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Showing papers on "Free boundary problem published in 2014"


Journal ArticleDOI
TL;DR: A numerical method for the solution of the elliptic Monge-Ampere Partial Differential Equation, with boundary conditions corresponding to the Optimal Transportation problem, is presented, leading to a fast solver comparable to solving the Laplace equation on the same grid several times.

211 citations


Journal ArticleDOI
TL;DR: The proposed Laplacian boundary value (LBV) method for background field removal retains data near the boundary and is computationally efficient and more accurate than two existing methods.
Abstract: The removal of the background magnetic field is a critical step in generating phase images and quantitative susceptibility maps, which have recently been receiving increasing attention. Although it is known that the background field satisfies Laplace's equation, the boundary values of the background field for the region of interest have not been explicitly addressed in the existing methods, and they are not directly available from MRI measurements. In this paper, we assume simple boundary conditions and remove the background field by explicitly solving the boundary value problems of Laplace's or Poisson's equation. The proposed Laplacian boundary value (LBV) method for background field removal retains data near the boundary and is computationally efficient. Tests on a numerical phantom and an experimental phantom showed that LBV was more accurate than two existing methods.

205 citations


Journal ArticleDOI
TL;DR: Recent developments in the non-standard asymptotics of the narrow escape problem are reviewed, which are based on several ingredients: a better resolution of the singularity of Neumann's function,resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks, and the breakup of composite domains into simpler components.
Abstract: The narrow escape problem in diffusion theory is to calculate the mean first passage time of a diffusion process to a small target on the reflecting boundary of a bounded domain. The problem is equivalent to solving the mixed Dirichlet--Neumann boundary value problem for the Poisson equation with small Dirichlet and large Neumann parts. The mixed boundary value problem, which goes back to Lord Rayleigh, originates in the theory of sound and is closely connected to the eigenvalue problem for the mixed problem and for the Neumann problem in domains with bottlenecks. We review here recent developments in the non-standard asymptotics of the problem, which are based on several ingredients: a better resolution of the singularity of Neumann's function, resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks, and the breakup of composite domains into simpler components. The new methodology applies to two- and higher-dimensional problems. Selected applications are r...

175 citations


Journal ArticleDOI
TL;DR: An exhaustive study of the sign of the related Green's function is made and the exact values for which it is positive on the whole square of definition are obtained.

112 citations


Journal ArticleDOI
TL;DR: In this paper, a free boundary problem for a predator-predator model in higher space dimensions and a heterogeneous environment is studied, where the free boundary represents the spreading front of the predator species and is described by Stefan-like condition.
Abstract: This paper is concerned with a free boundary problem for a prey–predator model in higher space dimensions and heterogeneous environment. Such a model may be used to describe the spreading of an invasive or new predator species in which the free boundary represents the spreading front of the predator species and is described by Stefan-like condition. For simplicity, we assume that the environment and solutions are radially symmetric. We prove a spreading–vanishing dichotomy for this model, namely the predator species either successfully spreads to infinity as t → ∞ and survives in the new environment, or it fails to establish and dies out in the long run while the prey species stabilizes at a positive equilibrium state. The criteria for spreading and vanishing are given.

105 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied the Rayleigh-Taylor instability for two incompressible, immiscible, viscous magnetohydrodynamic flows with zero resistivity and surface tension evolving with a free interface under presence of a uniform gravitational field.
Abstract: We study the Rayleigh-Taylor instability problem for two incompressible, immiscible, viscous magnetohydrodynamic (MHD) flows with zero resistivity and surface tension (or without surface tension), evolving with a free interface under presence of a uniform gravitational field. First, we reformulate the MHD free boundary problem in an infinite slab as a Navier-Stokes system in Lagrangian coordinates with a force term induced by the fluid flow map. Then, we analyze the linearized problem around the steady state which describes a denser immiscible fluid lying above a light one with a free interface separating the two fluids, and both fluids being in (unstable) equilibrium. By studying a family of modified variational problems, we construct smooth (when restricted to each fluid domain) solutions to the linearized problem that grow exponentially fast in time in Sobolev spaces, thus leading to an global instability result for the linearized problem. Finally, using these pathological solutions, we prove the globa...

