The mini workshop will be conducted both online and offline. If you would like to attend this workshop off-campus, please join it Here.

Speakers

Ki-Ahm Lee (Seoul National University, Korea)

Rolando Magnanini (University of Florence, Italy)

Hayk Mikayelyan (University of Nottingham Ningbo China)

Georg Weiss (University of Duisburg-Essen, Germany)

Program (Beijing time)

15:00-15:50 Rolando Magnanini (University of Florence, Italy)

Title: The location of hot spots and other extremal points.

Abstract: In a domain of the Euclidean space, we estimate from below the distance to the boundary of global maximum points of solutions of elliptic and parabolic equations with homogeneous Dirichlet boundary values. As reference cases, we first consider the torsional rigidity function of a bar, the first mode of a vibrating membrane, and the temperature of a heat conductor grounded to zero at the boundary. Our main results are presented for domains with a mean convex boundary and entail the ratio between that distance and the inradius of the relevant domain. For the torsional rigidity function, the bound for that ratio only depends on the space dimension. In case of domains which are not mean convex, the bound also depends on some geometrical quantities such as the diameter and the radius of the largest exterior osculating ball to the relevant domain, or the minimum of the mean curvature of the boundary.

Also in the case of the first mode, the relevant bound only depends on the space dimension. The bound related to the temperature depends on time and the initial distribution of temperature. Such a bound is substantially consistent with what one obtains in the stationary situation.

The methods employed are based on elementary arguments and existing literature, and can be extended to other situations that entail quasilinear equations, isotropic and anisotropic, and also certain classes of semi-linear equations.

16:00-16:50 Hayk Mikayelyan (University of Nottingham Ningbo China)

Title: The normalized fractional obstacle problem

Abstract: We generalize some classical results in optimal rearrangement theory for fractional Laplace operator and Gagliardo–Nirenberg semi-norm. As a result we rediscover the unstable fractional obstacle problem, recently considered by Mark Allen et al., as well as derive the fractional version of the normalized obstacle problem with a coincidence set. (Joint work with Julian Bonder and Zhiwei Cheng)

17:00-17:50 Georg Weiss (University of Duisburg-Essen, Germany)

Title: On global solutions of the obstacle problem

Abstract: Recently great progress has been achieved by Alessio Figalli et al. doing a fine analysis of the singular set in the obstacle problem. Much less is known about the behavior of the regular set close to singularities. The structure of this set may be very complicated, for example it may contain infinitely many connected components approaching the singularity, funnel-type singularities etc.

The issue is related to the characterization of global solutions of the obstacle problem, that is solutions in the entire space. This characterization in turn is related to conjectures in potential analysis and an open problem since the 1980s. In this talk we will discuss progress in the characterization of global solutions and the application to singularities.

18:00-18:50 Ki-Ahm Lee (Seoul National University, Korea)

Title: Boundary regularity of local and nonlocal equations.

Abstract: In this talk, we are going to discuss boundary regularities of various degenerate local equation and nonlocal equations. Diffusion rates deform undefined geometry related to diffusion and the corresponding distance function makes important role in the theory of regularity. And then we will also discuss the possible applications.