Lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth

In this paper we study asymptotic behavior of solutions for a free boundary problem modelling tumor growth. We first establish a general result for differential equations in Banach spaces possessing a Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either DA(θ) or DA(θ,∞) type. By using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem, we prove that if the surface tension coefficient γ is larger than a threshold value γ^* then the unique stationary solution is asymptotically stable modulo translations, provided the constant c is sufficiently small, whereas if γ<γ^* then this stationary solution is unstable.     

Bifurcation for a free boundary problem modeling the growth of multi-layer tumors

In this paper we study bifurcations for a free boundary problem modeling the growth of multi-layer tumors under the action of inhibitors. An important feature of this problem is that the surface tension effect of the free boundary is taken into account. By reducing this problem into an abstract bifurcation equation in a Banach space, overcoming some technical difficulties and finally using the Crandall–Rabinowitz bifurcation theorem, we prove that this problem has infinitely many branches of bifurcation solutions bifurcating from the flat solution.     

L q-Estimates for a Solution to the Problem about the Evolution of an Isolated Amount of a Fluid

We construct the L _q-theory of the nonstationary free boundary problem for the Navier–Stokes equations under the action of surface tension forces. Although the main theorem is proved for some finite time interval, it can be used for extending the solution to a larger interval since the smoothness of the solution is preserved for t < 0. Bibliography: 10 titles.     

Asymptotic Behaviour of Solutions of a Multidimensional Moving Boundary Problem Modeling Tumor Growth

We study a moving boundary problem modeling the growth of multicellular spheroids or in vitro tumors. This model consists of two elliptic equations describing the concentration of a nutrient and the distribution of the internal pressure in the tumor's body, respectively. The driving mechanism of the evolution of the tumor surface is governed by Darcy's law. Finally surface tension effects on the moving boundary are taken into account which are considered to counterbalance the internal pressure. To put our analysis on a solid basis, we first state a local well-posedness result for general initial data. However, the main purpose of our study is the investigation of the asymptotic behaviour of solutions as time goes to infinity. As a result of a centre manifold analysis, we prove that if the initial domain is sufficiently close to a Euclidean ball in the C ^m+μ-norm with m ≥ 3 and μ ∈ (0,1), then the solution exists globally and the corresponding domains converge exponentially fast to some (possibly shifted) ball, provided the surface tension coefficient γ is larger than a positive threshold value γ_*. In the case 0 < γ < γ_* the radially symmetric equilibrium is unstable.     

A Hopf Bifurcation in a Generalized Mckean Type of Free Boundary Problem Satisfying the Dirichlet Boundary Condition

In this paper, we consider the free boundary problem satisfying the Dirichlet boundary condition. This problem is derived from the reaction diffusion equations with the generalized McKean reaction dynamics. We shall show a Hopf bifurcation occurs at some critical point τ when the stationary solution (v^⋅ (x), s^⋅) satisfies 1/3 < s^⋅ < 1.     

The Spin-Coating Process: Analysis of the Free Boundary Value Problem

In this paper, an accurate model of the spin-coating process is presented and investigated from the analytical point of view. More precisely, the spin-coating process is being described as a one-phase free boundary value problem for Newtonian fluids subject to surface tension and rotational effects. It is proved that for T > 0 there exists a unique, strong solution to this problem in (0, T) belonging to a certain regularity class provided the data and the speed of rotation are small enough in suitable norms. The strategy of the proof is based on a transformation of the free boundary value problem to a quasilinear evolution equation on a fixed domain. The keypoint for solving the latter equation is the so-called maximal regularity approach. In order to pursue in this direction one needs to determine the precise regularity classes for the associated inhomogeneous linearized equations. This is being achieved by applying the Newton polygon method to the boundary symbol.     

Existence of Solutions of a Two-Point Boundary Value Problem

A two-point boundary value problem with a non-negative parameter Q arising in the study of surface tension induced flow of a liquid metal or semiconductor is studied. We prove that the problem has at least one solution for Q≥0.This improves a recent result that the problem has at least one solution for 0 ≤Q≤13.21.     

