Showing papers on "Free boundary problem published in 2003"
TL;DR: In this paper, a rational set of boundary conditions for flame-boundary interactions of an ideal, multicomponent, compressible gas has been determined by combining analyses of incompletely parabolic systems with those based on the hyperbolic Euler equations.
173 citations
TL;DR: In this paper, the singular elliptic boundary value problem is studied and the existence, multiplicity and asymptotic behavior of solutions of this problem are discussed by combining variational and sub-supersolution methods.
166 citations
TL;DR: In this article, the authors provided a unique smooth solution for a class of one and two-phase Stefan problems with Gibbs-Thomson correction in arbitrary space dimensions, and showed that the moving interface depends analytically on the temporal and spatial variables.
Abstract: We provide existence of a unique smooth solution for a class of oneand two-phase Stefan problems with Gibbs-Thomson correction in arbitrary space dimensions. In addition, it is shown that the moving interface depends analytically on the temporal and spatial variables. Of crucial importance for the analysis is the property of maximal Lpregularity for the linearized problem, which is fully developed in this paper as well.
141 citations
TL;DR: The Maxwell equations in a domain with Lipschitz boundary and the boundary integral operator A occuring in the Calderón projector are considered and an inf-sup condition for A is proved using a Hodge decomposition to prove quasioptimal convergence of the resulting boundary element methods.
Abstract: We consider the Maxwell equations in a domain with Lipschitz boundary and the boundary integral operator A occuring in the Calderon projector. We prove an inf-sup condition for A using a Hodge decomposition. We apply this to two types of boundary value problems: the exterior scattering problem by a perfectly conducting body, and the dielectric problem with two different materials in the interior and exterior domain. In both cases we obtain an equivalent boundary equation which has a unique solution. We then consider Galerkin discretizations with Raviart-Thomas spaces. We show that these spaces have discrete Hodge decompositions which are in some sense close to the continuous Hodge decomposition. This property allows us to prove quasioptimal convergence of the resulting boundary element methods.
134 citations
TL;DR: In this article, stable approximation schemes for the one-dimensional linear Schrodinger equation set in an unbounded domain were developed for the initial boundary value problem in a bounded domain with a transparent boundary condition.
129 citations
TL;DR: In this paper, a nonlinear evolution code was proposed for the initial-boundary value problem for Einstein's equations in harmonic coordinates, which is well posed for homogeneous boundary data and for boundary data that is small in a linearized sense.
Abstract: Maximally dissipative boundary conditions are applied to the initial-boundary value problem for Einstein's equations in harmonic coordinates to show that it is well posed for homogeneous boundary data and for boundary data that is small in a linearized sense. The method is implemented as a nonlinear evolution code, which satisfies convergence tests in the nonlinear regime and is stable in the weak field regime. A linearized version has been stably matched to a characteristic code to compute the gravitational wave form radiated to infinity.
113 citations
TL;DR: It is proved the existence, uniqueness, and regularity of the solution for a small time interval of p,q t,q T, the densities of cells within the tumor $\Omega_t in proliferating and quiescent states.
Abstract: We consider a system of two hyperbolic equations for p,q and two elliptic equations for $c,\sigma$, where p,q are the densities of cells within the tumor $\Omega_t$ in proliferating and quiescent states, respectively, c is the concentration of nutrients, and $\sigma$ is the pressure. The pressure is a result of the transport of cells which proliferate or die. The motion of the free boundary $\partial \Omega_t$ is given by the continuity condition, and $\sigma$ at the free boundary is proportional to the surface tension. We prove the existence, uniqueness, and regularity of the solution for a small time interval $0\leq t\leq T$.
110 citations
TL;DR: In this article, the Dirichlet problem for p-harmonic functions and p-energy minimizers in bounded domains in proper, path-connected metric measure spaces equipped with a doubling measure and supporting a Poincare inequality was studied.
