Showing papers on "Free boundary problem published in 1996"
TL;DR: In this article, it was shown that the coefficient -y(x) of the elliptic equation Vie (QyVu) = 0 in a two-dimensional domain is uniquely determined by the corresponding Dirichlet-to-Neumann map on the boundary.
Abstract: We show that the coefficient -y(x) of the elliptic equation Vie (QyVu) = 0 in a two-dimensional domain is uniquely determined by the corresponding Dirichlet-to-Neumann map on the boundary, and give a reconstruction pro
973 citations
TL;DR: In this paper, the authors proposed boundary conditions that are the equivalent of the frequency-domain impedance boundary condition for both single frequency and model broadband time domain impedance boundary conditions, together with linearized Euler equations, form well-posed initial boundary value problems.
Abstract: It is an accepted practice in aeroacoustics to characterize the properties of an acoustically treated surface by a quantity known as impedance. Impedance is a complex quantity. As such, it is designed primarily for frequency-domain analysis. Time-domain boundary conditions that are the equivalent of the frequency-domain impedance boundary condition are proposed. Both single frequency and model broadband time-domain impedance boundary conditions are provided. It is shown that the proposed boundary conditions, together with the linearized Euler equations, form well-posed initial boundary value problems. Unlike ill-posed problems, they are free from spurious instabilities that would render time-marching computational solutions impossible.
195 citations
Journal Article•
TL;DR: New methods for studying the Navier-Stokes equations in thin domains are developed and the behavior of the average of the strong solution in the thin direction when the thickness of the domain goes to zero is studied.
Abstract: In this work we develop methods for studying the Navier-Stokes equations in thin domains. We consider various boundary conditions and establish the global existence of strong solutions when the initial data belong to "large sets." Our work was inspired by the recent interesting results of G. Raugel and G. Sell [22, 23, 24] which, in the periodic case, give global existence for smooth solutions of the 3D Navier-Stokes equations in thin domains for large sets of initial conditions. We extend their results in several ways, we consider numerous boundary conditions and as it will appear hereafter, the passage from one boundary condition to another one is not necessarily straightforward. The proof of our improved results is based on precise estimates of the dependence of some classical constants on the thickness $\epsilon$ of the domain, e.g. Sobolev-type constants and the regularity constant for the corresponding Stokes problem.
134 citations
TL;DR: In this article, a variational formulation for the bean critical state model was derived and used to solve two-dimensional and axially symmetric critical-state problems numerically, which is a free boundary problem and its solution is of interest in applied superconductivity.
127 citations
110 citations
110 citations
TL;DR: In this paper, the authors studied the regularity properties of the free boundary of a parabolic two-phase free boundary problem with respect to the Stefan model and showed that the solution of the Stefan problem is as smooth and classical as possible.
Abstract: In this paper we start the study of the regularity properties of the free boundary, for parabolic two-phase free boundary problems. May be the best known example of a parabolic two-phase free boundary problem is the Stefan problem, a simplified model describing the melting (or solidification) of a material with a solid-liquid interphase. The concept of solution can be stated in several ways (classical solution, weak so- lution on divergence form, or viscosity solution) and as usual, one would like to prove that the (weak) solutions that may be constructed, are in fact as smooth and classical as possible. Locally, a classical solution of the Stefan problem may be described as following: On the unit cylinder Q1 =B1 “ (-1, 1) we have two complementary domains, ~ and QI\~, separated by a smooth surface S=(OI2)NQ1. In fl and QI\~ we have two smooth solutions, Ul and u2, of the heat equations
89 citations
TL;DR: In this article, the authors study the differentiability properties of strongly singular and vector valued boundary integral operators in Holder spaces and prove that the solution to the electromagnetic scattering problem depends infinitely differentiable on the boundary of the obstacle.
Abstract: Within the integral equation approach we study the dependence of the solution to the electromagnetic scattering problem from a perfect conductor with respect to the obstacle. We study the differentiability properties of strongly singular and vector valued boundary integral operators in Holder spaces. We prove that the solution to the scattering problem depends infinitely differentiable on the boundary of the obstacle. We give a characterization of the first derivative as a solution to a boundary value problem.
80 citations
TL;DR: In this article, a proof of the existence and uniqueness of a solution of a mixed problem with boundary integral conditions for a certain parabolic equation is given based on an energy inequality and on the fact that the range of the operator generated by the problem is dense.
Abstract: The present article is devoted to a proof of the existence and uniqueness of a solution of a mixed problem with boundary integral conditions for a certain parabolic equation. The proof is based on an energy inequality and on the fact that the range of the operator generated by the problem is dense.
80 citations
TL;DR: In this paper, the Frechet differentiability of boundary integral operators in the spaces of continuous functions is investigated. But the Freche differentiability is not shown for the case of the boundary integral operator in this paper.