The first Steklov eigenvalue, conformal geometry, and minimal surfaces

We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue σ₁ of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Σ with genus γ and k boundary components we obtain the upper bound σ₁L(∂Σ)?2(γ+k)π. For γ=0 and k=1 this result was obtained by Weinstock in 1954, and is sharp. We attempt to find the best constant in this inequality for annular surfaces (γ=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that σ₁(Σ)L(∂Σ) is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary. Motivated by the annulus case, we show that a proper submanifold of the ball is immersed by Steklov eigenfunctions if and only if it is a free boundary solution. We then prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free boundary solution in two dimensions has area at least π, and we observe that this implies the sharp isoperimetric inequality for free boundary solutions in the two-dimensional case.     

Dynamical stability of compressible drops and stars

Interest is directed to linearized free boundary motion of a compressible liquid subject to surface tension and self‐gravitation respectively. Linearization relative to an a‐priori given solution to the non‐linear equations leads to a non‐local second order evolution problem to be posed in a space‐time cylinder with variable cross section subject to Fréchet boundary conditions along the lateral boundary part. Well‐posedness of the corresponding initial value problem in a natural weak formulation is proved. Copyright © 2005 John Wiley & Sons, Ltd.     

On the uniqueness of a solution of a two-phase free boundary problem

In this paper, we study the uniqueness problem of a two-phase elliptic free boundary problem arising from the phase transition problem subject to given boundary data. We show that in general the comparison principle between the sub- and super-solutions does not hold, and there is no uniqueness of either a viscosity solution or a minimizer of this free boundary problem by constructing counter-examples in various cases in any dimension. In one-dimension, a bifurcation phenomenon presents and the uniqueness problem has been completely analyzed. In fact, the critical case signifies the change from uniqueness to non-uniqueness of a solution of the free boundary problem. Non-uniqueness of a solution of the free boundary problem suggests different physical stationary states caused by different processes, such as melting of ice or solidification of water, even with the same prescribed boundary data. However, we prove that a uniqueness theorem is true for the initial-boundary value problem of an ε-evolutionary problem which is the smoothed two-phase parabolic free boundary problem.     

Analysis of an elliptic-parabolic free boundary problem modelling the growth of non-necrotic tumor cord

We study a free boundary problem modelling the growth of a tumor cord in which tumor cells live around and receive nutrient from a central blood vessel. The evolution of the tumor cord surface is governed by Darcy's law together with a surface tension equation. The concentration of nutrient in the tumor cord satisfies a reaction-diffusion equation. In this paper we first establish a well-posedness result for this free boundary problem in some Sobolev-Besov spaces with low regularity by using the analytic semigroup theory. We next study asymptotic stability of the unique radially symmetric stationary solution. By making delicate spectrum analysis for the linearized problem, we prove that this stationary solution is locally asymptotically stable provided that the constant c representing the ratio between the diffusion time of nutrient and the birth time of new cells is sufficiently small.     

Bifurcation for a free boundary problem with surface tension modeling the growth of multi-layer tumors

This paper is devoted to the study of the bifurcation of a free boundary problem modeling the growth of tumors with the effect of surface tension being considered. The existence of infinitely many branches of bifurcation solutions is proved. The method of analysis is based on reducing the problem to an operator equation in certain Hölder space with a nonlinear Fredholm operator of index 0. The desired result then follows from the Crandall-Rabinowitz bifurcation theorem.     

Computing steady-state solutions for a free boundary problem modeling tumor growth by Stokes equation

We consider a free boundary problem modeling tumor growth where the model equations include a diffusion equation for the nutrient concentration and the Stokes equation for the proliferation of tumor cells. For any positive radius R$R$, it is known that there exists a unique radially symmetric stationary solution. The proliferation rate μ$\mu$ and the cell-to-cell adhesiveness γ$\gamma$ are two parameters for characterizing “aggressiveness” of the tumor. We compute symmetry-breaking bifurcation branches of solutions by studying a polynomial discretization of the system. By tracking the discretized system, we numerically verified a sequence of μ/γ$\mu /\gamma$ symmetry breaking bifurcation branches. Furthermore, we study the stability of both radially symmetric and radially asymmetric stationary solutions.     

Well-posedness and stability for an elliptic-parabolic free boundary problem modeling the growth of multi-layer tumors

In this paper we study a free boundary problem modeling the growth of multi-layer tumors. This free boundary problem contains one parabolic equation and one elliptic equation, defined on an unbounded domain in R2 of the form 0 〈 y 〈p（x,t）, where p（x,t） is an unknown function. Unlike previous works on this tumor model where unknown functions are assumed to be periodic and only elliptic equations are evolved in the model, in this paper we consider the case where unknown functions are not periodic functions and both elliptic and parabolic equations appear in the model. It turns out that this problem is more difficult to analyze rigorously. We first prove that this problem is locally well-posed in little H61der spaces. Next we investigate asymptotic behavior of the solution. By using the principle of linearized stability, we prove that if the surface tension coefficient y is larger than a threshold value y〉0, then the unique flat equilibrium is asymptotically stable provided that the constant c representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficiently small.     

