Showing papers on "Free boundary problem published in 2001"

Boundary Harnack principle and Martin boundary for a uniform domain

Abstract: We establish a uniform boundary Harnack principle for a uniform domain. As applications we study the Holder continuity of the ratios of positive harmonic functions, the Martin boundary and the Fatou theorem for a uniform domain.

Two-dimensional linear partial differential equations in a convex polygon

Abstract: A method is introduced for solving boundary‐value problems for linear partial differential equations (PDEs) in convex polygons. It consists of three algorithmic steps. (1) Given a PDE , construct two compatible eigenvalue equations. (2) Given a polygon , perform the simultaneous spectral analysis of these two equations. This yields an integral representation in the complex k ‐plane of the solution q (x1,x2) in terms of a function q ( k ), and an integral representation in the (x1, x2)‐plane of q( k ) in terms of the values of q and of its derivatives on the boundary of the polygon. These boundary values are in general related, thus only some of them can be prescribed. (3) Given appropriate boundary conditions , express the part of q ( k ) involving the unknown boundary values in terms of the boundary conditions. This is based on the existence of a simple global relation formulated in the complex k ‐plane, and on the invariant properties of this relation. As an illustration, the following integral representations are obtained: (a) q (x, t ) for a general dispersive evolution equation of order n in a domain bounded by a linearly moving boundary; (b) q (x,y) for the Laplace, modified Helmholtz and Helmholtz equations in a convex polygon. These general formulae and the analysis of the associated global relations are used to discuss typical boundary‐value problems for evolution equations and for elliptic equations.

Nonlinear boundary value problems with concave nonlinearities near the origin

Abstract: In this paper we consider the effect of concave nonlinearities for the solution structure of nonlinear boundary value problems such as Dirichlet and Neumann boundary value problems of elliptic equations and periodic boundary value problems for Hamiltonian systems and nonlinear wave equations.

Free surface Stokes flow over topography

Abstract: We study two-dimensional viscous flow over topographical features under the action of an external body force. The Stokes equations are written as a set of harmonic and biharmonic equations for vorticity and stream function. A direct biharmonic boundary integral equation method is used to transform these equations to a pair of integral equations on the boundary of the domain. These equations are solved with a preassumed free surface profile to obtain values of the flow variables on the boundary. The location of the free boundary is then updated by considering the normal stress condition and using an iteration technique. We have studied the flow over steps and trenches of different depths and for a wide range of capillary numbers, Ca. Our computation shows that for small Ca, the free surface develops a ridge before the entrance to a step down and a depression region right before a step up. The magnitude and location of these features depend on the capillary number and the step depth. For large capillary numbers the free surface nearly follows the topography and the ridge and depression are found to be exponentially small in the capillary number. On the other hand, our results agree well with lubrication theory for small capillary numbers on the order of 10−2 or less, even for steep features.

Large-Time Behaviors of Solutions to an Inflow Problem in the Half Space for a One-Dimensional System¶of Compressible Viscous Gas

Abstract: The “inflow problem” for a one-dimensional compressible barotropic flow on the half-line R += (0,+∞) is investigated. Not only classical waves but also the new wave, which is called the “boundary layer solution”, arise. Large time behaviors of the solutions to be expected have been classified in terms of the boundary values by [A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, to appear in Proceedings of IMS Conference on Differential Equations from Mechanics, Hong Kong, 1999]. In this paper we give the rigorous proofs of the stability theorems on both the boundary layer solution and a superposition of the boundary layer solution and the rarefaction wave.

Numerical solution of a Cauchy problem for the Laplace equation

Abstract: We consider a two-dimensional steady state heat conduction problem. The Laplace equation is valid in a domain with a hole. Temperature and heat-flux data are specified on the outer boundary, and we wish to compute the temperature on the inner boundary. This Cauchy problem is ill-posed, i.e. the solution does not depend continuously on the boundary data, and small errors in the data can destroy the numerical solution. We consider two numerical methods for solving this problem. A standard approach is to discretize the differential equation by finite differences, and use Tikhonov regularization on the discrete problem, which leads to a large sparse least squares problem. We propose to use a conformal mapping that maps the region onto an annulus, where the equivalent problem is solved using a technique based on the fast Fourier transform. The ill-posedness is dealt with by filtering away high frequencies in the solution. Numerical results using both methods are given.

