Showing papers on "Neumann boundary condition published in 2017"

Nonlocal problems with Neumann boundary conditions

Abstract: We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model. We can consider both elliptic and parabolic equations in any domain. In addition, we formulate problems with nonhomogeneous Neumann conditions, and also with mixed Dirichlet and Neumann conditions, all of them having a clear probabilistic interpretation. We prove that solutions to the fractional heat equation with homogeneous Neumann conditions have the following natural properties: conservation of mass inside Ω, decreasing energy, and convergence to a constant as t→∞. Moreover, for the elliptic case we give the variational formulation of the problem, and establish existence of solutions. We also study the limit properties and the boundary behavior induced by this nonlocal Neumann condition. For concreteness, one may think that our nonlocal analogue of the classical Neumann condition ∂νu=0 on~∂Ω consists in the nonlocal prescription ∫Ωu(x)−u(y)|x−y|n+2sdy=0for x∈Rn∖Ω. We made an effort to keep all the arguments at the simplest possible technical level, in order to clarify the connections between the different scientific fields that are naturally involved in the problem, and make the paper accessible also to a wide, non-specialistic public (for this scope, we also tried to use and compare different concepts and notations in a somehow more unified way).

Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions

Abstract: Latterly, many problems arising in different fields of science and engineering can be reduced, by applying some appropriate discretization, to a series of time-fractional partial differential equations. Unlike the normal case derivative, the differential order in such equations is with a fractional order, which will lead to new challenges for numerical simulation. The purpose of this analysis is to introduce the reproducing kernel Hilbert space method for treating classes of time-fractional partial differential equations subject to Neumann boundary conditions with parameters derivative arising in fluid-mechanics, chemical reactions, elasticity, anomalous diffusion, and population growth models. The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. Numerical experiments with different order derivatives degree are performed to support the theoretical analyses which are acquired by interrupting the n-term of the exact solutions. Finally, the obtained outcomes showed that the proposed method is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional Neumann problems.

Stability, Steady‐State Bifurcations, and Turing Patterns in a Predator–Prey Model with Herd Behavior and Prey‐taxis

Abstract: In this paper, we consider a predator–prey model with herd behavior and prey-taxis subject to the homogeneous Neumann boundary condition. First, by analyzing the characteristic equation, the local stability of the positive equilibrium is discussed. Then, choosing prey-tactic sensitivity coefficient as the bifurcation parameter, we obtain a branch of nonconstant solutions bifurcating from the positive equilibrium by an abstract bifurcation theory, and find the stable bifurcating solutions near the bifurcation point under suitable conditions. We have shown that prey-taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns. Furthermore, some numerical simulations to illustrate the theoretical analysis are also carried out, Turing patterns such as spots pattern, spots–strip pattern, strip pattern, stable nonconstant steady-state solutions, and spatially inhomogeneous periodic solutions are obtained, which also expand our theoretical results.

On filter boundary conditions in topology optimization

Abstract: Most research papers on topology optimization involve filters for regularization Typically, boundary effects from the filters are ignored Despite significant drawbacks the inappropriate homogeneous Neumann boundary conditions are used, probably because they are trivial to implement In this paper we define three requirements that boundary conditions must fulfill in order to eliminate boundary effects Previously suggested approaches are briefly reviewed in the light of these requirements A new approach referred to as the “domain extension approach” is suggested It effectively eliminates boundary effects and results in well performing designs The approach is intuitive, simple and easy to implement

Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis

Abstract: We consider a diffuse interface model for tumor growth consisting of a Cahn--Hilliard equation with source terms coupled to a reaction-diffusion equation, which models a tumor growing in the presence of a nutrient species and surrounded by healthy tissue. The well-posedness of the system equipped with Neumann boundary conditions was found to require regular potentials with quadratic growth. In this work, Dirichlet boundary conditions are considered, and we establish the well-posedness of the system for regular potentials with higher polynomial growth and also for singular potentials. New difficulties are encountered due to the higher polynomial growth, but for regular potentials, we retain the continuous dependence on initial and boundary data for the chemical potential and for the order parameter in strong norms as established in the previous work. Furthermore, we deduce the well-posedness of a variant of the model with quasi-static nutrient by rigorously passing to the limit where the ratio of the nutrient diffusion time-scale to the tumor doubling time-scale is small.