71 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a general rigid bottom in a three-dimensional horizontally periodic setting.
Abstract: We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a general rigid bottom in a three-dimensional horizontally periodic setting. We establish the global well-posedness of the problem both with and without surface tension. We prove that without surface tension the solution decays to the equilibrium state at an almost exponential rate; with surface tension, we show that the solution decays at an exponential rate. Our results include the case in which a heavier fluid lies above a lighter one, provided that the surface tension at the free internal interface is above a critical value, which we identify. This means that sufficiently large surface tension stabilizes the Rayleigh–Taylor instability in the nonlinear setting. As a part of our analysis, we establish elliptic estimates for the two-phase stationary Stokes problem.

68 citations


Journal ArticleDOI
TL;DR: In this article, an existence result for variational boundary value problems for quasilinear elliptic equations in the Musielak-Orlicz spaces has been proved.
Abstract: In this paper we prove an existence result for some class of variational boundary value problems for quasilinear elliptic equations in the Musielak-Orlicz spaces. Some results concerning the Trace mapping have also been provided, as well as existence results for some strongly nonlinear elliptic equations in Musielak-Orlicz spaces.

67 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered a nonlinear fractional boundary value problem defined on a star graph and used a transformation to obtain an equivalent system of boundary value problems with mixed boundary conditions, then the existence and uniqueness of solutions are investigated by fixed point theory.
Abstract: In this paper, the authors consider a nonlinear fractional boundary value problem defined on a star graph. By using a transformation, an equivalent system of fractional boundary value problems with mixed boundary conditions is obtained. Then the existence and uniqueness of solutions are investigated by fixed point theory.

56 citations


Journal ArticleDOI
TL;DR: Christodoulou and Lindblad as mentioned in this paper proved the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid magnetohydrodynamics equations in all physical spatial dimensions n = 2 and 3.
Abstract: In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid magnetohydrodynamics equations in all physical spatial dimensions n = 2 and 3 by adopting a geometrical point of view used in Christodoulou and Lindblad (Commun Pure Appl Math 53:1536‐1602, 2000), and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition thattheouternormalderivativeofthetotalpressureincludingthefluidandmagnetic pressuresisnegativeonthefreeboundary,whichissimilartothephysicalcondition (Taylor sign condition) for the incompressible Euler equations of fluids.

55 citations


Journal ArticleDOI
TL;DR: This work investigates the existence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with coupled integral boundary conditions.

Journal ArticleDOI
TL;DR: In this paper, the free boundary problem for the plasma vacuum interface model in ideal incompressible magneto-hydrodynamics is considered and a suitable a priori estimate, uniform with respect to the small parameter of the regularization, is derived in the anisotropic Sobolev space.
Abstract: We consider the free boundary problem for the plasma vacuum interface model in ideal incompressible magneto-hydrodynamics. Under a suitable stability condition on the initial discontinuity, the well-posedness of the linearized problem, around a non constant basic state sufficiently smooth, is investigated. Since the latter amounts to be a non standard initial-boundary value problem of mixed hyperbolic-elliptic type, for its resolution we introduce a fully ”hyperbolic” regularized problem. For the regularized problem, a suitable a priori estimate, uniform with respect to the small parameter of the regularization, is derived in the anisotropic Sobolev space H 1 .

Journal ArticleDOI
TL;DR: A recently derived numerical algorithm for one-dimensional one-phase Stefan problems is extended for the purpose of two-phase moving boundary problems in which the second phase first appears only after a finite delay time; this can occur if the phase change is caused by a heat-flux boundary condition.

