.- 5.2 Definitions and Preliminary Facts.- 5.3 The Main Result.- 5.4 The Semigroup Solution.- 5.5 Strong Solutions for Data in L2(?).- 5.6 Existence and Uniqueness for Data in L1(?).- 5.7 Regularity for Positive Initial Data.- 6 Parabolic Equations Minimizing Linear Growth Functionals: L2-Theory.- 6.1 Introduction.- 6.2 Preliminaries.- 6.3 The Existence and Uniqueness Result.- 6.4 Strong Solution for Data in L2(?)).- 6.5 Asymptotic Behaviour.- 6.6 Proof of the Approximation Lemma.- 7 Parabolic Equations Minimizing Linear Growth Functionals: L1-Theory.- 7.1 Introduction.- 7.2 The Main Result.- 7.3 The Semigroup Solution.- 7.4 Existence and Uniqueness for Data in L1(?).- 7.5 A Remark for Strictly Convex Lagrangians.- 7.6 The Cauchy Problem.- A Nonlinear Semigroups.- A.1 Introduction.- A.2 Abstract Cauchy Problem.- A.3 Mild Solutions.- A.4 Accretive Operators.- A.5 Existence and Uniqueness Theorem.- A.6 Regularity of Mild Solutions.- A.7 Completely Accretive Operators.- B Functions of Bounded Variation.- B.2 Approximation by Smooth Functions.- B.3 Traces and Extensions.- B.4 Sets of Finite Perimeter and the Coarea Formula.- B.5 Some Isoperimetric Inequalities.- B.6 The Reduced Boundary.- B.7 Connected Components of Sets of Finite Perimeter.- C Pairings Between Measures and Bounded Functions.- C.1 Trace of the Normal Component of Certain Vector Fields.- Dankwoord/ Acknowledgements.