Exclusion Process with Slow Boundary

Abstract: We study the hydrodynamic and the hydrostatic behavior of the simple symmetric exclusion process with slow boundary. The term slow boundary means that particles can be born or die at the boundary sites, at a rate proportional to $$N^{-\theta }$$ , where $$\theta > 0$$ and N is the scaling parameter. In the bulk, the particles exchange rate is equal to 1. In the hydrostatic scenario, we obtain three different linear profiles, depending on the value of the parameter $$\theta $$ ; in the hydrodynamic scenario, we obtain that the time evolution of the spatial density of particles, in the diffusive scaling, is given by the weak solution of the heat equation, with boundary conditions that depend on $$ \theta $$ . If $$\theta \in (0,1)$$ , we get Dirichlet boundary conditions, (which is the same behavior if $$\theta =0$$ , see Farfan in Hydrostatics, statical and dynamical large deviations of boundary driven gradient symmetric exclusion processes, 2008); if $$\theta =1$$ , we get Robin boundary conditions; and, if $$\theta \in (1,\infty )$$ , we get Neumann boundary conditions.

A Neumann boundary term for gravity

Abstract: The Gibbons-Hawking-York (GHY) boundary term makes the Dirichlet problem for gravity well-defined, but no such general term seems to be known for Neumann boundary conditions. In this paper, we view Neumann not as fixing the normal derivative of the metric (''velocity'') at the boundary, but as fixing the functional derivative of the action with respect to the boundary metric (''momentum''). This leads directly to a new boundary term for gravity: the trace of the extrinsic curvature with a specific dimension-dependent coefficient. In three dimensions, this boundary term reduces to a ``one-half'' GHY term noted in the literature previously, and we observe that our action translates precisely to the Chern-Simons action with no extra boundary terms. In four dimensions, the boundary term vanishes, giving a natural Neumann interpretation to the standard Einstein-Hilbert action without boundary terms. We argue that in light of AdS/CFT, ours is a natural approach for defining a ``microcanonical'' path integral for gravity in the spirit of the (pre-AdS/CFT) work of Brown and York.

Lyapunov functions and global stability for a spatially diffusive SIR epidemic model

Abstract: This paper deals with the problem of global asymptotic stability for equilibria of a spatially diffusive SIR epidemic model with homogeneous Neumann boundary condition. By discretizing the model with respect to the space variable, we first construct Lyapunov functions for the corresponding ODEs model, and then broaden the construction method into the PDEs model in which either susceptible or infective populations are diffusive. In both cases, we obtain the standard threshold dynamical behaviors, that is, if , then the disease-free equilibrium is globally asymptotically stable and if , then the (strictly positive) endemic equilibrium is so. Numerical examples are given to illustrate the effectiveness of the theoretical results.

Iterative methods for solving coefficient inverse problems of wave tomography in models with attenuation

Abstract: We develop efficient iterative methods for solving inverse problems of wave tomography in models incorporating both diffraction effects and attenuation. In the inverse problem the aim is to reconstruct the velocity structure and the function that characterizes the distribution of attenuation properties in the object studied. We prove mathematically and rigorously the differentiability of the residual functional in normed spaces, and derive the corresponding formula for the Frechet derivative. The computation of the Frechet derivative includes solving both the direct problem with the Neumann boundary condition and the reversed-time conjugate problem. We develop efficient methods for numerical computations where the approximate solution is found using the detector measurements of the wave field and its normal derivative. The wave field derivative values at detector locations are found by solving the exterior boundary value problem with the Dirichlet boundary conditions. We illustrate the efficiency of this approach by applying it to model problems. The algorithms developed are highly parallelizable and designed to be run on supercomputers. Among the most promising medical applications of our results is the development of ultrasonic tomographs for differential diagnosis of breast cancer.

The KPZ Limit of ASEP with Boundary

Abstract: It was recently proved in [Corwin-Shen, 2016] that under weak asymmetry scaling, the height functions for open ASEP on the half-line and on a bounded interval converge to the Hopf-Cole solution of the KPZ equation with Neumann boundary conditions. In their assumptions [Corwin-Shen, 2016] chose positive values for the Neumann boundary conditions, and they assumed initial data which is close to stationarity. By developing more extensive heat-kernel estimates, we extend their results to negative values of the Neumann boundary parameters, and we also show how to generalize their results to narrow-wedge initial data (which is very far from stationarity). As a corollary via [Barraquand-Borodin-Corwin-Wheeler, 2017], we obtain the Laplace transform of the one-point distribution for half-line KPZ, and use this to prove $t^{1/3}$-scale GOE Tracy-Widom long-time fluctuations.

Tweaking one-loop determinants in AdS3

Abstract: We revisit the subject of one-loop determinants in AdS3 gravity via the quasi-normal mode method. Our goal is to evaluate a one-loop determinant with chiral boundary conditions for the metric field; chirality is achieved by imposing Dirichlet boundary conditions on certain components while others satisfy Neumann. Along the way, we give a generalization of the quasinormal mode method for stationary (non-static) thermal backgrounds, and propose a treatment for Neumann boundary conditions in this framework. We evaluate the graviton one-loop determinant on the Euclidean BTZ background with parity-violating boundary conditions (CSS), and find excellent agreement with the dual warped CFT. We also discuss a more general falloff in AdS3 that is related to two dimensional quantum gravity in lightcone gauge. The behavior of the ghost fields under both sets of boundary conditions is novel and we discuss potential interpretations.

