Meng Yang (CMA, FCT-NOVA)

Title: Gradient Estimate for the Heat Kernel on Some Fractal-Like Cable Systems

Dates/times:

07/02/2022 (Monday), at 14:00–16:00

10/02/2022 (Thursday), at 14:00–16:00

10/02/2022 (Thursday), at 14:00–16:00

14/02/2022 (Monday), at 14:00–16:00

17/02/2022 (Thursday), at 14:00–16:00

Location: Seminar room, building VII

Abstract:

In this course, we will give an introduction to some recent results about gradient estimate for the heat kernel, in particular, the sub-Gaussian estimate on some fractal-like cable systems.

First, we will introduce some basic theory of Dirichlet forms. In particular, we will give the construction of strongly local regular Dirichlet forms on cable systems.

Second, we will introduce some basic theory of analysis on fractals. We mainly consider two simple fractals, that is, the Vicsek set and the Sierpinski gasket (SG). We will give the construction of strongly local regular Dirichlet forms and consider harmonic functions on these two fractals.

Third, we will consider Poisson equation on metric measure spaces and cable systems. We will introduce some functional inequalities to give the existence, the uniqueness, the regularity and the gradient estimate of the solutions of Poisson equation. In particular, we will introduce a new inequality, that is, the so-called generalized reverse Holder inequality.

Fourth, we will verify some functional inequalities, especially the generalized reverse Holder inequality on the Vicsek and the SG cable systems to obtain the desired gradient estimate for the heat kernel.

First, we will introduce some basic theory of Dirichlet forms. In particular, we will give the construction of strongly local regular Dirichlet forms on cable systems.

Second, we will introduce some basic theory of analysis on fractals. We mainly consider two simple fractals, that is, the Vicsek set and the Sierpinski gasket (SG). We will give the construction of strongly local regular Dirichlet forms and consider harmonic functions on these two fractals.

Third, we will consider Poisson equation on metric measure spaces and cable systems. We will introduce some functional inequalities to give the existence, the uniqueness, the regularity and the gradient estimate of the solutions of Poisson equation. In particular, we will introduce a new inequality, that is, the so-called generalized reverse Holder inequality.

Fourth, we will verify some functional inequalities, especially the generalized reverse Holder inequality on the Vicsek and the SG cable systems to obtain the desired gradient estimate for the heat kernel.

**Prerequisites**: Basic functional analysis, basic measure theory and basic PDE theory at undergraduate level.