2023 Workshop on Elliptic & Parabolic PDEs and Related Topics - Title & Abstract

2023 Workshop on Elliptic & Parabolic PDEs and Related Topics

Lectures

Park, Jiewon (KAIST)
Hessian estimates, monotonicity formulae, and applications
This lecture is an introduction to Hessian estimates for problems in geometry analysis, also known as Li–Yau–Hamilton estimates, and their applications. These estimates are often called differential Harnack estimates as well, since they imply Harnack estimates by integration along space or spacetime paths. We will focus on how several of these matrix estimates imply monotonicity formulae, which in turn has geometric consequences.
Kim, Sunghan (Uppsala University)
Minimizing constraint maps
My lectures are concerned with constraint maps, where the theory of harmonic maps intersects with free boundary problems. The vectorial maps naturally develop discontinuous singularities, as so do harmonic maps, but also give rise to free boundaries due to the constraints. The basic features were studied back in the 80s, but many important issues have been left uncovered to this day. Over the last couple of years, my collaborators and I revisited the problem, asking ourselves: “would the set of singularities meet the free boundaries?”, a central question yet highly tantalizing to answer. Very recently, we successfully tackled this issue for minimizing constraint maps, more specifically, the constraint maps locally minimizing either Dirichlet or Alt–Caffarelli energy. In my lectures, I will provide some background for the vectorial maps, present some interesting examples, and go through the novel ideas by which we resolved the issue of singularities. The lectures will be based on the joint works by Alessio Figalli (ETH), André Guerra (ETH) and Henrik Shahgholian (KTH).

Invited Talks

Yun, Hyungsung (SNU)
Generalized Schauder theory and its application to degenerate/singular parabolic equations
This talk presents the generalized Schauder theory and the fractional version of Taylor expansion for the degenerate/singular parabolic equations. To prove the higher regularity of solutions, we establish generalized Schauder theory which approximates coefficients of the operator with s-polynomials rather than constants. The generalized Schauder theory not only recovers the proof for uniformly parabolic equations but is also applicable to other operators that are difficult to apply the bootstrap argument to obtain higher regularity.
Kim, Takwon (SNU)
Regularity theory for the porous medium equation in bounded domains
In this talk, we study the porous medium equation given in a bounded domain with nonnegative initial data. The solution of the porous medium equation has the Hölder optimal regularity due to the degeneracy of the equation along the boundary of the domain. To analyze the solution near the boundary, we establish approximation theory with the fractional order polynomial, named as generalized Schauder theory. Applying the generalized Schauder theory, we obtain the smoothness of solutions for the porous medium equation.
Jang, Hyo Seok (SNU)
Conformal heat flow of maps under Yamabe flow of metrics
We study the gradient flow of the sum of the Dirichlet energy of a map from a low-dimensional closed manifold to an arbitrary-dimensional Riemannian manifold, and the normalized Yamabe energy on the domain. We show that a smooth solution to the system exists for short time, and that, with small initial data, the solution exists for all time.
Lee, Se-Chan (SNU)
C¹˒ᵅ-regularity for solutions of degenerate/singular fully nonlinear parabolic equations
This talk is concerned with a priori estimates for a class of fully nonlinear parabolic equations whose ellipticity constants depend on the gradient of solutions. For this purpose, we develop a new type of Bernstein method with approximations via difference quotients. As an application, we establish regularity results for some degenerate/singular fully nonlinear parabolic equations.

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