Sharp geometric condition for null-controllability of the heat equation on $\mathbb{R}^d$ and consistent estimates on the control cost

Abstract: In this note we study the control problem for the heat equation on $\mathbb{R}^d$, $d\geq 1$, with control set $\omega\subset\mathbb{R}^d$. We provide a necessary and sufficient condition (called $(\gamma, a)$-\emph{thickness}) on $\omega$ such that the heat equation is null-controllable in any positive time. We give an estimate of the control cost with explicit dependency on the characteristic geometric parameters of the control set. Finally, we derive a control cost estimate for the heat equation on cubes with periodic, Dirichlet, or Neumann boundary conditions, where the control sets are again assumed to be thick. We show that the control cost estimate is consistent with the $\mathbb{R}^d$ case.

Fractional Operators with Inhomogeneous Boundary Conditions: Analysis, Control, and Discretization

Abstract: In this paper we introduce new characterizations of spectral fractional Laplacian to incorporate nonhomogeneous Dirichlet and Neumann boundary conditions. The classical cases with homogeneous boundary conditions arise as a special case. We apply our definition to fractional elliptic equations of order $s \in (0,1)$ with nonzero Dirichlet and Neumann boundary condition. Here the domain $\Omega$ is assumed to be a bounded, quasi-convex Lipschitz domain. To impose the nonzero boundary conditions, we construct fractional harmonic extensions of the boundary data. It is shown that solving for the fractional harmonic extension is equivalent to solving for the standard harmonic extension in the very-weak form. The latter result is of independent interest as well. The remaining fractional elliptic problem (with homogeneous boundary data) can be realized using the existing techniques. We introduce finite element discretizations and derive discretization error estimates in natural norms, which are confirmed by numerical experiments. We also apply our characterizations to Dirichlet and Neumann boundary optimal control problems with fractional elliptic equation as constraints.

Boundary Controllability of the Korteweg--de Vries Equation on a Bounded Domain

Abstract: This paper studies boundary controllability of the Korteweg--de Vries equation posed on a finite interval, in which, because of the third-order character of the equation, three boundary conditions are required to secure the well-posedness of the system. We consider the cases where one, two, or all three of those boundary data are employed as boundary control inputs. The system is first linearized around the origin and the corresponding linear system is proved to be exactly boundary controllable if using two or three boundary control inputs. In the case where only one control input is allowed to be used, the linearized system is known to be only null controllable if the single control input acts on the left end of the spatial domain. By contrast, if the single control input acts on the right end of the spatial domain, the linearized system is shown to be exactly controllable if and only if the length of the spatial domain does not belong to a set of critical values. Moreover, the nonlinear system is shown ...

Tweaking one-loop determinants in AdS$_3$

Abstract: We revisit the subject of one-loop determinants in AdS$_3$ gravity via the quasinormal mode method. Our goal is to evaluate a one-loop determinant with chiral boundary conditions for the metric field; chirality is achieved by imposing Dirichlet boundary conditions on certain components while others satisfy Neumann. Along the way, we give a generalization of the quasinormal mode method for stationary (non-static) thermal backgrounds, and propose a treatment for Neumann boundary conditions in this framework. We evaluate the graviton one-loop determinant on the Euclidean BTZ background with parity-violating boundary conditions (CSS), and find excellent agreement with the dual warped CFT. We also discuss a more general falloff in AdS$_3$ that is related to two dimensional quantum gravity in lightcone gauge. The behavior of the ghost fields under both sets of boundary conditions is novel and we discuss potential interpretations.

A Mathematical and Numerical Framework for Bubble Meta-screens

Abstract: The aim of this paper is to provide a mathematical and numerical framework for the analysis and design of bubble meta-screens. An acoustic meta-screen is a thin sheet with patterned subwavelength structures, which nevertheless has a macroscopic effect on acoustic wave propagation. In this paper, periodic subwavelength bubbles mounted on a reflective surface (with Dirichlet boundary condition) are considered. It is shown that the structure behaves as an equivalent surface with Neumann boundary condition at the Minnaert resonant frequency which corresponds to a wavelength much greater than the size of the bubbles. An analytical formula for this resonance is derived. Numerical simulations confirm its accuracy and show how it depends on the ratio between the periodicity of the lattice, the size of the bubble, and the distance from the reflective surface. The results of this paper formally explain the superabsorption behavior observed in [V. Leroy et al., Phys. Rev. B, 19 (2015), 02031].

