# AIS - Elliptic and Parabolic PDEs

## Venue: IISc, Bangalore

## Dates: 4 Dec 2023 to 23 Dec 2023

**Convener(s)**

Name: |
Arka Mallick | Swarnendu Sil |

Mailing Address: |
Assistant Professor Department of Mathematics Indian Institute of Science Bangalore 560012 |
Assistant Professor Department of Mathematics Indian Institute of Science Bangalore 560012 |

Email: |
arkamallick at iisc.ac.in | swarnendusil at iisc.ac.in |

**Full Adress of Venue**

Lecture Hall - I

Department of Mathematics

Indian Institute of Science

Bengaluru 560012

Partial differential Equations ( PDEs ) are the heart of mathematical analysis. Among the PDEs that are important to mathematics and other sciences, a substantial fraction are of elliptic and parabolic type. Most problems in mathematical physics where one has a variational problem and practically every PDE that is important in geometry and topology are either elliptic or parabolic. Some examples include the Laplace-Beltrami operator on a Riemannian manifold, Yamabe equation, Hardy-Sobolev operator, harmonic map equations, Monge-Ampere equations, Heat flow, Mean curvature flow and Ricci flow equation. These PDEs are distinguished from Hyperbolic PDEs by the remarkable property known as hypoelliptic regularity. In short, at least in the linear setting, solutions of these type of PDEs are typically more regular than the given data. The knowledge of basic existence and regularity theory for linear elliptic and parabolic equations are thus vital to graduate students and researchers in a large number of fields in analysis, geometry and topology. In this school, we intend to give a through exposition to the existence and regularity theory of linear elliptic and parabolic second order equations, along with a through treatment of the machinery of Sobolev spaces that are

needed.

The lectures are meant for an audience of students interested in Elliptic and Parabolic PDEs. Each day of lectures is accompanied by tutorial sessions (in the afternoon).