Abstract: We study the Dirichlet problem for p-harmonic functions (and p-energy minimizers) in bounded domains in proper, pathconnected metric measure spaces equipped with a doubling measure and supporting a Poincare inequality. The Dirichlet problem has previously been solved for Sobolev type boundary data, and we extend this result and solve the problem for all continuous boundary data. We study the regularity of boundary points and prove the Kellogg property, i.e. that the set of irregular boundary points has zero p-capacity. We also construct p-capacitary, p-singular and p-harmonic measures on the boundary. We show that they are all absolutely continuous with respect to the p-capacity. For p = 2 we show that all the boundary measures are comparable and that the singular and harmonic measures coincide. We give an integral representation for the solution to the Dirichlet problem when p = 2, enabling us to extend the solvability of the problem to L-1 boundary data in this case. Moreover, we give a trace result for Newtonian functions when p = 2. Finally, we give an estimate for the Hausdorff dimension of the boundary of a bounded domain in Ahlfors Q-regular spaces. (Less)
100 citations
TL;DR: In this article, the free boundary problem for a nonlinear parabolic partial dif-ferential equation with a quadratic nonlinearity is considered and a series solution is proposed.
Abstract: In this paper we consider a free boundary problem for a nonlinear parabolic partial dif- ferential equation. In particular, we are concerned with the inverse problem, which means we know the behavior of the free boundary ap rioriand would like a solution, e.g. a convergent series, in order to determine what the trajectories of the system should be for steady-state to steady-state boundary control. In this paper we combine two issues: the free boundary (Stefan) problem with a quadratic nonlinearity. We prove convergence of a series solution and give a detailed parametric study on the series radius of convergence. Moreover, we prove that the parametrization can indeed can be used for motion planning purposes; computation of the open loop motion planning is straightforward. Simu- lation results are given and we prove some important properties about the solution. Namely, a weak maximum principle is derived for the dynamics, stating that the maximum is on the boundary. Also, we prove asymptotic positiveness of the solution, a physical requirement over the entire domain, as the transient time from one steady-state to another gets large. Mathematics Subject Classication. 93C20, 80A22, 80A23.
98 citations
TL;DR: In this article, the blow-up rate of large solutions of a class of sublinear elliptic boundary value problems with a weight function in front of the nonlinearity that vanishes on the boundary of the underlying domain, Ω, at different rates according to the point of the boundary, x ∞ ∈∂Ω.
96 citations
TL;DR: In this paper, a finite difference method is used to solve the one-dimensional Stefan problem with periodic Dirichlet boundary condition, and the temperature distribution, position of the moving boundary and its velocity are evaluated.
TL;DR: In this paper, the authors study the problem of stabilizing a 3D Navier-Stokes system given in a bounded domain with the help of feedback control defined on a part of the boundary.
Abstract: We study the problem of stabilization a solution to 3D Navier-Stokes
system given in a bounded domain $\Omega$. This stabilization is carried
out with help of feedback control defined on a part $\Gamma$ of boundary
$\partial \Omega$. We assume that $\Gamma$ is closed 2D manifold without
boundary. Here we continuer investigation begun in [6], [7] where
stabilization problem for parabolic equation and for 2D Navier-Stokes system
was studied.
TL;DR: In this paper, an optimal stability estimate for an inverse Robin boundary value problem arising in corrosion detection by electrostatic boundary measurements is presented, where the authors prove that the stability of the estimate is optimal.
Abstract: We prove an optimal stability estimate for an inverse Robin boundary value problem arising in corrosion detection by electrostatic boundary measurements.
TL;DR: In this paper, boundary value problems for elliptic systems in a domain complementary to a smooth surface with boundary are considered, where a crack with its edge is modeled as a boundary.
Abstract: We consider boundary value problems for elliptic systems in a domain complementary to a smooth surface with boundary, which models a crack with its edge. The same boundary conditions are prescribed...
TL;DR: In this paper, the authors convert a (linear abstract) initial boundary value problem into an abstract Cauchy problem on some product space and use spectral theory to discuss stability under boundary feedback.
Abstract: In this paper we convert a (linear abstract) initial boundary value
problem into an abstract Cauchy problem on some product space and use
semigroup methods to solve it. In particular, we apply spectral theory in
order to discuss stability under boundary feedback.
Journal Article•
TL;DR: In this paper, the authors considered free boundary problems for systems of partial differential equations and established the existence of symmetry-breaking bifurcation branches of solutions with free boundary of the form r = R + eF.