Abstract: Using integral equation methods to solve the time harmonic acoustic scattering problem with Neumann boundary condition it is possible to reduce the solution of the scattering problem to the solution of a boundary integral equation of the second kind. We show the Frechet differentiability of the boundary integral operators which occur. They are considered in dependence of the boundary as integral operators in the spaces of continuous functions. Then we use this to prove the Frechet differentiability of the scattered fields. Finally we characterize the Frechet derivatives of the scattered fields by a suitable boundary value problem.
76 citations
TL;DR: In this article, a hierarchy of effective boundary conditions for a Helmholtz equation with Dirichlet or Neumann boundary conditions is derived, which approximate the effect of thin dielectric layers with perfectly conducting inclusions covering an object.
TL;DR: In this article, the first-order Hamilton-Jacobi-Bellman equation associated with the state constraint problem for optimal control is studied, and a new and appropriate boundary condition for the PDE is proposed.
Abstract: The first-order Hamilton--Jacobi--Bellman equation associated with the state constraint problem for optimal control is studied. Instead of the boundary condition which Soner introduced, a new and appropriate boundary condition for the PDE is proposed. The uniqueness and Lipschitz continuity of viscosity solutions for the boundary value problem are obtained.
TL;DR: In this article, the authors consider boundary value problems on infinite intervals governed by a third-order ordinary differential equation and highlight a novel approach to define the truncated boundary, which is an unknown free boundary and has to be determined as part of the solution.
Abstract: The classical numerical treatment of two-point boundary value problems on infinite intervals is based on the introduction of a truncated boundary (instead of infinity) where appropriate boundary conditions are imposed. Then, the truncated boundary allowing for a satisfactory accuracy is computed by trial. Motivated by several problems of interest in boundary layer theory, here we consider boundary value problems on infinite intervals governed by a third-order ordinary differential equation. We highlight a novel approach to define the truncated boundary. The main result is the convergence of the solution of our formulation to the solution of the original problem as a suitable parameter goes to zero. In the proposed formulation, the truncated boundary is an unknown free boundary and has to be determined as part of the solution. For the numerical solution of the free boundary formulation, a noniterative and an iterative transformation method are introduced. Furthermore, we characterize the class of free boundary value problems that can be solved noniteratively. A nonlinear flow problem involving two physical parameters and belonging to the characterized class of problems is then solved. Moreover, the Falkner--Skan equation with relevant boundary conditions is considered and representative results, obtained by the iterative transformation method, are listed for the Homann flow. All the obtained numerical results clearly indicate the effectiveness of our approach. Finally, we discuss the possible extensions of the proposed approach and for the question of a priori error analysis.
TL;DR: In this article, the authors considered the free boundary problem in the case of an incompressible fluid-air flow in a homogeneous and isotropic porous medium and introduced a class of admissible interfaces.
Abstract: Of concern is a class of free boundary problems which arise, for instance, in connection with the flow of an incompressible fluid in porous media. More precisely, we consider the following situation: Let F0 denote a fixed, impermeable layer in a homogeneous and isotropic porous medium. We assume that some part of the region above F0 is occupied with an incompressible Newtonian fluid. In addition, we suppose that there is a sharp interface, Ff, separating the wet region I2f enclosed by F0 and Ff, respectively, from the dry part, i.e., we consider a saturated fluid-air flow. The fluid moves under the influence of gravity and we assume that the motion is governed according to Darcy's law. The standard model encompassing this situation consists of an elliptic equation for a velocity potential, to be solved in a domain with a free boundary, and of an evolution equation for the free boundary. In order to give a concise mathematical description let us introduce the following class of admissible interfaces:
TL;DR: In this paper, a posteriori error estimates for elliptic variational inequalities are derived for the solution of corresponding scalar local subproblems, and some upper bounds for the effectivity rates and the numerical properties are illustrated by typical examples.
Abstract: We derive hierarchical a posteriori error estimates for elliptic variational inequalities. The evaluation amounts to the solution of corresponding scalar local subproblems. We derive some upper bounds for the effectivity rates and the numerical properties are illustrated by typical examples.
TL;DR: In this paper, the authors presented a new fictitious domain formulation for the solution of strongly elliptic boundary value problem with Neumann boundary conditions for a bounded domain in a finite-dimensional Euclidean space with a smooth (possibly only Lipschitz) boundary.
TL;DR: In this article, an open supersymmetric t -J chain with boundary fields is studied by means of the Bethe ansatz. But the boundary susceptibilities are calculated as functions of the boundary fields, and the effects of boundary on excitations are investigated by constructing the exact boundary S-matrix.