The motion of the rigid body in the viscous fluid including collisions. Global solvability result

Abstract: We shall consider the problem of the motion of a rigid body in an incompressible viscous fluid filling a bounded domain. This problem was studied by several authors. They mostly considered classical non-slip boundary conditions, which gave them very paradoxical result of no collisions of the body with the boundary of the domain. Only recently there are results when the Navier type of boundary is considered. In our paper we shall consider the Navier condition on the boundary of the body and the non-slip condition on the boundary of the domain. This case admits collisions of the body with the boundary of the domain. We shall prove the global existence of weak solution of the problem.

A time-dependent optimal harvesting problem with measure-valued solutions

Abstract: The paper is concerned with the optimal harvesting of a marine park, which is described by a parabolic heat equation with Neumann boundary conditions and a nonlinear source term. We consider a cost functional, which is linear with respect to the control; hence the optimal solution can belong to the class of measure-valued control strategies. For each control function, we prove existence and stability estimates for solutions of the parabolic equation. Moreover, we prove the existence of an optimal solution. Finally, some numerical simulations conclude the paper.

Stabilization in a higher-dimensional quasilinear Keller–Segel system with exponentially decaying diffusivity and subcritical sensitivity

Abstract: The quasilinear chemotaxis system ( ⋆ ) { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) , v t = Δ v − v + u , is considered under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n , n ≥ 2 , with smooth boundary, where the focus is on cases when herein the diffusivity D ( s ) decays exponentially as s → ∞ . It is shown that under the subcriticality condition that (0.1) S ( s ) D ( s ) ≤ C s α for all s ≥ 0 with some C > 0 and α 2 n , for all suitably regular initial data satisfying an essentially explicit smallness assumption on the total mass ∫ Ω u 0 , the corresponding Neumann initial–boundary value problem for ( ⋆ ) possesses a globally defined bounded classical solution which moreover approaches a spatially homogeneous steady state in the large time limit. Viewed as a complement of known results on the existence of small-mass blow-up solutions in cases when in (0.1) the reverse inequality holds with some α > 2 n , this confirms criticality of the exponent α = 2 n in (0.1) with regard to the singularity formation also for arbitrary n ≥ 2 , thereby generalizing a recent result on unconditional global boundedness in the two-dimensional situation. As a by-product of our analysis, without any restriction on the initial data, we obtain boundedness and stabilization of solutions to a so-called volume-filling chemotaxis system involving jump probability functions which decay at sufficiently large exponential rates.

Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion

Abstract: The coupled quasilinear Keller-Segel-Navier-Stokes system is considered under Neumann boundary conditions for $n$ and $c$ and no-slip boundary conditions for $u$ in three-dimensional bounded domains $\Omega\subseteq \mathbb{R}^3$ with smooth boundary, where $m>0,\kappa\in \mathbb{R}$ are given constants, $\phi\in W^{1,\infty}(\Omega)$. If $ m> 2$, then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(KSNF)$ possesses a globally defined weak solution.

Homogenization of the Brush Problem with a Source Term in L^1

Abstract: We consider a domain which has the form of a brush in 3D or the form of a comb in 2D, i.e. an open set which is composed of cylindrical vertical teeth distributed over a fixed basis. All the teeth have a similar fixed height; their cross sections can vary from one teeth to another one and are not supposed to be smooth; moreover the teeth can be adjacent, i.e. they can share parts of their boundaries. The diameter of every tooth is supposed to be less than or equal to e, and the asymptotic volume fraction of the teeth (as e tends to zero) is supposed to be bounded from below away from zero, but no periodicity is assumed on the distribution of the teeth. In this domain we study the asymptotic behavior (as e tends to zero) of the solution of a second order uniformly elliptic equation with a zeroth order term which is bounded from below away from zero, when the homogeneous Neumann boundary condition is satisfied on the whole of the boundary. First, we revisit the problem where the source term belongs to L^2. This is a classical problem, but our homogenization result takes place in a geometry which is more general that the ones which have been considered before. Moreover we prove a corrector result which is new. Then, we study the case where the source term belongs to L^1. Working in the framework of renormalized solutions and introducing a definition of renormalized solutions for degenerate elliptic equations where only the vertical derivative is involved (such a definition is new), we identify the limit problem and prove a corrector result.

Representation of the Green’s function of the exterior Neumann problem for the Laplace operator

Abstract: We represent the Green’s function of the classical Neumann problem for the exterior of the unit ball of arbitrary dimension. We show that the Green’s function can be expressed through elementary functions. The explicit form of the function is written out.