José M. Mazón

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 Lo(?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.

Nonlocal Diffusion Problems

Nonlocal diffusion problems arise in a wide variety of applications, including biology, image processing, particle systems, coagulation models, and mathematical finance. These types of problems are also of great interest for their purely mathematical content. This book presents recent results on nonlocal evolution equations with different boundary conditions, starting with the linear theory and moving to nonlinear cases, including two nonlocal models for the evolution of sandpiles. Both existence and uniqueness of solutions are considered, as well as their asymptotic behaviour. Moreover, the authors present results concerning limits of solutions of the nonlocal equations as a rescaling parameter tends to zero. With these limit procedures the most frequently used diffusion models are recovered: the heat equation, the

Existence and uniqueness for a degenerate parabolic equation with ¹-data

p

A NONLOCAL p-LAPLACIAN EVOLUTION EQUATION WITH NONHOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS

-Laplacian evolution equation, the porous media equation, the total variation flow, a convection-diffusion equation and the local models for the evolution of sandpiles due to Aronsson-Evans-Wu and Prigozhin. Readers are assumed to be familiar with the basic concepts and techniques of functional analysis and partial differential equations. The text is otherwise self-contained, with the exposition emphasizing an intuitive understanding and results given with full proofs. It is suitable for graduate students or researchers. The authors cover a subject that has received a great deal of attention in recent years. The book is intended as a reference tool for a general audience in analysis and PDEs, including mathematicians, engineers, physicists, biologists, and others interested in nonlocal diffusion problems.

MINIMIZING TOTAL VARIATION FLOW

In this paper we study existence and uniqueness of solutions for the boundary-value problem, with initial datum in L1(Ω), ut = div a(x, Du) in (0,∞)×Ω, − ∂u ∂ηa ∈ β(u) on (0,∞)× ∂Ω, u(x, 0) = u0(x) in Ω, where a is a Caratheodory function satisfying the classical Leray-Lions hypothesis, ∂/∂ηa is the Neumann boundary operator associated to a, Du the gradient of u and β is a maximal monotone graph in R× R with 0 ∈ β(0).

Porous medium equation with absorption and a nonlinear boundary condition

In this paper we study the nonlocal p-Laplacian-type diffusion equation

A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions

u_t(t,x)=\int_{\mathbb{R}^N}J(x-y)|u(t,y)-u(t,x)|^{p-2}(u(t,y)-u(t,x))\,dy

QUALITATIVE PROPERTIES OF THE SOLUTIONS OF A NONLINEAR FLUX-LIMITED EQUATION ARISING IN THE TRANSPORT OF MORPHOGENS

,

Radially Symmetric Solutions of a Tempered Diffusion Equation. A Porous Media, Flux-Limited Case

(t,x)\in]0,T[\times\Omega

THE MINIMIZING TOTAL VARIATION FLOW WITH MEASURE INITIAL CONDITIONS

, with