Homogenization of one-phase Stefan-type problems in periodic and random media
Transactions of the American Mathematical Society, 2010
Page 1. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 8, August 2010, Page... more Page 1. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 8, August 2010, Pages 4161–4190 S 0002-9947(10)04945-7 Article electronically published on March 24, 2010 HOMOGENIZATION ...
Regularity of one-phase Stefan problem near Lipschitz initial data
American Journal of Mathematics, 2010
Page 1. Regularity of one-phase Stefan problem near Lipschitz initial domain Sunhi Choi∗and Inwon... more Page 1. Regularity of one-phase Stefan problem near Lipschitz initial domain Sunhi Choi∗and Inwon C. Kim† Abstract In this paper we show that, starting from Lipschitz initial free boundary with small Lipschitz constant, the solution ...
Regularity of one-phase Stefan problem near Lipschitz initial data
American Journal of Mathematics, 2010
Page 1. Regularity of one-phase Stefan problem near Lipschitz initial domain Sunhi Choi∗and Inwon... more Page 1. Regularity of one-phase Stefan problem near Lipschitz initial domain Sunhi Choi∗and Inwon C. Kim† Abstract In this paper we show that, starting from Lipschitz initial free boundary with small Lipschitz constant, the solution ...
Regularity of one-phase Stefan problem near Lipschitz initial data
American Journal of Mathematics, 2010
Page 1. Regularity of one-phase Stefan problem near Lipschitz initial domain Sunhi Choi∗and Inwon... more Page 1. Regularity of one-phase Stefan problem near Lipschitz initial domain Sunhi Choi∗and Inwon C. Kim† Abstract In this paper we show that, starting from Lipschitz initial free boundary with small Lipschitz constant, the solution ...
Homogenization of one-phase Stefan-type problems in periodic and random media
Transactions of The American Mathematical Society, 2010
Page 1. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 8, August 2010, Page... more Page 1. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 8, August 2010, Pages 4161–4190 S 0002-9947(10)04945-7 Article electronically published on March 24, 2010 HOMOGENIZATION ...
We investigate the homogenization limit of a free boundary problem with space-dependent free boun... more We investigate the homogenization limit of a free boundary problem with space-dependent free boundary velocities. The problem under consideration has a well-known obstacle problem transformation, formally obtained by integrating with respect to the time variable. By making rigorous the link between these two problems, we are able to derive an explicit formula for the homogenized free boundary velocity, and we establish the uniform convergence of the free boundaries.
We investigate the regularizing behavior of two-phase Stefan problem near initial data. The main ... more We investigate the regularizing behavior of two-phase Stefan problem near initial data. The main step in the analysis is to establish that in any given scale, the scaled solution is very close to a Lipschitz profile in space-time. We introduce a new decomposition argument to generalize the preceding ones by Choi, Jerion and Kim([CJK1], [CJK2]) and by Choi and Kim([CK]) on one-phase problems.
This article presents a local regularity theorem for the one-phase Hele-Shaw flow. We prove that ... more This article presents a local regularity theorem for the one-phase Hele-Shaw flow. We prove that if the Lipschitz constant of the initial free boundary in a unit ball is small, then for small uniform positive time the solution is smooth. This result improves on our earlier results in [CJK] because it is scale-invariant. As a consequence we obtain existence, uniqueness and regularity properties of global solutions with Lipschitz initial free boundary.
Communications in Partial Differential Equations, 2010
We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates... more We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data.
We investigate the homogenization limit of a free boundary problem with space-dependent free boun... more We investigate the homogenization limit of a free boundary problem with space-dependent free boundary velocities. The problem under consideration has a well-known obstacle problem transformation, formally obtained by integrating with respect to the time variable. By making rigorous the link between these two problems, we are able to derive an explicit formula for the homogenized free boundary velocity, and we establish the uniform convergence of the free boundaries.
Annales De L Institut Henri Poincare-analyse Non Lineaire, 2009
In this paper we consider a free boundary problem which describes contact angle dynamics on inhom... more In this paper we consider a free boundary problem which describes contact angle dynamics on inhomogeneous surface. We obtain an estimate on convergence rate of the free boundaries to the homogenization limit in periodic media. The method presented here also applies to more general class of free boundary problems with oscillating boundary velocities.
In this paper we investigate the waiting time phenomena for the one-phase Hele-Shaw and Stefan pr... more In this paper we investigate the waiting time phenomena for the one-phase Hele-Shaw and Stefan problems. We identify a general criterion on the growth rate of the initial data for the Hele-Shaw problem which determines the occurrence of a waiting time. For the Stefan problem we show that the waiting time phenomena depends on the balance between the initial data and the geometry of the initial positive phase.
In this paper we consider a free boundary problem which is used to describe the motion of contact... more In this paper we consider a free boundary problem which is used to describe the motion of contact lines of a liquid droplet on a flat surface. The elliptic nature of the equation for droplet shape and the monotonic dependence of contact line velocity on contact angle allows us to introduce a notion of "viscosity" solutions for this problem. Unlike similar free boundary problems, a comparison principle is only available for a modified short-time approximation because of the constraint that conserves volume. We use this modified problem to construct viscosity solutions to the original problem under a weak geometric restriction on the free boundary shape. We also prove uniqueness provided there is an upper bound on front velocity.
Communications in Partial Differential Equations, 2005
In this paper we are interested in a free boundary problem whith a motion law involving the mean ... more In this paper we are interested in a free boundary problem whith a motion law involving the mean curvature term of the free boundary. Viscosity solutions are introduced as a notion of global-time solutions past singularities. We show the comparison principle for viscosity solutions, which yields the existence of minimal and maximal solutions for given initial data. We also prove uniqueness of the solution for several classes of initial data and discuss the possibility of nonunique solutions. *
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