Neumann Boundary Problem Research Articles

Stabilization in a chemotaxis system modelling T-cell dynamics with simultaneous production and consumption of signals

Abstract In a smoothly bounded domain $\Omega \subset \mathbb{R}^n$ , $n\ge 1$ , this manuscript considers the homogeneous Neumann boundary problem for the chemotaxis system \begin{eqnarray*} \left \{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v), \\[5pt] v_t = \Delta v + u - \alpha uv, \end{array} \right . \end{eqnarray*} with parameter $\alpha \gt 0$ and with coincident production and uptake of attractants, as recently emphasized by Dallaston et al. as relevant for the understanding of T-cell dynamics. It is shown that there exists $\delta _\star =\delta _\star (n)\gt 0$ such that for any given $\alpha \ge \frac{1}{\delta _\star }$ and for any suitably regular initial data satisfying $v(\cdot, 0)\le \delta _\star$ , this problem admits a unique classical solution that stabilizes to the constant equilibrium $(\frac{1}{|\Omega |}\int _\Omega u(\cdot, 0), \, \frac{1}{\alpha })$ in the large time limit.

Multiple solutions to nonlocal Neumann boundary problem with sign‐changing coefficients

We consider the following Neumann boundary problem with a nonlocal Kirchhoff term where is a bounded smooth domain and and are sign‐changing coefficients which play important impact on the study. By using variational method and dividing corresponding Nehari manifold into two disjoint subsets, we study the existence of two positive solutions. To show the multiplicity of positive solutions, we find a set whose elements can be mapped to two subsets of Nehari manifold, which is also the key to proving strong convergence for minimizing sequences. Finally, the asymptotic behavior of solutions is studied as parameter tends to 0. Different from Dirichlet boundary problem, we show that the two solutions of Neumann problem tend to two different positive constants, respectively.

Numerical Analysis of the Cylindrical Shell Pipe with Preformed Holes Subjected to a Compressive Load Using Non-Uniform Rational B-Splines and T-Splines for an Isogeometric Analysis Approach

In this paper, we implement the finite detail technique primarily based on T-Splines for approximating solutions to the linear elasticity equations in the connected and bounded Lipschitz domain. Both theoretical and numerical analyses of the Dirichlet and Neumann boundary problems are presented. The Reissner–Mindlin (RM) hypothesis is considered for the investigation of the mechanical performance of a 3D cylindrical shell pipe without and with preformed hole problems under concentrated and compression loading in the linear elastic behavior for trimmed and untrimmed surfaces in structural engineering problems. Bézier extraction from T-Splines is integrated for an isogeometric analysis (IGA) approach. The numerical results obtained, particularly for the displacement and von Mises stress, are compared with and validated against the literature results, particularly with those for Non-Uniform Rational B-Spline (NURBS) IGA and the finite element method (FEM) Abaqus methods. The obtained results show that the computation time of the IGA based on the T-Spline method is shorter than that of the IGA NURBS and FEM Abaqus/CAE (computer-aided engineering) methods. Furthermore, the highlighted results confirm that the IGA approach based on the T-Spline method shows more success than numerical reference methods. We observed that the NURBS IGA method is very limited for studying trimmed surfaces. The T-Spline method shows its power and capability in computing trimmed and untrimmed surfaces.

Existence and bifurcation of positive solutions to a class of predator-prey models with mutual interference among the predators

In this work, we study a class of diffusive predator-prey models with mutual interference among the predators while searching for food. Under Dirichlet boundary condition, by the fixed point index theory, we provide the necessary and sufficient conditions for the existence of coexistence states (i.e., stationary pattern). This is a strong contrast to the corresponding Neumann boundary problem, which exhibits bistability and admits no patterns.

Solving Neumann Boundary Problem with Kernel-Regularized Learning Approach

Solving Neumann Boundary Problem with Kernel-Regularized Learning Approach

The first eigencurve for a Neumann boundary problem involving p-Laplacian with essentially bounded weights

This article is intended to prove the existence and uniqueness of the first eigencurve, for a homogeneous Neumann problem with singular weights associated with the equation \[-\Delta_{p} u=\alpha m_{1}|u|^{p-2}u+\beta m_{2}|u|^{p-2}u\] in a bounded domain \(\Omega \subset \mathbb{R}^{N}\). We then establish many properties of this eigencurve, particularly the continuity, variational characterization, asymptotic behavior, concavity and the differentiability.

