Variational and Diffusion Problems in Random Walk Spaces (Progress in Nonlinear Differential Equations and Their Applications, 103)

Preface
Contents
1 Random Walks
	1.1 Markov Chains
		1.1.1 φ-Essential Irreducibility
	1.2 Random Walk Spaces
	1.3 Examples
	1.4 The Nonlocal Gradient, Divergence and Laplace Operators
	1.5 The Nonlocal Boundary, Perimeter and Mean Curvature
	1.6 Poincaré-Type Inequalities
		1.6.1 Global Poincaré-Type Inequalities
		1.6.2 Poincaré-Type Inequalities on Subsets
2 The Heat Flow in Random Walk Spaces
	2.1 The m-Heat Flow
	2.2 Infinite Speed of Propagation
	2.3 Asymptotic Behaviour
	2.4 Ollivier-Ricci Curvature
	2.5 The Bakry-Émery Curvature-Dimension Condition
	2.6 Logarithmic-Sobolev Inequalities
	2.7 Transport Inequalities
	2.8 The m-Heat Content
		2.8.1 Probabilistic Interpretation
		2.8.2 The Spectral m-Heat Content
3 The Total Variation Flow in Random Walk Spaces
	3.1 The m-Total Variation
	3.2 The m-1-Laplacian and m-The Total Variation Flow
	3.3 Asymptotic Behaviour
	3.4 m-Cheeger and m-Calibrable Sets
	3.5 The Eigenvalue Problem for - m1
	3.6 Isoperimetric Inequality
	3.7 The m-Cheeger Constant
	3.8 The m-Cheeger Constant and the m-Eigenvalues of -1m
4 ROF-Models in Random Walk Spaces
	4.1 The m-ROF Model with L2-Fidelity Term
		4.1.1 The Gradient Descent Method
	4.2 The m-ROF-Model with L1-Fidelity Term
		4.2.1 The Geometric Problem
		4.2.2 Regularity of Solutions in Terms of the NonlocalCurvature
		4.2.3 Thresholding Parameters
		4.2.4 The Gradient Descent Method
5 Least Gradient Functions in Random Walk Spaces
	5.1 The Nonlocal Least Gradient Problem
	5.2 Nonlocal Median Value Property
	5.3 Nonlocal Poincaré Inequality
6 Doubly Nonlinear Nonlocal Stationary Problems of Leray-Lions Type with Nonlinear Boundary Conditions
	6.1 Nonlocal Diffusion Operators of Leray-Lions Type and Nonlocal Neumann Boundary Operators
	6.2 Nonlocal Stationary Problems with Neumann Boundary Conditions of Gunzburger-Lehoucq Type
		6.2.1 Existence of Solutions of an Approximate Problem
		6.2.2 Some Estimates on the Solutions of the Approximate Problems
		6.2.3 Monotonicity of the Solutions of the ApproximateProblems
		6.2.4 An Lp-Estimate for the Solutions of the Approximate Problems
		6.2.5 Proof of the Existence Result
	6.3 Neumann Boundary Conditions of Dipierro-Ros-Oton-Valdinoci Type
7 Doubly Nonlinear Nonlocal Diffusion Problems of Leray-Lions Type with Nonlinear Boundary Conditions
	7.1 Evolution Problems with Neumann Boundary Conditions of Gunzburger-Lehoucq Type
		7.1.1 Nonlinear Dynamical Boundary Conditions
		7.1.2 The Evolution Problem for a Nonlocal Dirichlet-to-Neumann Operator
		7.1.3 Doubly Nonlinear Boundary Conditions
		7.1.4 Nonhomogeneous Boundary Conditions
	7.2 Evolution Problems Under Neumann Boundary Conditions of Dipierro-Ros-Oton-Valdinoci Type
A Nonlinear Semigroups
	A.1 Introduction
	A.2 Abstract Cauchy Problems
	A.3 Mild Solutions
	A.4 Accretive Operators
	A.5 Existence and Uniqueness Theorem
	A.6 Regularity of the Mild Solution
	A.7 Completely Accretive Operators
	A.8 Yosida Approximation of Maximal Monotone Graphs in RR
Bibliography
Index
Index of notations