Stabilization of periodic Stokesian Hele–Shaw flows of ferrofluids

We consider an incompressible ferrofluid in a vertical Hele–Shaw cell and develop a proper analytic framework for the free interface and the velocity potential of the fluid in a periodic geometry. The flow is assumed to obey a non-Newtonian Darcy law. The forces influencing the fluid are gravity, surface tension and the response to a magnetic field induced by a current. In addition, the flow is stabilized at the lower boundary component by an external source b. We prove a well-posedness result for the flow near flat solutions. Moreover, we find conditions on the parameters and on the slope of b for the exponential stability and instability of flat interfaces. Furthermore, we identify values for the current's intensity ι where critical bifurcation of nontrivial finger-shaped solutions from the branch of trivial (flat) solutions takes place.     

Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator

In this paper we study the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lipschitz free boundaries are C1,γ-smooth for some γ∈(0,1). As part of our argument, and which is of independent interest, we establish a Hopf boundary type principle for non-negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain.     

On the nonstationary motion of a viscous incompressible liquid over a rotating ball

We consider the free boundary problem for the Navier–Stokes equations governing a nonstationary motion of a layer of a viscous incompressible liquid that covers the surface of a rigid ball rotating around a fixed axis with constant angular velocity ω. The liquid is subject to the gravitation force generated by the mass of the ball. The self-gravitation forces between the liquid particles and capillary forces on the free surface are not taken into account. We consider the problem of stability of the regime of the rigid rotation of the liquid with the same angular velocity and prove that it is stable if |ω| is less than a certain constant. Bibliography: 10 titles. Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 91–145.     

A conformal mapping algorithm for the Bernoulli free boundary value problem

We propose a new numerical method for the solution of the Bernoulli free boundary value problem for harmonic functions in a doubly connected domain D in where an unknown free boundary Γ₀ is determined by prescribed Cauchy data on Γ₀ in addition to a Dirichlet condition on the known boundary Γ₁. Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar, and Kress for the solution of a related inverse boundary value problem. For this, we interpret the free boundary Γ₀ as the unknown boundary in the inverse problem to construct Γ₀ from the Dirichlet condition on Γ₀ and Cauchy data on the known boundary Γ₁. Our method for the Bernoulli problem iterates on the missing normal derivative on Γ₁ by alternating between the application of the conformal mapping method for the inverse problem and solving a mixed Dirichlet–Neumann boundary value problem in D. We present the mathematical foundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach. Copyright © 2015 John Wiley & Sons, Ltd.     

On the existence of incompressible current–vortex sheets: study of a linearized free boundary value problem

We study the initial boundary value problem resulting from the linearization of the equations of ideal incompressible magnetohydrodynamics and the jump conditions on the hypersurface of tangential discontinuity (current–vortex sheet) about an unsteady piecewise smooth solution. Under some assumptions on the unperturbed flow, we prove an energy a priori estimate for the linearized problem. Since the so‐called loss of derivatives in the normal direction to the boundary takes place even for the constant coefficients linearized problem, for the variable coefficients problem and non‐planar current–vortex sheets the natural functional setting is provided by the anisotropic weighted Sobolev space W₂^1,σ. The result of this paper is a necessary step to prove the local in time existence of solutions of the original non‐linear free boundary value problem. The uniqueness of the regular solution of this problem follows already from the a priori estimate we obtain for the linearized problem. Copyright © 2005 John Wiley & Sons, Ltd.     

On a resolvent estimate of the Stokes system in a half space arising from a free boundary problem for the Navier–Stokes equations

In this paper we consider a resolvent problem of the Stokes operator with some boundary condition in the half space, which is obtained as a model problem arising in evolution free boundary problems for viscous, incompressible fluid flow. We show standard resolvent estimates in the L_q framework (1 < q < ∞), applying some kernel estimates to concrete solution formulas. The Volevich trick in [21] plays a fundamental role in estimating solutions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)