Journal ArticleDOI
TL;DR: In this article, a Hele-Shaw type free boundary problem for a tumor growing under the combined effects of pressure forces, cell multiplication and active motion is formulated, which is considered as a standard diffusion process, and the free boundary model is derived from a description at the cell level using the asymptotic of a stiff pressure limit.
Abstract: We formulate a Hele-Shaw type free boundary problem for a tumor growing under the combined effects of pressure forces, cell multiplication and active motion, the latter being the novelty of the present paper. This new ingredient is considered here as a standard diffusion process. The free boundary model is derived from a description at the cell level using the asymptotic of a stiff pressure limit. Compared to the case when active motion is neglected, the pressure satisfies the same complementarity Hele-Shaw type formula. However, the cell density is smoother (Lipschitz continuous), while there is a deep change in the free boundary velocity, which is no longer given by the gradient of the pressure, because some kind of ‘mushy region’ prepares the tumor invasion.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the potential theoretic aspects of the normalized Laplacian evolution and showed that the Dirichlet problem can be solved in cylinders whose section is regular for the p-Laplacians.
Abstract: In this paper, we study the potential theoretic aspects of the normalized $p$-Laplacian evolution, see (1.1) below. A systematic study of such equation was recently started in [1], [4] and [25]. Via the classical Perron approach, we address the question of solvability of the Cauchy-Dirichlet problem with "very weak" assumptions on the boundary of the domain. The regular boundary points for the Dirichlet problem are characterized in terms of barriers. For $p \geq 2 $, in the case of space - time cylinder $G \times (0,T)$, we show that $(x,t) \in \partial G \times (0, T]$ is a regular boundary point if and only if $x \in \partial G$ is a a regular boundary point for the p-Laplacian. This latter operator is the steady state corresponding to the evolution (1.1) below. Consequently, when $p\geq 2$ the Cauchy- Dirichlet problem for (1.1) can be solved in cylinders whose section is regular for the $p$-Laplacian. This can be thought of as an analogue of the results obtained in [17] for the standard parabolic $p$-Laplacian div$(|Du|^{p-2}Du) - u_t = 0 $.

Journal ArticleDOI
TL;DR: In this article, the authors re-examine the boundary-valued problem of wave scattering and diffraction in elastic half-space from an applied mathematics points of view and redefine the proper form of the orthogonal cylindrical-wave functions for both the longitudinal P- and shear SV-waves so that they can together simultaneously satisfy the zero-stress boundary conditions at the halfspace surface.

Journal ArticleDOI
TL;DR: In this paper, a geometric control of the position of a liquid-solid interface in a melting process of a material known as Stefan problem is addressed, and a control law is designed using the concept of characteristic index, from geometric control theory, directly issued from the hybrid model without any reduction of the partial differential equation.

Journal ArticleDOI
TL;DR: In this article, it was shown that for any continuous Riemannian manifold f: (\Sigma, \partial \Sigma) \rightarrow (N, M)$ for which the induced homomorphism on certain fundamental groups is injective, there exists a branched minimal immersion of f solving the free boundary problem.
Abstract: Let N be a complete, homogeneously regular Riemannian manifold of dimension greater than 2 and let M be a compact submanifold of N. Let $\Sigma$ be a compact orientable surface with boundary. We show that for any continuous $f: (\Sigma, \partial \Sigma) \rightarrow (N, M)$ for which the induced homomorphism on certain fundamental groups is injective, there exists a branched minimal immersion of $\Sigma$ solving the free boundary problem $(\Sigma, \partial \Sigma) \rightarrow (N, M)$, and minimizing area among all maps which induce the same action on the fundamental groups as f. Furthermore, under certain nonnegativity assumptions on the curvature of a 3-manifold N and convexity assumptions on M which is the boundary of N, we derive bounds on the genus, number of boundary components and area of any compact two-sided minimal surface solving the free boundary problem with low index.

29 Sep 2014
TL;DR: This paper investigates optimal boundary control problems for Cahn-Hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator and proves existence of optimal controls and derive first-order necessary conditions of optimality.
Abstract: In this paper, we investigate optimal boundary control problems for Cahn-Hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy, which follows the lines of the recent approach by Colli, Farshbaf-Shaker, Sprekels (see the preprint arXiv:1308.5617) to the (simpler) Allen-Cahn case, is the following: we use the results that were recently established by Colli, Gilardi, Sprekels in the preprint arXiv:1407.3916 [math.AP] for the case of (differentiable) logarithmic potentials and perform a so-called "deep quench limit". Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (non-differentiable) double obstacle potentials.

Journal ArticleDOI
TL;DR: In this article, the authors give a general monotonicity formula for local minimizers of free discontinuity problems which have a critical deviation from minimality, of order d − 1.
Abstract: We give a general monotonicity formula for local minimizers of free discontinuity problems which have a critical deviation from minimality, of order d − 1. This result allows us to prove partial regularity results (that is closure and density estimates for the jump set) for a large class of free discontinuity problems involving general energies associated to the jump set, as for example free boundary problems with Robin conditions. In particular, we give a short proof to the De Giorgi–Carriero–Leaci result for the Mumford–Shah functional.

