Showing papers on "Free boundary problem published in 2005"

A Geometric Approach to Free Boundary Problems

Abstract: Elliptic problems: An introductory problem Viscosity solutions and their asymptotic developments The regularity of the free boundary Lipschitz free boundaries are $C^{1,\gamma}$ Flat free boundaries are Lipschitz Existence theory Evolution problems: Parabolic free boundary problems Lipschitz free boundaries: Weak results Lipschitz free boundaries: Strong results Flat free boundaries are smooth Complementary chapters: Main tools: Boundary behavior of harmonic functions Monotonicity formulas and applications Boundary behavior of caloric functions Bibliography Index.

Well-posedness for the motion of an incompressible liquid with free surface boundary

Abstract: We study the motion of an incompressible perfect liquid body in vacuum This can be thought of as a model for the motion of the ocean or a star The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface This leads to a free boundary problem for Euler?s equations, where the regularity of the boundary enters to highest order We prove local existence in Sobolev spaces assuming a ?physical condition?, related to the fact that the pressure of a fluid has to be positive

On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation

Abstract: Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second-order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one-dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

Homogenization of Partial Differential Equations

Abstract: * Preface * Introduction * The Dirichlet Boundary Value Problem in Strongly Perforated Domains with Fine-Grained Boundary * The Dirichlet Boundary Value Problem in Strongly Perforated Domains with Complex Boundary * Strongly Connected Domains * The Neumann Boundary Value Problems in Strongly Perforated Domains * Nonstationary Problems and Spectral Problems * Differential Equations with Rapidly Oscillating Coefficients * Homogenized Conjugation Conditions * References * Index

On the American option problem

Abstract: We show how the change-of-variable formula with local time on curves derived recently in Peskir (2002) can be used to prove that the optimal stopping boundary for the American put option can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium representation. This settles the question raised in Myneni (1992) and dating back to McKean (1965).

Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector

Abstract: This paper discusses boundary conditions appropriate to a theory of single-crystal plasticity (Gurtin, J. Mech. Phys. Solids 50 (2002) 5) that includes an accounting for the Burgers vector through energetic and dissipative dependences on the tensor G=curlHp, with Hp the plastic part in the additive decomposition of the displacement gradient into elastic and plastic parts. This theory results in a flow rule in the form of N coupled second-order partial differential equations for the slip-rates γ˙α(α=1,2…,N), and, consequently, requires higher-order boundary conditions. Motivated by the virtual-power principle in which the external power contains a boundary-integral linear in the slip-rates, hard-slip conditions in which (A) γ˙α=0 on a subsurface Shard of the boundary for all slip systems α are proposed. In this paper we develop a theory that is consistent with that of (Gurtin, 2002), but that leads to an external power containing a boundary-integral linear in the tensor H˙ijpɛjrlnr, a result that motivates replacing (A) with the microhard condition (B) H˙ijpɛjrlnr=0 on the subsurface Shard. We show that, interestingly, (B) may be interpreted as the requirement that there be no flow of the Burgers vector across Shard. What is most important, we establish uniqueness for the underlying initial/boundary-value problem associated with (B); since the conditions (A) are generally stronger than the conditions (B), this result indicates lack of existence for problems based on (A). For that reason, the hard-slip conditions (A) would seem inappropriate as boundary conditions. Finally, we discuss conditions at a grain boundary based on the flow of the Burgers vector at and across the boundary surface.

Partial Differential Equations: With Fourier Series and Boundary Value Problems

Abstract: Course Description: This course is an introduction to partial differential equations (PDEs), their applications in the sciences and the techniques that have proved useful in analyzing them. The techniques include separation of variables, Fourier series and Fourier transforms, orthogonal functions and eigenfunction expansions, Bessel functions, and Legendre polynomials. The student will see how the sciences motivate the formulation of partial differential equations as well as the formulation of boundary conditions and initial conditions. Parabolic, hyperbolic, and elliptic PDEs will be studied.

Certified Real-Time Solution of Parametrized Partial Differential Equations

Abstract: Engineering analysis requires the prediction of (say, a single) selected “output” se relevant to ultimate component and system performance:* typical outputs include energies and forces, critical stresses or strains, flowrates or pressure drops, and various local and global measures of concentration, temperature, and flux. These outputs are functions of system parameters, or “inputs”, μ, that serve to identify a particular realization or configuration of the component or system: these inputs typically reflect geometry, properties, and boundary conditions and loads; we shall assume that μ is a P-vector (or P-tuple) of parameters in a prescribed closed input domain D ⊂ ℝp. The input-output relationship se(μ): D → ℝ thus encapsulates the behavior relevant to the desired engineering context.

Remarks on an elliptic equation of Kirchhoff type

Abstract: We present a short survey on the nonlocal elliptic boundary value problem - M ∫ Ω | ∇ u | 2 d x Δ u = f ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain of R N , M is a positive function, and f has subcritical growth.

