Neumann Type Boundary Value Problems Research Articles

Multiplicity results for Hamiltonian systems with Neumann-type boundary conditions

We prove some multiplicity results for Neumann-type boundary value problems associated with a Hamiltonian system. Such a system can be seen as the weak coupling of two systems, the first of which has some periodicity properties in the Hamiltonian function, the second one presenting the existence of a well-ordered pair of lower/upper solutions.

Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems

The chemotaxis system [Formula: see text] is considered in a ball [Formula: see text], [Formula: see text], where the positive function [Formula: see text] reflects suitably weak diffusion by satisfying [Formula: see text] for some [Formula: see text]. It is shown that whenever [Formula: see text] is positive and satisfies [Formula: see text] as [Formula: see text], one can find a suitably regular nonlinearity [Formula: see text] with the property that at each sufficiently large mass level [Formula: see text] there exists a globally defined radially symmetric classical solution to a Neumann-type boundary value problem for (⋆) which satisfies [Formula: see text]

Boundary Value Problems of Thermoelasticity for Porous Sphere and for A Space with Spherical Cavity

This article is concerned with the coupled linear quasi-static theory of thermoelasticity for porous materials under local thermal equilibrium. The system of equations is based on the constitutive equations, Darcy's law of the flow of a fluid through a porous medium, Fourier's law of heat conduction, the equations of equilibrium, fluid mass conservation and heat transfer. The system of governing equations is expressed in terms of displacement vector field, the change of volume fraction of pores, the change of fluid pressure in pore network and the variation of temperature of porous material. The present paper is devoted to construct explicit solutions of the quasi-static boundary value problems (BVPs) of coupled theory of thermoelasticity for a porous elastic sphere and for a space with a spherical cavity. In this research the regular solution of the system of equations for an isotropic porous material is constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. The basic boundary value problems (the Dirichlet type boundary value problem for a sphere and the Neumann type boundary value problem for a space with a spherical cavity) are solved explicitly. The obtained solutions are given by means of the harmonic, bi-harmonic and meta-harmonic functions. For the harmonic functions the Poisson type formulas are obtained. The bi-harmonic and meta-harmonic functions are presented as absolutely and uniformly convergent series.

The Neumann Type Boundary Value Problem in the Theory of Thermoelasticity with Microtemperatures for a Plane with Circular Hole

This paper studies the linear theory of thermoelastic materials with inner structure whose particles,in addition to the classical displacement and temperature fields, possess microtemperatures. The present work considers the 2D equilibrium theory of thermoelasticity for solids with microtemperatures. This paper is devoted to the explicit solution of the Neumann type boundary value problem for an elastic plane, with microtemperatures having a circular hole. Special representations of the regular solutions of the considered equations are constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. Using the Fourier method, we presented the solution of the Neumann type boundary value problem for the plane with circular hole in the explicit form.

Rigidity theorem for harmonic maps with complex normal boundary conditions

Let [Formula: see text] be the open unit ball in [Formula: see text] and let [Formula: see text] be a Kähler manifold with strongly negative or strongly semi-negative curvature. In this paper, we study Siu type rigidity theorem for the harmonic map [Formula: see text] satisfying the boundary condition that [Formula: see text] on [Formula: see text]. We also prove the existence and uniqueness theorem for some Neumann type boundary value problem for harmonic functions on [Formula: see text].

An Augmented Regularized Combined Source Integral Equation for Nonconforming Meshes

We present a new version of the regularized combined source integral equation (CSIE-AR) for the solution of electromagnetic scattering problems in the presence of perfectly conducting bodies. The integral equation is of the second kind and has no spurious resonances. It is well conditioned at all frequencies for simply connected geometries. Reconstruction of the magnetic field, however, is subject to catastrophic cancelation due to the need for computing a scalar potential from magnetic currents. Here, we show that by solving an auxiliary (scalar) integral equation, we can avoid this form of low-frequency breakdown. The auxiliary scalar equation is used to solve a Neumann-type boundary value problem using data corresponding to the normal component of the magnetic field. This scalar equation is also of the second kind, nonresonant, and well conditioned at all frequencies. A principal advantage of our approach, by contrast with the hypersingular electric field integral equation, the combined field integral equation, or CSIE formulations, is that the standard loop-star and related basis function constructions are not needed, and preconditioners are not required. This permits an easy coupling to fast algorithms such as the fast multipole method. Furthermore, the formalism is compatible with nonconformal mesh discretization and works well with singular (sharp) boundaries.

A stabilized RBF collocation scheme for Neumann type boundary value problems

The numerical solutionof partial dif- ferential equations (PDEs) with Neumann bound- ary conditions (BCs) resulted from strong form collocation scheme are typically much poorer in accuracy compared to those with pure Dirichlet BCs. In this paper, we show numerically that the reason of the reduced accuracy is that Neu- mann BC requires the approximation of the spa- tial derivatives at Neumann boundaries which are significantly less accurate than approximation of main function. Therefore, we utilize boundary treatment schemes that based upon increasing the accuracy of spatial derivatives at boundaries. In- creased accuracy of the spatial derivative approx- imation can be achieved by h-refinement reduc- ing the spacing between discretization points or byincreasingthemultiquadricshape parameter, c. Increasing the MQ shape parameter is very com- putationally cost effective, but leads to increased ill-conditioning. We have implemented an im- proved version of the truncated singular value de- composition (IT-SVD) originated by Volokh and Vilnay (2000) that projects very small singular values into the null space, producing a well con- ditioned system of equations. To assess the pro- posed refinement scheme, elliptic PDEs with dif- ferent boundary conditions are analyzed. Com- parisons that made with analytical solution reveal superior accuracy and computationalefficiency of the IT-SVD solutions.

