Ki-Ahm Lee

All Time Smooth Solutions of the One-Phase Stefan Problem and the Hele-Shaw Flow

Abstract We consider the one phase free boundary problem of Stefan (or Hele-Shaw) type: find {u, Ω} such that Ω = {u > 0} and with Q T = ℝ n × (0, T), T > 0. Under the condition that u o is C 1, 1 and log u o is concave on Ω o , we show that the concavity of log u(⋅, t) is preserved under the flow. As a consequence, we show that there exists a solution which is smooth at all time, up to the interface. In particular, the interface is smooth.

Hölder regularity of solutions of degenerate elliptic and parabolic equations

We establish the Alexandroff–Bakelman–Pucci estimate, the Harnack inequality, and the Holder continuity of solutions to degenerate parabolic equations of the non-divergence form (∗)Lu≔xa11uxx+2xa12uxy+a22uyy+b1ux+b2uy−ut=g on x⩾0, with bounded measurable coefficients. We also establish similar regularity results in the corresponding elliptic case.

Regularity of a free boundary for viscosity solutions of nonlinear elliptic equations

Our objective in this paper is to analyze the regularity of the boundary of a set Omega2 that admits a solution u to the following overdetermined problem: F(D(2)u) = chi (Omega) in B, u = \ delu \ = 0 in B \ Omega. Here F is a convex, uniformly elliptic operator of homogeneity one, B is the unit ball in R-n, and the equation is satisfied in the viscosity sense. We also consider the case of concave F in R-2.

Free-boundary regularity on the focusing problem for the Gauss Curvature Flow with flat sides

Abstract. We consider the motion of a compact weakly convex two-dimensional surface of revolution

Harnack inequality for nondivergent parabolic operators on Riemannian manifolds

\Sigma

Homogenization of Oscillating Free Boundaries: The Elliptic Case

under the Gauss Curvature Flow. We assume that the initial surface has a flat side and as a consequence the parabolic equation describing the motion of the hypersurface becomes degenerate at points where the curvature is zero. Expressing the strictly convex part of the surface near the interface as the graph of a function

FLAME PROPAGATION IN ONE-DIMENSIONAL STATIONARY ERGODIC MEDIA

z=f(r,t)

The evolution of complete non-compact graphs by powers of Gauss curvature

, we show that if at ti me

Higher Order Convergence Rates in Theory of Homogenization: Equations of Non-divergence Form

t=0, g=\sqrt f

ROLLING STONES WITH NONCONVEX SIDES II: ALL TIME REGULARITY OF INTERFACE AND SURFACE

vanishes linearly at the flat side, then