Some linear integro-differential operators have old and classical representations as the Dirichlet-to-Neumann operators for linear elliptic equations, such as the 1/2-Laplacian or the generator of the boundary process of a reflected diffusion. In this work, we make some extensions of this theory to the case of a nonlinear Dirichlet-to-Neumann mapping that is constructed using a solution to a fully nonlinear elliptic equation in a given domain, mapping Dirichlet data to its normal derivative of the resulting solution. Here we begin the process of giving detailed information about the Lévy measures that will result from the integro-differential representation of the Dirichlet-to-Neumann mapping. We provide new results about both linear and nonlinear Dirichlet-to-Neumann mappings. Information about the Lévy measures is important if one hopes to use recent advancements of the integro-differential theory to study problems involving Dirichlet-to-Neumann mappings.
Optimal Control, Numerics, and Applications of Fractional PDEs
This chapter provides a brief review of recent developments on two nonlocal operators: fractional Laplacian and fractional time derivative. We start by accounting for several applications of these operators in imaging science, geophysics, harmonic maps, and deep (machine) learning. Various notions of solutions to linear fractional elliptic equations are provided and numerical schemes for fractional Laplacian and fractional time derivative are discussed. Special emphasis is given to exterior optimal control problems with a linear elliptic equation as constraints. In addition, optimal control problems with interior control and state constraints are considered. We also provide a discussion on fractional deep neural networks, which is shown to be a minimization problem with fractional in time ordinary differential equation as constraint. The paper concludes with a discussion on several open problems.
- Trélat, E.; Zuazua, E.
- Publication Date:
- NSF-PAR ID:
- Journal Name:
- Handbook of numerical analysis
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- Sponsoring Org:
- National Science Foundation
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