20th Internet Seminar

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The 20th Internet Seminar on Evolution Equations is devoted to the study of parabolic operators, paying also attention to the case of unbounded coefficients.

The lectures are at master or PhD students level and aim at introducing them to varying subjects related to evolution equations. In particular we hope to open areas of research and a fruitful collaboration between operator theorists, and experts in differential and stochastic differential equations.

The concept of the “Internet Seminar” originates in 1998 when Rainer Nagel (Tübingen) organized the first Internet Seminar. Since then, many schools have been organized in the same spirit and the experience of the previous editions has shown that these schools are much more effective than traditional schools where participants have a much more passive role. The course is organised in three phases.

  • In Phase 1 (October-February), a weekly lecture will be freely accessible over the internet via the ISEM website. The aim of the lectures is to present the theoretic background which lies behind current ongoing research.
  • In Phase 2 (March-May), the participants will form small international groups to work on diverse projects which supplement the theory of Phase 1 and provide some applications of it.
  • Finally, Phase 3 (19-23 June 2017) consists in a final workshop in Baronissi, Sa (Italy), where the teams will present their projects and additional lectures will be delivered by leading experts.

ISEM team 2016/17 :

Virtual lecturers
Luca Lorenzi (Parma)
Abdelaziz Rhandi (Salerno)
Davide Addona (Parma)
Tiziana Durante (Salerno)
Federica Gregorio (Salerno)
Rosanna Manzo (Salerno)
Cristian Tacelli (Salerno)
Carlo Troiano (Salerno)

Description of the course

We intend to propose a course on parabolic equations (paying attention both to the classical case of bounded coefficients and to the case when the coefficients are unbounded). Clearly, to face operators with unbounded coefficients, one should first set the basis of the classical theory of bounded coefficients. For this reason, we plan to split the project into two parts.

  • In the first part of the lectures we will illustrate the main results of the classical theory of PDE's. More specifically, we address the following topics:
    • classical maximum principles,
    • existence/uniqueness and (local and global) Hölder regularity for solutions to parabolic equations D_t u-Au=f in sufficiently smooth domains, when A is a uniformly elliptic operator.

    To present the main results of the first part we will use the book by Evans "Partial differential equations". We will start by the Heat equation in \mathbb{R}^N and in \mathbb{R}^N_+ with homogeneous Dirichlet boundary conditions, where everything is explicit and using some perturbation arguments (e.g., continuity method) we address more general operators with coefficients defined in \mathbb{R}^N and in \mathbb{R}^N_+. These result will be then used to study parabolic operators in bounded and smooth domains via local charts.

  • The second part of the course is devoted to elliptic and parabolic problems with unbounded coefficients. Here, we will follow the book by L. Lorenzi: "Analytical methods for Kolmogorov equations" starting from the easiest case when A is the Ornstein-Uhlenbeck operator, which is the prototype of an elliptic operator with unbounded coefficients. The Ornstein-Uhlenbeck operator is easier to treat than more general operators with unbounded coefficients since everything can be done "by hands": a representation formula for the solution of the parabolic equation D_t u-Au=f is available and this simplifies a lot the analysis. Moreover, it is a very nice example to illustrate the main peculiarities of solution to parabolic equations associated with elliptic operators A with unbounded coefficients. Based on this, we will subsequently address the general case. We confine ourselves to the autonomous case (but some comments to the nonautonomous case will be also provided) and illustrate the C_b-theory.

Some basic facts of Functional analysis which participants should know will be provided in appendices to make the course as much self contained as possible.

You can download the program, the poster and the announcement for the Internet Seminar here: