Introduction to Nonlocal Operators

In this course, we will study the well-posedness of certain classes of nonlocal problems involving integrodifferential operators of Lévy type, that generalize the fractional Laplace operator. The prototype problems we will study, include IntegroDifferential Equations (IDEs) with Dirichlet type condition and/or Neumann type condition. In the area of Partial Differential Equations (PDEs), these two prototype problems are very similar to the Dirichlet and Neumann boundary problems for the Laplace operator. Classical Sobolev spaces play a decisive role in the study of weak solutions associated with the aforementioned boundary values problems and many other PDEs. However, due to the fact that integrodifferential operators are nonlocal (i.e., do not preserve the support), the corresponding Dirichlet and/or Neumann type conditions are assigned on the complement of the underlying domains. Hereby contrasting with the Dirichlet and the Neumann problems for the Laplace operator where the conditions are set on the boundary. Thus, the type of IDEs under consideration in this course underscores the need to introduce and to study some new tailored made "nonlocal Sobolev type spaces".  Another objective of this lecture consists of showing that solutions to a certain class of elliptic PDEs are limits of solutions to elliptic IDEs.

Prerequisites: (Will be offered in the first part of the lecture)

• Basics on Sobolev spaces,
• Basics on functional analysis and variational methods (e.g. Lax-Milgram).

Topics: (Main part)

• Characterization of (elliptic) integrodifferential operators,
• Nonlocal Sobolev spaces and applications to IntegroDierential Equations (IDEs),
• Convergence from nonlocal elliptic IDEs to local elliptic PDEs.

Main references:

The lecture will be based on the recent (open access) monograph: \(L^2\)-Theory for nonlocal operators on domains.

For the prerequisites we will use Brezis' book:

Haim Brezis. Functional analysis, Sobolev spaces, and partial differential equations. Springer. Science & Business Media, 2010.

Target group: Master students (mathematics, physics) and PhD-students.

Language: The course will be taught in English.

Teaching concept: The lecture will be taught remotely via Zoom, in the synchronous format 3+1 (3 h lectures + 1h exercise).

Please register on the OPAL platform where further detailed information will be posted.