Abstract: In this paper we consider free boundary problems for systems of partial differential equations. The system has solutions which are spherically symmetric with free boundary r = R, for any value of a parameter γ. We establish the existence of symmetry-breaking bifurcation branches of solutions with free boundary of the form r = R + eF .
TL;DR: In this paper, the authors prove the first genuine partial differential equation result on a conjecture concerning the number of solutions of second-order elliptic boundary value problems with a nonlinearity which grows superlinearly at + ∞.
TL;DR: In this paper, the analysis of problems with mixed boundary conditions is dealt with, and results about existence, multiplicity and a priori estimates about multiplicity are proved about existence and multiplicity.
TL;DR: In this paper, a free boundary problem for a system of two partial differential equations, one parabolic and other elliptic, was studied and the existence and uniqueness of a solution for some time interval was established.
Abstract: In this article, we study a free boundary problem for a system of two partial differential equations, one parabolic and other elliptic. The system models the growth of a tumor with arbitrary initial shape. We establish the existence and uniqueness of a solution for some time interval. In the special case where we only have the elliptic equation, the problem coincides with the Hele–Shaw problem.
TL;DR: In this article, the authors considered the nonlocal boundary value problem in an arbitrary Banach space E with the positive operator A and established the well-posedness of this boundary-value problem in the spaces of smooth functions.
Abstract: The nonlocal boundary value problem in an arbitrary Banach space E with the positive operator A is considered. The well-posedness of this boundary value problem in the spaces of smooth functions is established. The new exact Schauder's estimates of solutions of the boundary value problems for elliptic equations are obtained.
TL;DR: This contribution deals with an efficient method for the numerical realization of the exterior and interior Bernoulli free boundary problems based on a shape optimization approach.
Abstract: This contribution deals with an efficient method for the numerical realization of the exterior and interior Bernoulli free boundary problems. It is based on a shape optimization approach. The state problems are solved by a fictitious domain solver using boundary Lagrange multipliers.
TL;DR: In this paper, a boundary element formulation is developed for the static analysis of two and three-dimensional solids and structures characterized by a linear elastic material behavior taking into account microstructural effects.
TL;DR: In this article, the authors established the behavior of the solutions of the degenerate parabolic equation u t = Δ(u m ), m > 1, posed in the whole space when the initial data are nonnegative, continuous and compactly supported.
Abstract: We establish the behavior of the solutions of the degenerate parabolic equation u t = Δ(u m ), m > 1, posed in the whole space when the initial data are nonnegative, continuous and compactly supported. We prove that, after a finite time, the pressure v = u m-1 becomes a concave function in the space variable which converges to all orders of differentiability to a truncated parabolic shape, so-called Barenblatt profile. In particular, the support of the solution is a convex subset of R N which converges to a ball. Estimates are optimal. The results are extended to the heat equation (log-concavity) and fast diffusion (pressure-convexity).
TL;DR: In this article, the Leray-Schauder alternative is used to solve the nonlinear anti-periodic first order problem, and several new existence results for the problem are presented.
Abstract: We prove several new existence results for a nonlinear anti-periodic first order problem using a Leray-Schauder alternative. Two definitions of lower and upper solutions are presented and we show in this paper the validity of the lower and upper solution method. Also, we give a method to generate a sequence of approximate solutions converging to a solution of the anti-periodic problem. ∗First and second authors were supported in part by D.G.E.S.I.C. (Spain), project PB97 – 0552. †This is the preprint version of the paper published in Journal of Mathematical Inequalities and Applications, Vol. 6 (2003) 477–485. ‡Corresponding author
TL;DR: In this paper, the authors consider a free boundary problem for a coupled system consisting of an elliptic equation Ap + μ(σ -?) = 0 (p > 0) for p and a parabolic equation for σ.
Abstract: We consider a free boundary problem for a coupled system consisting of an elliptic equation Ap + μ(σ -?) = 0 (p > 0) for p and a parabolic equation for σ. The problem is motivated by a model of tumor growth whereby p represents the pressure of the proliferating cells and (7 is the concentration of nutrients. On the boundary Γ(t) of the tumor region, p is equal to the surface tension, and the flux of p is equal to the normal velocity of Γ(t). In the case p = 0, the system decouples into a Hele-Shaw problem for p and a standard parabolic equation for σ. For the Hele-Shaw problem it is known that there are stationary radially symmetric solutions and each one is asymptotically stable in the following sense: If we take for initial data a small perturbation of a radially symmetric solution, then the corresponding Hele-Shaw problem has a unique global solution and its free boundary converges to a sphere as t → ∞. In this paper we prove a similar result for the coupled elliptic-parabolic problem provided p is small. The asymptotic stability result is generally false if p is not small.