Nonlinear integral equations and the iterative solution for an inverse boundary value problem

Abstract: Determining the shape of a perfectly conducting inclusion within a conducting medium from voltage and current measurements on the accessible boundary of the medium can be modelled as an inverse boundary value problem for harmonic functions. We present a novel solution method for such inverse boundary value problems via a pair of nonlinear and ill-posed integral equations for the unknown boundary that can be solved by linearization, i.e., by regularized Newton iterations. We present a mathematical foundation of the method and illustrate its feasibility by numerical examples.

Compressible Flow in a Half-Space with Navier Boundary Conditions

Abstract: We prove the global existence of weak solutions of the Navier–Stokes equations of compressible flow in a half-space with the boundary condition proposed by Navier: the velocity on the boundary is proportional to the tangential component of the stress. This boundary condition allows for the determination of the scalar function in the Helmholtz decomposition of the acceleration density, which in turn is crucial in obtaining pointwise bounds for the density. Initial data and solutions are small in energy-norm with nonnegative densities having arbitrarily large sup-norm. These results generalize previous results for solutions in the whole space and are the first for solutions in this intermediate regularity class in a region with a boundary.

Well-Posed Boundary Conditions for the Navier--Stokes Equations

Abstract: In this article we propose a general procedure that allows us to determine both the number and type of boundary conditions for time dependent partial differential equations. With those, well-posedness can be proven for a general initial-boundary value problem. The procedure is exemplified on the linearized Navier--Stokes equations in two and three space dimensions on a general domain.

Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations

Abstract: Perturbation of the boundary is a rather neglected topic in the study of PDEs for two main reasons. First, on the surface it appears trivial, merely a change of variables and an application of the chain rule. Second, carrying out such a change of variables frequently results in long and difficult calculations. In this book, first published in 2005, the author carefully discusses a calculus that allows the computational morass to be bypassed, and he goes on to develop more general forms of standard theorems, which help answer a wide range of problems involving boundary perturbations. Many examples are presented to demonstrate the usefulness of the author's approach, while on the other hand many tantalizing open questions remain. Anyone whose research involves PDEs will find something of interest in this book.

Well Posedness for the Motion of a Compressible Liquid with Free Surface Boundary

Abstract: We study the motion of a compressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a ``physical condition'', related to the fact that the pressure of a fluid has to be positive.

Boundary element-free method for elastodynamics

Abstract: The moving least-square approximation is discussed first. Sometimes the method can form an ill-conditioned equation system, and thus the solution cannot be obtained correctly. A Hilbert space is presented on which an orthogonal function system mixed a weight function is defined. Next the improved moving least-square approximation is discussed in detail. The improved method has higher computational efficiency and precision than the old method, and cannot form an ill-conditioned equation system. A boundary element-free method (BEFM) for elastodynamics problems is presented by combining the boundary integral equation method for elastodynamics and the improved moving least-square approximation. The boundary element-free method is a meshless method of boundary integral equation and is a direct numerical method compared with others, in which the basic unknowns are the real solutions of the nodal variables and the boundary conditions can be applied easily. The boundary element-free method has a higher computational efficiency and precision. In addition, the numerical procedure of the boundary element-free method for elastodynamics problems is presented in this paper. Finally, some numerical examples are given.

The Russian option: Finite horizon

Abstract: We show that the optimal stopping boundary for the Russian option with finite horizon can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium representation (an explicit formula for the arbitrage-free price in terms of the optimal stopping boundary having a clear economic interpretation). The results obtained stand in a complete parallel with the best known results on the American put option with finite horizon. The key argument in the proof relies upon a local time-space formula.

What is a Quadrature Domain

Abstract: We give an overview of the theory of quadrature domains with indications of some if its ramifications.

The initial-boundary-value problem for the 1D nonlinear Schrödinger equation on the half-line

Abstract: We prove, by adapting the method of Colliander-Kenig [9], local well posedness of the initial-boundary-value problem for the one-dimensional nonlinear Schrodinger equation $i\partial_tu +\partial_x^2u +\lambda u|u|^{\alpha-1}=0$ on the half-line under low boundary regularity assumptions

Boundary value problems for Dirac-type equations

Abstract: Abstract We prove regularity for a class of boundary value problems for first order elliptic systems, with boundary conditions determined by spectral decompositions, under coefficient differentiability conditions weaker than previously known. We establish Fredholm properties for Dirac-type equations with these boundary conditions. Our results include sharp solvability criteria, over both compact and non-compact manifolds; weighted Poincaré and Schrödinger-Lichnerowicz inequalities provide asymptotic control in the noncompact case.