Liouville type theorem of integral equation with anisotropic structure

In this paper, we classify all nonnegative solutions for the following integral equation:(0.1)u(x)=∫R+nKb(x,y)ynbf(u(y))dy, where b>1 is a constant. Here Kb(x,y) is the Green function of the following homogeneous Neumann boundary problem(0.2){−div(xnb∇u)=finR+n∂u∂xn=0on∂R+n. By using the method of moving planes in an integral form, we derive the symmetry of nonnegative solutions. We also establish the Liouville type theorem of (0.1). Similarly, the results can be generalized to the integral system.

Multiplicity of solutions for a nonlocal nonhomogeneous Neumann boundary problem with two critical exponents

This article concerns the multiplicity of solutions for a class of nonlocal and nonhomogeneous Neumann boundary value problems involving the p ( x )-Laplacian, in which both nonlinear terms assume critical growth. We use variational method, exploring an important truncation argument and properties of the genus.

Asymptotic behavior of the occupancy density for obliquely reflected Brownian motion in a half-plane and Martin boundary

Let π be the occupancy density of an obliquely reflected Brownian motion in the half plane and let (ρ,α) be the polar coordinates of a point in the upper half plane. This work determines the exact asymptotic behavior of π(ρ,α) as ρ→∞ with α∈(0,π). We find explicit functions a, b, c such that π(ρ,α)∼ρ→∞a(α)ρb(α)e−c(α)ρ. This closes an open problem first stated by Professor J. Michael Harrison in August 2013. We also compute the exact asymptotics for the tail distribution of the boundary occupancy measure and we obtain an explicit integral expression for π. We conclude by finding the Martin boundary of the process and giving all of the corresponding harmonic functions satisfying an oblique Neumann boundary problem.

Concentration solutions to the singularly prescribed Gaussian and geodesic curvatures problem

We consider the following Liouville-type equation with exponential Neumann boundary condition:{−Δu˜=ε2K(x)e2u˜,x∈D,∂u˜∂n+1=εκ(x)eu˜,x∈∂D, where D⊂R2 is the unit disk, ε2K(x)>0 and εκ(x)>0 stand for the prescribed Gaussian curvature and geodesic curvature of the boundary, respectively. We prove the existence of concentration solutions if κ(x)+K(x)+κ(x)2 (x∈∂D) has a local extremum point, which is a new result for exponential Neumann boundary problems.

Neumann Boundary Problems for Parabolic Partial Differential Equations with Divergence Terms

In this paper, we give a probabilistic interpretation for solutions to the Neumann boundary problems for a class of semi-linear parabolic partial differential equations (PDEs for short) with singular non-linear divergence terms. This probabilistic approach leads to the study on a new class of backward stochastic differential equations (BSDEs for short). A connection between this class of BSDEs and semi-linear PDEs is established.

Singular Neumann Boundary Problems for a Class of Fully Nonlinear Parabolic Equations in One Dimension

In this paper, we discuss a singular Neumann boundary problem for a class of nonlinear parabolic equations in one space dimension. Our boundary problem describes the motion of a planar curve sliding along the boundary with a zero contact angle, which can be viewed as a limiting model for the capillary phenomenon. We study the uniqueness and existence of solutions by using the viscosity solution theory. We also show the convergence of the solution to a traveling wave as time proceeds to infinity when the initial value is assumed to be convex.

Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions

We consider a class of equations in divergence form with a singular/degenerate weight Under suitable regularity assumptions for the matrix A and f (resp. F) we prove Hölder continuity of solutions which are even in and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the and a priori bounds for approximating problems in the form as Finally, we derive and bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems.