TL;DR: In this paper, a diffuse interface model for the one-phase Hele-Shaw problem is derived from a gradient flow characterization due to Otto (1998 Arch. Mech. 141 63), which yields a generalized form of Darcy's law and reduces to a degenerate version of the well-known Cahn-Hilliard equation.
Abstract: A diffuse interface model for the one-phase Hele–Shaw problem is derived from a gradient flow characterization due to Otto (1998 Arch. Rat. Mech. Anal. 141 63). The resulting dynamical model yields a generalized form of Darcy's law, and reduces to a degenerate version of the well-known Cahn–Hilliard equation. Formal asymptotics illustrate the connection to the classical Hele–Shaw free boundary problem. Some example computations are carried out to demonstrate the flexibility of the modelling framework.
TL;DR: In this article, the radial basis functions were applied as a meshless method for solving diffusion type problems under free boundary condition, and the numerical solution of the Black-Scholes equation for pricing American options was obtained and compared with the traditional binomial method for numerical verification.
Abstract: This paper gives an order of convergence in applying the radial basis functions as a meshless method for solving diffusion type problems under free boundary condition. For illustration, the numerical solution of the Black–Scholes equation for pricing American options, which is a classical heat diffusion equation under free boundary value condition, is obtained and compared with the traditional binomial method for numerical verification.
TL;DR: In this article, a free boundary problem with three cell populations, namely proliferating cells, quiescent cells and dead cells, was studied, where the densities of these cells satisfy a system of nonlinear first order hyperbolic equations.
Abstract: In this paper we study a free boundary problem modeling the growth of tumors with three cell populations: proliferating cells, quiescent cells and dead cells. The densities of these cells satisfy a system of nonlinear first order hyperbolic equations in the tumor, with tumor surface as a free boundary. The nutrient concentration satisfies a diffusion equation, and the free boundary r = R(t) satisfies an integro-differential equation. We consider the radially symmetric case of this free boundary problem, and prove that it has a unique global solution for all the three cases 0 0, while limt!1 R(t) = 1 in the case KR = 0.
TL;DR: In this article, it was shown that for simple polygons and for a large class of boundary conditions, the above Riemann-Hilbert problem can either be reduced to a triangular RH problem which can be solved in closed form or bypassed, and the ρ j can be obtained using only algebraic manipulations.
Abstract: Let q(x, y) satisfy a boundary value problem for the Laplace equation in an arbitrary convex polygon with n sides An integral representation in the complex k-plane is given for q(x, y) in terms of n functions ρ j (k), j = 1,,n The function ρ j consists of an integral over the jth side involving both q x and q y , thus each ρ j involves one unknown boundary value The functions ρ j are not independent but they satisfy the important global relation that their sum vanishes The solution of a given boundary value problem reduces to the analysis of this single relation for the n unknown ρ j For a general polygon with general Poincare boundary conditions, this gives rise to a matrix Riemann-Hilbert problem In this paper it is shown that for simple polygons and for a large class of boundary conditions, the above Riemann-Hilbert problem (a) can either be reduced to a triangular RH problem which can be solved in closed form or (b) can be bypassed, and the ρ j can be obtained using only algebraic manipulations As an illustration of these 'triangular' and 'algebraic' cases we solve the Laplace equation in the quarter-plane, the semi-infinite strip and the right isosceles triangle with certain Poincare boundary conditions These boundary value problems, which include the Dirichlet and the Neumann problems as particular cases, cannot be solved by conformal mappings
TL;DR: The approach transforms the question of continuity of the input/output map of a boundary control system into boundedness of the solution to a related elliptic problem.
Abstract: Continuity of the input/output map for boundary control systems is shown through the system transfer function. Our approach transforms the question of continuity of the input/output map of a boundary control system to uniform boundedness of the solution to a related elliptic problem. This is shown for a class of boundary control systems with Dirichlet, Neumann, or Robin boundary control.