Program

We characterize concavity properties preserved by the Dirichlet heat flow in convex domains of the Euclidean space. (This is a joint work with Paolo Salani and Asuka Takatsu.) 

Next, we show that no concavity properties are preserved by the Dirichlet heat flow in a totally convex domain of a Riemannian manifold unless the sectional curvature vanishes everywhere on the domain. (This is a joint work with Asuka Takatsu and Haruto Tokunaga.)

We discuss in this talk optimal regularity estimates for solutions to degenerate/singular elliptic equations, with particular emphasis on the associated matrix weights with small log-BMO.

In this talk, we are concerning the asymptotic behavior of noncompact orbits for dynamical systems. First we introduce a semilinear parabolic problem defined onentire spatial domain as a typical example and give a typical behavior which comes from the noncompactness of the spatial domain. The possibility to develop a general dynamical system theory allowing the noncompactness of the orbit (i.e., a generalization of the LaSalle principle) will be also discussed. The method is based on the profile decomposition.

The Ricci flow is a semi-parabolic equation on Riemannian metrics. However, the RF can be considered as a system of two parabolic equations, the Ricci De-Turck flow and the harmonic map heat flow.

In this talk, we first discuss the asymptotic behavior of parabolic PDEs near a self-similar solutions. Then, we talk about its analogue for the ancient Ricci flows.

We investigate the motion of sets under anisotropic curvature with a volume constraint in the plane. We demonstrate the exponential convergence of the area-preserving anisotropic flat flow to a disjoint union of equal Wulff shapes, the critical points of the anisotropic perimeter functional. Furthermore, we prove that the flat flow preserves certain reflectional symmetries, allowing us to establish uniform bounds on the distance between the evolving profile and the initial data. This talk is based on joint work with Eric Kim (UCLA).

We consider radial solutions of the Dirichlet problem of the Henon equation with critical exponent on annuli in R^N, N\ge 3. For each n, the problem has exactly two radial solutions with n nodal domains. We obtain exact Morse indices of these solutions when the inner radius is small. If the exponent of the Henon equation is 3, then exact Morse indices of positive and negative solutions can be determined for every annulus. A variational method, an ODE technique with Emden transformation and multiplicities of eigenvalues of Laplace-Beltrami operator on S^N are used in proofs.

In this talk, we consider elliptic and parabolic systems in divergence form with degenerate or singular coefficients. Under the conormal boundary condition on the flat boundary, we establish boundary Schauder type estimates when the coefficients have partially Dini mean oscillation. Moreover, as an application, we achieve higher order boundary Harnack principles for certain uniformly elliptic and parabolic equations. This is based on ongoing work with Hongjie Dong.

In this talk, we consider the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term. We show the temporal decay estimates for the fractional heat semigroup in several Zygmund spaces, and establish the critical integral estimates for the inhomogeneous term and the nonlinear term by making use of the real interpolation method in weak Zygmund spaces. Then, we prove the local-in-time existence of solutions to the inhomogeneous fractional semilinear heat equation in uniformly local weak Zygmund spaces including the inhomogeneous functions with optimal singularities. This talk is based on the joint work with Professor Kazuhiro Ishige (The University of Tokyo) and Professor Tatsuki Kawakami (Ryukoku University).