On Realization of the Original Weyl–Titchmarsh Functions by Shrödinger L-systems

We study realizations generated by the original Weyl–Titchmarsh functions $$m_\infty (z)$$ and $$m_\alpha (z)$$ . It is shown that the Herglotz–Nevanlinna functions $$(-\,m_\infty (z))$$ and $$(1/m_\infty (z))$$ can be realized as the impedance functions of the corresponding Shrodinger L-systems sharing the same main dissipative operator. These L-systems are presented explicitly and related to Dirichlet and Neumann boundary problems. Similar results but related to the mixed boundary problems are derived for the Herglotz–Nevanlinna functions $$(-\,m_\alpha (z))$$ and $$(1/m_\alpha (z))$$ . We also obtain some additional properties of these realizations in the case when the minimal symmetric Shrodinger operator is non-negative. In addition to that we state and prove the uniqueness realization criteria for Shrodinger L-systems with equal boundary parameters. A condition for two Shrodinger L-systems to share the same main operator is established as well. Examples that illustrate the obtained results are presented in the end of the paper.

Harmonic Green and Neumann functions for domains bounded by two intersecting circular arcs

ABSTRACT The parqueting-reflection principle is shown to work for constructing the harmonic Green functions and harmonic Neumann functions for a class of domains, which are bounded by two intersecting circular arcs in with a particular angle . With the explicit expressions of the harmonic Green and Neumann functions, we solve the Dirichlet and Neumann boundary problems to the Poisson equation in these domains by applying the Green and Neumann representation formulas.

Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion

This paper deals with a homogeneous Neumann problem of a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms. We prove the existence of weak solutions using Galerkin’s approximation and we derive suitable energy estimates. To this end, we establish the needed Poincaré type inequality for variable exponents related to the Neumann boundary problem. Furthermore, we show that the investigated problem possesses a unique weak solution and satisfies a stability estimate, provided some additional assumptions are fulfilled. In addition, we show under which conditions the solution is nonnegative.

Convergence rate of multiscale finite element method for various boundary problems

In this paper, we examine the effectiveness of classic multiscale finite element method (MsFEM) (Hou and Wu, 1997; Hou et al., 1999) for mixed Dirichlet–Neumann, Robin and hemivariational inequality boundary problems. Constructing so-called boundary correctors is a common technique in existing methods to prove the convergence rate of MsFEM, while we think not reflects the essence of those problems. Instead, we focus on the first-order expansion structure. Through recently developed estimations in homogenization theory, our convergence rate is provided with milder assumptions and in neat forms.

Brosamler’s formula revisited and extensions

Brosamler’s formula gives a probabilistic representation of the solution of the Neumann problem for the Laplacian on a smooth bounded domain $$D\subset \mathbb {R}^n$$ in terms of the reflecting Brownian motion in D. The original proof, as well as other proofs in the literature (e.g., in the case of Lipschitz domains), are based on potential theory (transition densities of the reflecting Brownian motion). We give new proofs of Brosamler’s formula using (path trajectories of) stochastic processes. More precisely, we use a connection between the Dirichlet and the Neumann boundary problems, and the explicit description of the reflecting Brownian motion and its boundary local time in terms of the free Brownian motion. The results are obtained in the case of the Euclidean unit ball in any dimension and in the case of smooth $$C^{1,\alpha }$$ planar simply connected domains, for continuous boundary data, and then extended to the case of bounded measurable data, respectively integrable boundary data. A new Brosamler-type formula in terms of the free Brownian motion is also given.

Asymptotical Behaviors for Neumann Boundary Problem with Singular Data

In this paper, we will analyze the blow-up behaviors for solutions to the Laplacian equation with exponential Neumann boundary condition. In particular, the boundary value is with a kind of singular data. We show a Brezis–Merle type concentration-compactness theorem, calculate the blow up value at the blow-up point, and give a point-wise estimate for the profile of the solution sequence at the blow-up point.

On a nonlocal nonhomogeneous Neumann boundary problem with two critical exponents

ABSTRACTIn this paper, we are concerned with questions of the existence of solution for a class of nonlocal and nonhomogeneous Neumann boundary value problems involving the -Laplacian in which the nonlinear terms assume both critical growth. The main tools used are the Lions' Concentration-Compactness Principle [Lions PL. The concentration-compactness principle in the calculus of variations. The limit case I, part 1. Rev Mat Iberoam. 1985;1(1):145–201.] for variable exponent spaces and the Mountain Pass Theorem.