최범준 교수

On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace is known to lead to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error, showing the rate is either exponentially fast (with a rate constant predicted by the spectral gap), or algebraically slow (which is only possible in the presence of non-integrable zero modes).

In this paper we derive diffusion equations in a heterogeneous environment. We consider a system of discrete kinetic equations that consists of two phenotypes of different turning frequencies. The two phenotypes change their states according to state transition frequencies which depend on the environment. We show that the density of the total population of the two phenotypes converges to the solution of a Fokker--Planck type diffusion equation if turning frequencies are of higher order than the

This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in R 2 under the -curve shortening flow for exponents > 1 2 . We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under -curve shortening flow to the unique translating soliton whose ends are asymptotic to the same parallel lines. This is a new result even in the standard case = 1, and we prove for all exponents up to the critical case > 1 2 .

We address the classification of ancient solutions to the Gauss curvature flow under the assumption that the solutions are contained in a cylinder of bounded cross-section. For each cylinder of convex bounded cross-section, we show that there are only two ancient solutions which are asymptotic to this cylinder: the non-compact translating soliton and the compact oval solution obtained by gluing two translating solitons approaching each other from time $-\infty$ from two opposite ends.

We study the evolution of complete, noncompact, convex hypersurfaces in Rn+1 by the inverse mean curvature flow. We establish the long-time existence of solutions, and we provide the characterization of the maximal time of existence in terms of the tangent cone at infinity of the initial hypersurface. Our proof is based on an a priori pointwise estimate on the mean curvature of the solution from below in terms of the aperture of a supporting cone at infinity. The strict convexity of convex solut

Abstract This work concerns with the existence and detailed asymptotic analysis of type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up around the curvature maximum points, to a rotationally symmetric steady soliton. It is the first time that the steady soliton is shown to be a finite time singularity model

The fast diffusion equation is analyzed on a bounded domain with Dirichlet boundary conditions, for which solutions are known to extinct in finite time. We construct invariant manifolds that provide a finite-dimensional approximation near the vanishing solution to any prescribed convergence rate.

We classify the surfaces translating under the flows by sub-affine-critical powers of the Gauss curvature. This, in particular, lists all translating solitons possibly model Type II singularities for convex closed solutions in all positive powers. The surfaces are entire graphs, and therefore our result corresponds to the Liouville theorem for the degenerate Monge--Ampère equations $\det D^2 u=(1+|Du|^2)^{2-\frac{1}{2α}}$ on $\mathbb{R}^2$ in the range $0<α<1/4$. The result also reveals th

In this paper, we investigate generalized Carleman kinetic equation for n$\ge$2 and prove convergence towards the solution of equation with fast diffusion or porous medium type, $u_t=Δu^m$ ($0\le m\le2$), in its diffusive hydrodynamic limit. Using comparison principle of system combined with fixed speed propagation property of transport equation, we create a new barrier argument for this hyperbolic system. It is crucial to construct explicit local sub and solution of system and this is done by e

Abstract We generalize the Benjamini–Pemantle–Peres estimate relating hitting probability and Martin capacity to the setting of manifolds with Ricci curvature bounded below. As applications we obtain: (1) a sharp estimate for the probability that Brownian motion comes close to the high curvature part of a Ricci-flat manifold, (2) a proof of an unpublished theorem of Naber that every noncollapsed limit of Ricci-flat manifolds is a weak solution of the Einstein equations, (3) an effective intersec

It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean convex surfaces all limit flows are selfsimilar (static, shrinking, or translating) if and only if there are only finitely many spherical singularities. More generally, using the solution of the mean convex neighborhood conjecture for neck singularities, we establi

In this paper we reconcile several different approaches to Ricci flow through singularities that have been proposed over the last few years by Kleiner–Lott, Haslhofer–Naber and Bamler. Specifically, we prove that every noncollapsed limit of Ricci flows, as provided by Bamler’s precompactness theorem, as well as every singular Ricci flow of Kleiner–Lott, is a weak solution in the sense of Haslhofer–Naber. We also generalize all path-space estimates of Haslhofer–Naber to the setting of noncollapse