Foghem Gounoue, Guy Fabrice: $L^2$-Theory for nonlocal operators on domains. 2020
Inhalt
- Introduction
- Basics On Nonlocal Operators
- Characterization of a Lévy operator
- Case of the fractional Laplacian
- Renormalization constant of the fractional Laplacian
- Order of the fractional Laplace operator
- Nonlocal elliptic operators
- Mixed Lévy operators
- Nonlocal Sobolev-like Spaces
- Preliminaries
- Convolution product
- Approximation by smooth functions via convolution
- The Riesz-Fréchet-Kolmogorov theorem
- Classical Sobolev spaces
- Nonlocal Hilbert function spaces
- Nonlocal Sobolev-like spaces
- Fractional Sobolev spaces
- Approximations by smooth functions
- Compact embeddings and Poincaré type inequalities
- Complement Value Problems
- Review of variational principles
- Lagrange multipliers
- Integrodifferential equations (IDEs)
- Integrodifferential equations (IDEs) with Neumann condition
- Integrodifferential equations (IDEs) with Dirichlet condition
- Integrodifferential equations (IDEs) with mixed condition
- Integrodifferential equations (IDEs) with Robin condition
- Spectral decomposition of nonlocal operators
- Helmholtz equation for nonlocal operators
- Profiling solutions of evolution of IDEs
- Dirichlet-to-Neumann map for nonlocal operators
- Essentially self-adjointness for nonlocal operators
- From Nonlocal To Local
- Approximation of the Dirac mass by Lévy measures
- Characterization of classical Sobolev spaces
- Characterization of the spaces of bounded variation
- Asymptotically compactness
- Robust Poincaré inequalities
- Convergence of Hilbert spaces
- Mosco convergence of nonlocal to local quadratic forms
- Convergence of Dirichlet and Neumann problems
- Lebesgue Spaces
- Bibliography