Related papers: On the Geometry of Solutions of the Fully Nonlinea…
In this paper, we prove that flat free boundaries of solutions to inhomogeneous one-phase Stefan problem are $C^{1,\alpha}$. The method consists of employing a hodograph transform and deriving the regularity via a linearization technique,…
We study the regularity of the free boundary in one-phase Stefan problem with nonlinear operator. Using the Hodograph transform and a linearization technique, we prove that flat free boundaries are $C^{1, \alpha}$ in space and time. When…
We study solutions to a one-phase singular perturbation problem that arises in combustion theory and that formally approximates the classical one-phase free boundary problem. We introduce a natural density condition on the transition layers…
We provide perturbative estimates for the one-phase Stefan free boundary problem and obtain the regularity of flat free boundaries via a linearization technique in the spirit of the elliptic counterpart established by the first author.
We prove the nondegeneracy condition for stable solutions to the one-phase free boundary problem. The proof is by a De Giorgi iteration, where we need the Sobolev inequality of Michael and Simon and, consequently, an integral estimate for…
We propose a method to determine the smoothness of sufficiently flat solutions of one phase Hele-Shaw problems. The novelty is the observation that under a flatness assumption the free boundary --represented by the hodograph transform of…
We study self-similar solutions of a multi-phase Stefan problem, first in the case of one space variable, and then in the radial multidimensional case. In both these cases we prove that a nonlinear algebraic system for determination of the…
The classical one-phase Stefan problem (without surface tension) allows for a continuum of steady state solutions, given by an arbitrary (but sufficiently smooth) domain together with zero temperature. We prove global-in-time stability of…
We consider a one-phase free boundary problem governed by doubly degenerate fully non-linear elliptic PDEs with non-zero right hand side, which should be understood as an analog (non-variational) of certain double phase functionals in the…
In this paper we consider the model of phase relaxation introduced in [22], where an asymptotic analysis is performed toward an integral formulation of the Stefan problem when the relaxation parameter approaches zero. Assuming the natural…
In this chapter we consider different approximations for the one-dimensional one-phase Stefan problem corresponding to the fusion process of a semi-infinite material with a temperature boundary condition at the fixed face and non-linear…
We formulate a Stefan problem on an evolving hypersurface and study the well-posedness of weak solutions given $L^1$ data. To do this, we first develop function spaces and results to handle equations on evolving surfaces in order to give a…
We study the Stefan problem with surface tension and radially symmetric initial data. In this context, the notion of a so-called physical solution, which exists globally despite the inherent blow-ups of the melting rate, has been recently…
In this paper we describe invariant geometrical ~structures in the phase space of the Swift-Hohenberg equation in a neighborhood of its periodic stationary states. We show that in spite of the fact that these states are only marginally…
We consider the one-phase Stefan problem describing the evolution of melting ice. On the one hand, we focus on understanding the evolution of the free boundary near isolated singular points, and we establish for the first time upper and…
We investigate existence and regularity properties of one-phase free boundary graphs, in connection with the question of whether there exists a complete non-planar free boundary graph in high dimensions.
The classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing…
We consider viscosity solution to one-phase free boundary problems for general fully nonlinear operators and free boundary condition depending on the normal vector. We show existence of viscosity solutions via the Perron's method and we…
A one phase Stefan problem in nonlinear conduction is considered. The problem is shown to admit a unique solution for small times. An exact solution is obtained which is a travelling front moving with constant speed.
Free boundary problems appear naturally in numerous areas of mathematics, science and engineering. These problems present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of…