Abstract: An open supersymmetric t - J chain with boundary fields is studied by means of the Bethe ansatz. Ground state properties for the case of an almost half-filled band and a bulk magnetic field are determined. Boundary susceptibilities are calculated as functions of the boundary fields. The effects of the boundary on excitations are investigated by constructing the exact boundary S-matrix. From the analytic structure of the boundary S-matrices one deduces that holons can form boundary bound states for sufficiently strong boundary fields.
TL;DR: In this article, an estimate of Lipschitz type on the continuous dependence of an unknown linear crack from the boundary measurements is presented. Butler et al. consider the inverse boundary value problem of crack detection in a two-dimensional electrical conductor.
Abstract: We consider the inverse boundary value problem of crack detection in a two-dimensional electrical conductor. We prove an estimate of Lipschitz type on the continuous dependence of an unknown linear crack from the boundary measurements.
TL;DR: In this paper, the problem of determining part of the boundary of a domain where a potential satisfies the Laplace equation has been studied and a sufficient condition for the potential to be monotonic along the unknown boundary is established.
Abstract: We have studied the problem of determining part of the boundary of a domain where a potential satisfies the Laplace equation. The potential and its normal derivative have prescribed values on the known part of the boundary that encloses while its normal derivative must vanish on the remaining part. We establish a sufficient condition for the potential to be monotonic along the unknown boundary. This allows us to use the potential to parametrize the boundary. Two methods are presented that solve the problem under this assumption. The first one solves the problem in a closed form and it can be used to define a parameter that will describe the ill-posedness of the problem. The effect of this parameter on the second method presented has been determined for a particular numerical example.
TL;DR: In this paper, a numerical method for 2D orthogonal grid generation with control of the boundary point distribution is presented, based on the solution of a system of partial differential equations.
TL;DR: In this article, the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane was considered and a new boundary integral equation formulation for this problem was proposed, utilizing the Green's function for an impedance halfplane in place of the standard fundamental solution.
Abstract: We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle Finally, a result of continuous dependence of the solution on the boundary shape is obtained
TL;DR: In this paper, the authors used a path-following method for the two point boundary value problem governing the ignition of a solid reactant undergoing slow oxidation for symmetric class A geometries and showed the occurrence of multiplicity of steady states.
TL;DR: In this article, a boundary element method is derived for solving a class of boundary value problems governed by an elliptic second order linear partial differential equation with variable coefficients, and numerical results are given for a specific test problem.
Abstract: A boundary element method is derived for solving a class of boundary value problems governed by an elliptic second order linear partial differential equation with variable coefficients. Numerical results are given for a specific test problem.
TL;DR: The inverse problem of recovering the absorption coefficient and the collision kernel in the stationary linear Boltzmann equation in a bounded domain from the albedo operator on the boundary was studied in this article.
Abstract: We study the inverse problem of recovering the absorption coefficient and the collision kernel in the stationary linear Boltzmann equation in a bounded domain from the albedo operator on the boundary. We show that under some conditions on the coefficients that guarantee well-posedness of the direct problem, the inverse problem has a unique solution. Moreover, we provide explicit formulae for recovering , k.
TL;DR: The numerical solution of homogenization equations by the finite element (FE) method is explained briefly in this paper, where the issue of extracting boundary conditions from the periodicity assumption is addressed and a direct method utilizing symmetry is presented.
Abstract: The numerical solution of homogenization equations by the finite element (FE) method is explained briefly. The issue of extracting boundary conditions from the periodicity assumption is addressed and a direct method utilizing symmetry is presented. Using this method, the computation of the elements of the constitutive matrix of a composite material is reduced to a very conventional boundary value problem with known forces and boundary conditions which can be carried out with any FE code. Two examples are presented.
TL;DR: In this paper, the validity of the method of upper and lower solutions to obtain an existence result for a periodic boundary value problem of first order impulsive differential equations at variable times was shown.
TL;DR: The mode properties for spectral and mixed boundary conditions for massless spin-fields are derived for the d-ball in this paper, where the corresponding functional determinants and traced heat-kernel coefficients are presented, the latter as polynomials in d.
Abstract: The mode properties for spectral and mixed boundary conditions for massless spin- fields are derived for the d-ball. The corresponding functional determinants and traced heat-kernel coefficients are presented, the latter as polynomials in d.
TL;DR: Using the finite-element method to discretize the variational inequality problems related to some free boundary problems with nonlinear source terms, the resulting algebraic systems are solved by the combination of the quasi-Newton method and the domain decomposition techniques.
Abstract: Using the finite-element method to discretize the variational inequality problems related to some free boundary problems with nonlinear source terms, we then solve the resulting algebraic systems by the combination of the quasi-Newton method and the domain decomposition techniques. The numerical solution processes are completely parallelizable. Both the convergence of the finite-element approximation and of the parallel iterative processes are proved.