Schulze, Tim: Nonlocal operators with symmetric kernels. 2020
Inhalt
- Introduction
- Coercivity in fractional Sobolev spaces
- Local boundedness from above and elliptic Harnack inequalities
- Basics
- Basics from measure theory, integration theory and classical function spaces
- Generalized function spaces involving a symmetric kernel
- Elliptic partial differential equations
- (Local) Elliptic operators in divergence form
- Nonlocal operators corresponding to an integral kernel
- Dirichlet forms
- Coercivity in fractional Sobolev spaces
- Setting and Preliminaries
- Cones, bounded configurations and reference cones
- Admissible configurations
- Cubes and favored indices
- A result for discrete quadratic forms
- A chaining argument in the Euclidean space
- The graph on the set of lattice points
- The induction
- Renormalization: Blocks and Towns
- Connecting points at scale: The discrete heart
- Proof of Theorem 7.1
- The coercivity result
- A discrete comparability result on lattices
- The discrete version of the kernel
- The limiting argument and the proof of the main result
- Applications of the main result
- Local boundedness from above and elliptic Harnack inequalities
- A local prelude: Local boundedness for solutions of local elliptic PDEs
- Local boundedness and Harnack inequalities for nonlocal operators
- Nonlocal tail functions
- Assumptions on the kernel
- Behavior of the conditions under scaling
- A local tail estimate
- Local boundedness of solutions
- Harnack inequalities
- Examples and a counterexample
- Appendices
- Auxiliary results for the coercivity estimate
- The relative Kato condition, the Harnack inequality and Condition (C)
- Preliminaries
- Condition (C) implies the relative Kato condition
- The relative Kato condition does not imply Condition (C)
- Bibliography