Journal ArticleDOI
TL;DR: In this paper, the evolution problem for a membrane-based model of an electrostatically actuated microelectromechanical system is studied, where the model describes the dynamics of the membrane displacement and the electric potential.
Abstract: The evolution problem for a membrane based model of an electrostatically actuated microelectromechanical system is studied. The model describes the dynamics of the membrane displacement and the electric potential. The latter is a harmonic function in an angular domain, the deformable membrane being a part of the boundary. The former solves a heat equation with a right-hand side that depends on the square of the trace of the gradient of the electric potential on the membrane. The resulting free boundary problem is shown to be well-posed locally in time. Furthermore, solutions corresponding to small voltage values exist globally in time, while global existence is shown not to hold for high voltage values. It is also proven that, for small voltage values, there is an asymptotically stable steady-state solution. Finally, the small aspect ratio limit is rigorously justified.

Journal ArticleDOI
TL;DR: In this paper, a combined approach of boundary element method and precise integration method is presented for solving transient heat conduction problems with heat sources, where two domain integrals are involved in the derived integral equations.

Journal ArticleDOI
TL;DR: In this paper, the authors show that the boundary conditions for AdS3 are dual to two-dimensional quantum gravity in certain fixed gauges, and that an appropriate identification of the generator of Virasoro transformations leads to a vanishing total central charge in agreement with the theory at the boundary.
Abstract: We show that recently proposed free boundary conditions for AdS3 are dual to two-dimensional quantum gravity in certain fixed gauges. In particular, we note that an appropriate identification of the generator of Virasoro transformations leads to a vanishing total central charge in agreement with the theory at the boundary. We argue that this identification is necessary to match the bulk and boundary generators of Virasoro transformations and for consistency with the constraint equations.

Journal ArticleDOI
TL;DR: In this paper, the existence and multiplicity of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations, subject to integral boundary conditions, were investigated.
Abstract: We study the existence and multiplicity of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations, subject to integral boundary conditions. The nonsingular and singular cases for the nonlinearities are investigated.

Journal ArticleDOI
TL;DR: In this paper, the existence and uniqueness of solutions for a fractional differential equation of order with slit-strips type boundary conditions were discussed and the desired results were obtained by applying standard tools of the fixed point theory and are well illustrated with the aid of examples.
Abstract: We discuss the existence and uniqueness of solutions for a fractional differential equation of order with slit-strips type boundary conditions. The slit-strips type boundary condition states that the sum of the influences due to finite strips of arbitrary lengths is related to the value of the unknown function at an arbitrary position (nonlocal point) in the slit (a part of the boundary off the two strips). The desired results are obtained by applying standard tools of the fixed point theory and are well illustrated with the aid of examples. We also extend our discussion to the cases of arbitrary number of nonlocal points in the slit, the nonlocal multi-substrips conditions and Riemann-Liouville type slit-strips boundary conditions. MSC: 34A12, 34A40.

Journal ArticleDOI
TL;DR: In this paper, a wave equation with a viscoelastic boundary is considered, and the decay rate of the wave equation is investigated, and an explicit and general decay rate result that allows a larger class of relaxation functions is established.
Abstract: In this paper we consider a wave equation with a viscoelastic boundary damping localized on a part of the boundary. We establish an explicit and general decay rate result that allows a larger class of relaxation functions and generalizes previous results existing in the literature.

Journal ArticleDOI
TL;DR: The work of Park and Park (2011) is generalized to an arbitrary rate of decay with not necessarily of an exponential or polynomial one and without the assumption condition of the relaxation function due to Tatar.

Journal ArticleDOI
TL;DR: In this paper, the existence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with sign-changing nonlinearities with integral boundary conditions was studied.
Abstract: We study the existence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with sign-changing nonlinearities, subject to integral boundary conditions. MSC:34A08, 45G15.

Posted Content
TL;DR: In this paper, it was shown that stable cones for the one-phase free boundary problem are hyperplanes in dimension 4, and that both one and two-phase energy minimizing hypersurfaces are smooth in dimension 2.
Abstract: We show that stable cones for the one-phase free boundary problem are hyperplanes in dimension $4$. As a corollary, both one and two-phase energy minimizing hypersurfaces are smooth in dimension $4$.

Journal ArticleDOI
TL;DR: In this article, a non-reflecting boundary condition (NRBC) is proposed for 2D wave equation problems with far field sources, which is based on a known space-time integral equation defining a relationship between the solution of the differential problem and its normal derivative.