Transonic shock in a nozzle I: 2D case

Abstract: In this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two-dimensional nozzle with variable sections. The flow is governed by the inviscid potential equation and is supersonic upstream, has no-flow boundary conditions on the nozzle walls, and an appropriate boundary condition at the exit of the exhaust section. The transonic shock is a free boundary dividing two regions of C 1,1-δ 0 flow in the nozzle. The potential equation is hyperbolic upstream where the flow is supersonic, and elliptic in the downstream subsonic region. In particular, our results show that there exists a solution to the corresponding free boundary problem such that the equation is always subsonic in the downstream region of the nozzle when the pressure in the exit of the exhaustion section is appropriately larger than that in the entry. This problem is motivated by the conjecture of Courant and Friedrichs on the transonic phenomena in a nozzle [10]. Furthermore, the stability of the transonic shock is also proven when the upstream supersonic flow is a small steady perturbation for the uniform supersonic flow and the corresponding pressure at the exit has a small perturbation. The main ingredients of our analysis are a generalized hodograph transformation and multiplier methods for elliptic equation with mixed boundary conditions and comer singularities.

On Cauchy's problem: I. A variational Steklov–Poincaré theory

Abstract: In 1923 (Lectures on Cauchy's Problem in Linear PDEs (New York, 1953)), J Hadamard considered a particular example to illustrate the ill-posedness of the Cauchy problem for elliptic partial differential equations, which consists in recovering data on the whole boundary of the domain from partial but over-determined measures. He achieved explicit computations for the Laplace operator, due to the squared shape of the domain, to observe, in fine, that the solution does not depend continuously on the given boundary data. The primary subject of this contribution is to extend the result to general domains by proving that the Cauchy problem has a variational formulation that can be put under a (variational) pseudo-differential equation, set on the boundary where the data are missing, and defined by a compact Steklov?Poincar?-type operator. The construction of this operator is based on the Dirichlet-to-Neumann mapping, and its compactness is derived from the elliptic regularity theory. Next, using mathematical tools from the linear operator theory and the convex optimization, we provide a comprehensive analysis of the reduced problem which enables us to state that (i) the set of compatible data, for which existence and uniqueness are guaranteed, is dense in the admissible data space; (ii) when the existence fails, due to possible noisy data, the variational problem can be consistently approximated by the least-squares method, that is the incompatibility measure (the deviation indicator or the variational crime made on the Steklov?Poincar? equation) equals zero though all the minimizing sequences blow up.

A transform method for linear evolution PDEs on a finite interval

Abstract: We study initial boundary value problems for linear scalar evolution partial differential equations, with spatial derivatives of arbitrary order, posed on the domain {t > 0, 0 < x < L). We show that the solution can be expressed as an integral in the complex k-plane. This integral is defined in terms of an x-transform of the initial condition and a t-transform of the boundary conditions. The derivation of this integral representation relies on the analysis of the global relation, which is an algebraic relation defined in the complex k-plane coupling all boundary values of the solution. For particular cases, such as the case of periodic boundary conditions, or the case of boundary value problems for even-order PDEs, it is possible to obtain directly from the global relation an alternative representation for the solution, in the form of an infinite series. We stress, however, that there exist initial boundary value problems for which the only representation is an integral which cannot be written as an infinite series. An example of such a problem is provided by the linearized version of the KdV equation. Similarly, in general the solution of odd-order linear initial boundary value problems on a finite interval cannot be expressed in terms of an infinite series.

A new formulation of Signorini's type for seepage problems with free surfaces

Abstract: A new variational inequality formulation for seepage problems with free surfaces is presented, in which a boundary condition of Signorini's type is prescribed over the potential seepage surfaces. This makes the singularity of seepage points eliminated and the location of seepage points determined easily. Compared to other variational formulations, the proposed formulation can effectively overcome the mesh dependency and significantly improve the numerical stability. A very challenging engineering example with complicated geometry aid strong inhomogeneity is investigated in detail.

Modelling of topological derivatives for contact problems

Abstract: The problem of topology optimization is considered for free boundary problems of thin obstacle types. The formulae for the first term of asymptotics for energy functionals are derived. The precision of obtained terms is verified numerically. The topological differentiability of solutions to variational inequalities is established. In particular, the so-called outer asymptotic expansion for solutions of contact problems in elasticity with respect to singular perturbation of geometrical domain depending on small parameter are derived by an application of nonsmooth analysis. Such results lead to the topological derivatives of shape functionals for contact problems. The topological derivatives are used in numerical methods of simultaneous shape and topology optimization.

Analysis of a Free Boundary Problem Modeling Tumor Growth

Abstract: In this paper, we study a free boundary problem arising from the modeling of tumor growth. The problem comprises two unknown functions: R = R(t), the radius of the tumor, and u = u(r, t), the concentration of nutrient in the tumor. The function u satisfies a nonlinear reaction diffusion equation in the region 0 0, and the function R satisfies a nonlinear integrodi fferential equation containing u. Under some general conditions, we establish global existence of transient solutions, unique existence of a stationary solution, and convergence of transient solutions toward the stationary solution as t → ∞.