On a two-phase free boundary problem for reaction-diffusion equations in population biology

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.

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 with discontinuous velocity: Application to tumor model

We consider the optimization problem of minimizing \(\mathcal{J }_{\gamma }(u)=\int _{\Omega }(G(|\nabla u|)+\lambda _{+}(u^{+})^{\gamma }+\lambda _{-}(u^{-})^{\gamma }+fu)\,\text {d}x\) in the class of functions \(W^{1,G}(\Omega )\) with \(u-\varphi \in W^{1,G}_{0}(\Omega )\) for a given function \(\varphi \), where \(W^{1,G}(\Omega )\) is the class of weakly differentiable functions with \(\int _{\Omega }G(|\nabla u|)\,\text {d}x<\infty \). The conditions on the function \(G\) allow for a different behavior at \(0\) and at \(\infty \). We give a rather complete description of regularity theory for a family of two-phase variational free boundary problems. For \(0<\gamma \le 1\), we prove that every minimizer \(u_{\gamma }\) of \(\mathcal{J }_{\gamma }(u)\) is \(C^{1,\alpha }_{loc}\) continuous. For \(\gamma =0\), we obtain local Lipschitz continuity for any minimizer \(u_{0}\) of \(\mathcal{J }_{0}(u)\). Here we also consider an asymptotic problem as \(\gamma \rightarrow 0\). At last, we establish sharp geometric estimates for the free boundary corresponding to the minimizer \(u_{0}\) of \(\mathcal J _{0}(u)\).

A two-phase free boundary problem associated with the Burgers equation is considered. The problem is reduced to a system of nonlinear integral equations which is analysed and shown to have a unique solution. The system admits a two-component shock solution which travels with the same velocity as that of the free boundary. The stability analysis of such a solution shows the existence of stability and instability regions according to different values of the parameters characterizing the system.

In this paper, we study the limit as p goes to infinity of a minimizer of a variational problem that is a two-phase free boundary problem of phase transition for the p -Laplacian. Under a geometric compatibility condition, we prove that this limit is a solution of a free boundary problem for the \infty -Laplacian. When the compatibility condition does not hold, we prove that there still exists a uniform limit that is a solution of a minimization problem for the Lipschitz constant. Moreover, we provide, in the latter case, an example that shows that the free boundary condition can be lost in the limit.

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 continue the analysis of the two-phase free boundary problems initiated by ourselves, studying where we studied the linear growth of minimizers in a Bernoulli-type free boundary problem at the non-flat points and the related regularity of free boundary. There, among other things, we also defined the functionalwhere x0 is a free boundary point, i.e. and u is a minimizer of the functional for some bounded smooth domain and positive constants with . Here we show in discrete monotone at non-flat points x0, when N = 2 and p is sufficiently close to 2, and then establish the linear growth of u. A new feature of our approach is the anisotropic scaling argument discussed in Section 4.

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

Regularity for degenerate two-phase free boundary problems

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.

&lt;abstract&gt;&lt;p&gt;In this paper, we are concerned with the existence, boundedness, and integrability of minimizers of heterogeneous, two-phase free boundary problems $ \mathcal {J}_{\gamma}(u) = \int_{\Omega}\left(f(x, \nabla u)+\lambda_{+}(u^{+})^{\gamma}+\lambda_{-}(u^{-})^{\gamma}+gu\right)\text{d}x \rightarrow \text{min} $ under non-standard growth conditions. Included in such problems are heterogeneous jets and cavities of Prandtl-Batchelor type with $ \gamma = 0 $, chemical reaction problems with $ 0 &amp;lt; \gamma &amp;lt; 1 $, and obstacle type problems with $ \gamma = 1 $, respectively.&lt;/p&gt;&lt;/abstract&gt;

We prove Lipschitz continuity of viscosity solutions to a class of two-phase free boundary problems governed by fully non-linear operators.

In this paper we start to develop the regularity theory of general two-phase free boundary problems for parabolic equations. In particular we consider uniformly parabolic operators in nondivergence form and we are mainly concerned with the optimal regularity of the viscosity solutions. We prove that under suitable nondegenerate conditions the solution is Lipschitz across the free boundary.

We describe the theory we developed in recent times concerning two-phasefree boundary problems governed by elliptic operators with forcing terms.Our results range from existence of viscosity solutions to smoothness ofboth solutions and free boundaries. We also discuss some open questions,possible object of future investigation.