Twin solutions of boundary value problems for ordinary differential equations and finite difference equations

Abstract: A new twin fixed-point theorem is applied first to obtain the existence of at least two positive solutions for the right focal boundary value problem y″ + f(ity) = 0, 0 <- t <- 1, y(0) = y′(1) = 0. It is applied later to obtain the existence of at least two positive solutions for the analogous discrete right focal boundary value problem Δ2u(k) + g(u(k)) = 0, k ϵ {0, … ,N}, u(0) = Δu(N + 1) = 0.

An initial-boundary value problem for the Korteweg-de Vries equation posed on a finite interval

Abstract: The Korteweg--de Vries equation occurs as a model for unidirectional propagation of small amplitude long waves in numerous physical systems. The aim of this work is to propose a well-posed mixed initial--boundary value problem when the spacial domain is of finite extent. More precisely, we establish local existence of solutions for arbitrary initial data in the Sobolev space $H^1$ and global existence for small initial data in this space. In a second step we show global strong regularizing effects.

Sharp Results for the Regularity and Stability of the Free Boundary in the Obstacle Problem

Abstract: In this paper, we study the obstacle problem with obstacles whose Laplacians are not necessarily Holder continuous. We show that the free boundary at a regular point is C 1 if the Laplacian of the obstacle is negative and Dini continuous. We also show that this condition is sharp by giving a method to construct a counter-example when we weaken the requirement on the Laplacian of the obstacle by allowing it to have any modulus of continuity which is not Dini. In the course of proving optimal regularity we also improve some of the perturbation theory due to Caffarelli(1981). Since our methods depend on comparisonprinciples and regularity theory, and not on linearity, our stability results apply to a large class of obstacle problems with nonlinear elliptic operators. In the case of obstacles where the Laplacian is negative and has sufficiently small oscillation,we establishmeasure-theoretic analogues of the alternative proven by Caffarelli (1977). Specifically, if the Laplacian is continuous, then at a free boundary point either the contact set has density zero, or the free boundary is a Reifenberg vanishing set and the contact set has density equal to one half in a neighborhood of the point. If the Laplacian is not necessarily continuous, but has sufficiently small oscillation, then at a free boundary point either the contact set has density close to zero, or the free boundary is a �-Reifenberg set and the contact set has density close to one half in a neighborhood of the point.

Boundary conditions as Dirac constraints

Abstract: In this article we show that boundary conditions can be treated as Lagrangian and Hamiltonian constraints. Using the Dirac method, we find that boundary conditions are equivalent to an infinite chain of second class constraints, which is a new feature in the context of constrained systems. Constructing the Dirac brackets and the reduced phase space structure for different boundary conditions, we show why mode expanding and then quantizing a field theory with boundary conditions is the proper way. We also show that in a quantized field theory subjected to the mixed boundary conditions, the field components are non-commutative.

Mathematical concepts of open quantum boundary conditions

Abstract: This paper is concerned with the derivation and the numerical discretization of open boundary conditions for the 1D Schrodinger equation in order to simulate quantum devices. New discrete transparent boundary conditions are presented that are able to handle the situation of a continuous plane wave influx into a device. Also, we give a review of various formulations of boundary conditions that are used within the Schrodinger and the Wigner formalism of quantum mechanics, and we discuss their mathematical properties.

An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary

Abstract: where λ+ 0 and λ+ + λ− > 0. Equation (1) is related to the time-dependent equation 0 = α∂t max(v, 0)+ β∂t min(v, 0)−∆v in (0, T ) × Ω which has been used to describe an instantaneous and complete reaction of two substances coming into contact at a surface Γ (see [2, 3] and [6]). The difficulty one confronts in this Stefan-like problem is that the interface {v = 0} consists in general of two parts—one where the gradient of v is nonzero and one where the gradient of v vanishes. At the latter part we expect the gradient of v to have linear growth in space. However, because of the decomposition into two different types of growth, it is not possible to derive a growth estimate by, for example, a Bernstein technique. Assuming that α > β > 0 and that the time derivative ∂tv is non-negative and Holder continuous near some free boundary point (t0, x0), v(t0) is a solution of (1) with Holder continuous coefficients λ+ and λ−. To that our result applies and yields the expected C1,1-regularity of v(t0) in a pointwise sense and, for positive time derivative, the Hausdorff dimension estimate dim(∂{v(t0) > 0} ∪ ∂{v(t0) < 0}) n − 1 (see Proposition 4.1, Remark 4.1, Corollary 5.1 and Remark 5.1).

