Inhalt

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   Introduction
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   Existence and uniqueness of variational solutions
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   Nonlocal to local phase transition
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   Homogenization of nonlocal Dirichlet problem
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   Connection of the three main parts
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   Outline
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  Function spaces
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  Classical function spaces
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   Sobolev and fractional Sobolev spaces
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   Asymtotics as s1
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   Further characterizations of Sobolev spaces
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  Function spaces with regularity over the boundary
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   Nonlocal generalization of H1()
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   Dirichlet forms associated to V(| Rd)
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   Generalization of fractional Sobolev spaces
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   Asymptotics for s1 in the generalized setting
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   Changing the asymptotics
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   Weighted L2-spaces
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  Function spaces with a general kernel as weight
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   Definition and basic properties
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   Poincaré-Friedrichs inequality
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   Existence and uniqueness of solutions for nonlocal boundary value problems
•  
   Setting
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   Variational formulation of the Dirichlet problem
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  Gårding inequality and Lax-Milgram Lemma
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   Gårding inequality
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   Application of the Lax-Milgram Lemma
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  Weak maximum principle and Fredholm alternative
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   Weak maximum principle
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   Fredholm alternative
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   Examples of kernels
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   Integrable kernels
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   Non-integrable kernels
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   Nonlocal to local phase transition
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   Setting and main result
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   Gamma-Convergence of the energies
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   Application of -Convergence
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  Homogenization of nonlocal Dirichlet Problem
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   Homogenization of second order elliptic equations
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   Homogenization for elliptic nonlocal operators
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   An application of Beurling Deny
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   Additivity and translation invariance of localized functionals
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   Open questions in the nonlocal case
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   Appendix
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  -Convergence
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   Definition and basic properties
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   Convergence of minimizers and compactness of -convergence
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   Dirichlet forms
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  Definitions and auxiliary results
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   Domains
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   Auxiliary computations
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   Bibliography