Regularity for degenerate two-phase free boundary problems

Purpose The main purpose of this paper is given below: To present a mathematical model of a two-phase Stefan problem including a moving phase change material and variable thermophysical properties. To find a numerical solution of the problem to discuss the dependence of considered phase change problem on variable thermal conductivity, variable specific heat and Peclet number. Design/methodology/approach In this paper, a numerical solution of the problem is obtained using the front-fixing method in tandem with the explicit finite difference scheme. The authors have also discussed the consistency and stability of proposed numerical scheme. Findings In this study, it is observed that the considered scheme is an efficient tool that provides sufficiently accurate results for exploring the behaviors of moving interface (free boundary) and temperature profile for a nonclassical two-phase free boundary problem. In this study, the authors have observed that the parameters α1 and α2 influence the temperature profiles of the liquid region and the solid region. It is also found that the free boundary propagates faster when the authors increase the parameter α1 or decrease the parameter α2. Originality/value From the literature, it is seen that most of the two-phase problems with free boundary in an infinite domain are considered by the authors with constant thermophysical properties. Because it is possible to establish an analytical solution of two-phase problems with free boundary in case of an infinite domain. Moreover, a two-phase problem in a finite domain involving moving phase change material with the unidirectional speed is not considered. Therefore, the authors have considered a two-phase free boundary problem with variable thermal coefficients.

Determination of one unknown coefficient in a two-phase free boundary problem in an angular domain with variable thermal conductivity and specific heat

A two-phase free boundary problem associated with a diffusion–convection equation is considered. The problem is reduced to a system of nonlinear integral equations, which admits a unique solution for small times. The system admits an explicit two-component solution corresponding to a two-component shock wave of the Burgers equation. The stability of such a solution is also discussed.

We provide an overview of some recent results about the regularity of the solution and the free boundary for so-called two-phase free boundary problems driven by uniformly elliptic equations.

We study the existence and structure of branch points in two-phase free boundary problems. More precisely, we construct a family of minimizers to an Alt–Caffarelli–Friedman-type functional whose free boundaries contain branch points in the strict interior of the domain. We also give an example showing that branch points in the free boundary of almost-minimizers of the same functional can have very little structure. This last example stands in contrast with recent results of De Philippis, Spolaor and Velichkov on the structure of branch points in the free boundary of stationary solutions.

A two-phase free boundary problem associated with nonlinear heat conduction is considered. The problem is mapped into two one-phase moving boundary problems for the linear heat equation, connected through a constraint on the relative motion of their moving boundaries. Existence and uniqueness of the solution is proved for small times and a particular exact solution is discussed.

In this paper we are interested in the study of a two-phase problem equipped with the $\Phi$-Laplacian operator $$ \Delta_\Phi u \coloneqq \mbox{div} \bigg(\varphi(|\nabla u|) \dfrac{\nabla u}{|\nabla u|}\bigg), $$% where $\Phi(s)=e^{s^2}-1$ and $\varphi=\Phi'$. We obtain the existence, boundedness, and Log-Lipschitz regularity of the minimizers of the energy functional associated to the two-phase problem. Furthermore, we also prove that the free boundaries of these minimizers have locally finite perimeter and Hausdorff dimension at most $(N-1)$.

A two-phase free boundary problem for coupled system including three elliptic equations is considered. The problem comes from the discussion of a growth model of self-maintaining protocell in multidimensional case. The global classical solution of the problem with free boundary Γ:y=g(x,t) between two domains is under search. The global existence and uniqueness of a quasi-stationary problem are proved in multidimensional case.

Classical solutions in the large of a two-phase free boundary problem. II

Explicit solutions for one-dimensional two-phase free boundary problems with either shrinkage or expansion

A two-phase free boundary problem for system of reaction-diffusion equations

The paper is concerned with optimal control problems for systems governed by multidimensional two-phase parabolic free boundary value problems of Stefan type. In the case of nonlinear process equations and nonlinear boundary conditions existence of optimal controls enclosed in these conditions and convergence of approximating optimal control problems including smoothed coefficients are shown.

Varifold solutions of the two-phase three-dimensional magnetohydrodynamic equations

We are concerned with the existence of a weak solution \(u \in W_0^{1,p}(\Omega)\) to the degenerate quasi-linear Dirichlet boundary value problem $$- \Delta_p u = \lambda |u|^{p-2} u + f(x) \quad \hbox{in}\ \Omega ;\qquad u = 0 \quad \hbox{on}\, \partial \Omega.\qquad\quad \hbox{(P)}$$ It is assumed that 1 0 small enough). Moreover, we obtain at least three distinct solutions if either p 2 and λ1 < λ ≤ λ1 + δ. The proofs use a minimax principle for the corresponding energy functional performed in the orthogonal decomposition \(W_0^{1,p}(\Omega) = {\rm lin} \{\varphi_1\} \oplus W_0^{1,p}(\Omega)^\perp\) induced by the inner product in L2(Ω). First, the global minimum is taken over \(W_0^{1,p}(\Omega)^\perp\), and then either a local minimum or a local maximum over lin {φ1}. If the latter is a local minimum, the local minimizer in \(W_0^{1,p}(\Omega)\) thus obtained provides a solution to problem (P). On the other hand, if it is a local maximum, one gets only a pair of sub- and supersolutions to problem (P), which is then used to obtain a solution by a topological degree argument.

The Si-adatom dynamics on a single-height-stepped Si{001} surface is studied via a combined molecular-dynamics, simplified transition-state theory and time-dependent lattice-gas description using Tersoff's potential for Si-Si interactions. The energies of adatom binding and migration near ${\mathit{S}}_{\mathit{A}}$, ${\mathit{S}}_{\mathit{B}}$(b) (bonded), and ${\mathit{S}}_{\mathit{B}}$(n) (nonbonded) step edges show that the global minima, 4.28 eV, are in the trenches at the ${\mathit{S}}_{\mathit{B}}$(n) edges. At all other sites, the binding energies are comparable to those on the flat Si{001}-(2\ifmmode\times\else\texttimes\fi{}1) surface. We find that if the adatoms are deposited on an ${\mathit{S}}_{\mathit{B}}$ terrace, reflection dominates at the ${\mathit{S}}_{\mathit{A}}$ edge, whereas at the ${\mathit{S}}_{\mathit{B}}$(b) and ${\mathit{S}}_{\mathit{B}}$(n) edges, adatoms step down and migrate in the trenches parallel to the edges. The deposition rate and surface-temperature-dependent reflection, step up, step down, and accommodation probabilities at all three step edges are calculated and used in a simple estimation of step-edge growth coefficients. We find that the allowable surface-temperature range in which a 400-\AA{}-wide ${\mathit{S}}_{\mathit{B}}$ terrace may grow at the ${\mathit{S}}_{\mathit{B}}$(b) and ${\mathit{S}}_{\mathit{B}}$(n) edges is above 800 K for the 0.30 ML/min (where ML denotes monolayer) film deposition rate and above 700 K for the slower 0.03 ML/min film deposition rate. A microscopic model of the growth of ${\mathit{S}}_{\mathit{B}}$ terraces is also presented in which we find that the nature of the growth, by accommodation, at the ${\mathit{S}}_{\mathit{B}}$(b) and ${\mathit{S}}_{\mathit{B}}$(n) edges, is different such that it maintains the roughness of the growing edge of an ${\mathit{S}}_{\mathit{B}}$ terrace.