Reflected forward-backward SDEs and obstacle problems with boundary conditions

Abstract: In this paper we study a class of forward-backward stochastic differential equations with reflecting boundary conditions (FBSDER for short). More precisely, we consider the case in which the forward component of the FBSDER is restricted to a fixed, convex region, and the backward component will stay, at each fixed time, in a convex region that may depend on time and is possibly random. The solvability of such FBSDER is studied in a fairly general way. We also prove that if the coefficients are all deterministic and the backward equation is one-dimensional, then the adapted solution of such FBSDER will give the viscosity solution of a quasilinear variational inequality (obstacle problem) with a Neumann boundary condition. As an application, we study how the solvability of FBSDERs is related to the solvability of an American game option.

Eigenvalues of the Dirac Operator on Manifolds¶with Boundary

Abstract: Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. For the local boundary conditions, limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary.

Multiple positive solutions to a third-order discrete focal boundary value problem

Abstract: We are concerned with the discrete focal boundary value problem Δ 3 x ( t − k ) = f ( x ( t )), x ( a ) = Δ x ( t 2 ) = Δ 2 x ( b + 1 = 0. Under various assumptions on f and the integers a , t 2 , and b we prove the existence of three positive solutions of this boundary value problem. To prove our results, we will apply a generalization of the Leggett-Williams fixed-point theorem.

Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem

Abstract: We consider a one-phase Stefan problem for the heat equation with a superlinear reaction term. It is known from a previous work (Ghidouche, Souplet, & Tarzia [5]) that all global solutions are bounded and decay uniformly to 0. Moreover, it was shown in Ghidouche, Souplet, & Tarzia [5] that either: (i) the free boundary converges to a finite limit and the solution decays at an exponential rate, or (ii) the free boundary grows up to infinity and the decay rate is at most polynomial, and it was also proved that small data solutions behave like (i). Here we prove that there exist global solutions with slow decay and unbounded free boundary, i.e. of type (ii). Also, we establish uniform a priori estimates for all global solutions. Moreover, we provide a correction to an error in the proof of decay from Ghidouche, Souplet, & Tarzia [5].

The Nishiura--Ohnishi Free Boundary Problem in the 1D Case

Abstract: A free boundary problem due to Nishiura and Ohnishi is solved in one space dimension. That problem was derived, during their study of phase separation phenomena in diblock copolymers, as an asymptotic limit of pattern-forming PDEs generalizing that of Cahn and Hilliard. The free boundary problem in one dimension reduces to a linear system of ODEs for the lengths of the intervals between interfaces. This system also arises in a completely different context as the spatial discretization of a simple heat equation in a medium with periodic properties. (The medium is homogeneous in an important special case.) The initial-value problem for this system is completely solved, and global stability results for stationary solutions (in which the interfaces are regularly spaced) are obtained. Nucleation phenomena are briefly discussed.

Stabilizability of a quasi-linear parabolic equation by means of a boundary control with feedback

Abstract: The problem of stabilizability from the boundary for a?parabolic equation given in a?bounded domain , consists in choosing a?boundary condition (a?control) such that the solution of the resulting mixed boundary-value problem tends as to a?given steady-state solution at a?prescribed rate . Furthermore, it is required that the control be with feedback, that is, that it react to unpredictable fluctuations of the system by suppressing the results of their action on the stabilizable solution. A?new mathematical formulation of the concept of feedback is presented and then used in solving the problem of stabilizability of linear as well as quasi-linear parabolic equations by means of a?control with feedback defined on part of the boundary.

Tailoring freeform lenses for illumination

Abstract: The general problem to find the shape of a refractive surface such as to produce a desired brightness distribution on a given target surface from a known point source leads to a boundary value problem with an elliptic partial differential equation of the Monge-Ampere type This equation has been described and analyzed in the literature The purpose of our contribution is to present a venue for a numerical solution as well as several solved examples The essence of our algorithm for a numerical solution appears to be the explicit incorporation of the condition for the existence of a pseudopotential for a normal vector field© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering Downloading of the abstract is permitted for personal use only