Heterogeneous Anisotropy Index and scaling in two-phase random polycrystals

Under consideration is the finite-size scaling of the elastic properties in two-phase random polycrystals with individual grains belonging to any arbitrary crystal class. These polycrystals are generated by Voronoi Tessellations with varying grain sizes and volume fractions. Any given realization of such a microstructure sampled randomly is highly anisotropic and heterogeneous. Using extremum principles in elasticity, we introduce the notion of a ‘Heterogeneous Anisotropy Index $$\left( A^U_H\right) $$ ’ and examine its role in the scaling of elastic properties at finite mesoscales ( $$\delta $$ ). The relationship between $$A^U_H$$ and the Universal Anisotropy Index $$A^U$$ by Ranganathan and Ostoja-Starzewski (Phys Rev Lett 101(5):055504, 2008) is established for special cases. The index $$A^U_H$$ turns out to be a function of 43 variables—21 independent components for each phase and the volume fraction of either phase. The scale-dependent bounds are then obtained by setting up and solving 9250 Dirichlet and Neumann type boundary value problems consistent with the Hill–Mandel homogenization condition. Subsequently, the concept of an elastic scaling function is introduced that takes a power-law form in terms of $$A^U_H$$ and ( $$\delta $$ ). Finally, a material scaling diagram is constructed by employing the elastic scaling function which captures the convergence to the effective properties for any two-phase elastic microstructure.

Boundary value problems of statics of thermoelasticity of bodies with microstructure and microtemperatures

The paper deals with boundary value problems of statics of the thermoelasticity theory of isotropic microstretch materials with microtemperatures and microdilatations. For the system of differential equations of equilibrium the fundamental matrix is constructed explicitly in terms of elementary functions. With the help of the corresponding Green identities the general integral representation formula of solutions by means of generalized layer and Newtonian potentials are derived. The basic Dirichlet and Neumann type boundary value problems are formulated in appropriate function spaces and the uniqueness theorems are proved. The existence theorems for classical solutions are established by using the potential method.

Mathematical problems of thermoelasticity of bodies with microstructure and microtemperatures

The paper deals with the linear theory of thermoelasticity for elastic isotropic microstretch materials with microtemperatures and microdilatations. For the differential equations of pseudo-oscillations the fundamental matrix is constructed explicitly in terms of elementary functions. With the help of the corresponding Green identities the general integral representation formula of solutions by means of generalized layer and Newtonian potentials are derived. The basic Dirichlet and Neumann type boundary value problems are formulated in appropriate function spaces and the uniqueness theorems are proved. The existence theorems for classical solutions are established by using the potential method.

РАЗРЕШИМОСТЬ ОДНОЙ ЗАДАЧИ ТИПА НЕЙМАНА ДЛЯ ТРИГАРМОНИЧЕСКОГО УРАВНЕНИЯ В ШАРЕ

A boundary-value problem for 3-harmonic equation in a unit ball, containing in the boundary conditions the Laplacian levels up to the second order inclusively, and the normal derivative, is considered. This problem is a natural Neumann-type continuation of the Riquier problem for a 3-harmonic equation. The problem is more general than the considered one, but it has been researched before by V.V. Karachick and B. Torebek for a biharmonic equation. By the means of reducing the initial boundary-value problem to a system of three differential equations of the third order in harmonic equations in a unit ball of functions, the necessary and sufficient condition for solvability of the initial Neumann-type boundary-value problem is discovered. This condition is obtained as a vanishing of the integral over the unit sphere from one of the boundary functions of the problem. Besides, the method of theorem proof allows framing the solution of the considered Neumann-type problem in an explicit form. Moreover, it is determined in the article that solution of the initial boundary-value problem is unique up to an arbitrary constant.

Representation Formula for General Solution of a Homogeneous System of Differential Equations

We consider the stationary oscillation case of the theory of linear thermoelasticity with microtemperatures of materials. The representation formula of a general solution of the homogeneous system of differential equations obtained in the paper is expressed by means of seven metaharmonic functions. These formulas are very convenient and useful in many particular problems for domains with concrete geometry. Here we demonstrate applications of these formulas to the Dirichlet- and Neumann-type boundary-value problems for a ball. Uniqueness theorems are proved. We construct explicit solutions in the form of absolutely and uniformly convergent series.

Blow-up of weak solutions to a chemotaxis system under influence of an external chemoattractant

We study nonnnegative radially symmetric solutions of the parabolic–elliptic Keller–Segel whole space system0, \\\\ 0=\\Delta v+u+f(x),~ & x\\in {{\\mathbb{R}}^{n}},t>0, \\\\ u(x,0)={{u}_{0}}(x),~ & x\\in {{\\mathbb{R}}^{n}}, \\end{array}\\right. \\end{eqnarray} ?>{ut=Δu−∇⋅(u∇v), x∈Rn,t>0,0=Δv+u+f(x), x∈Rn,t>0,u(x,0)=u0(x), x∈Rn,with prototypical external signal productionf(x):={f0|x|−α,if |x|⩽R−ρ,0,if |x|⩾R+ρ,for and , which is still integrable but not of class for some . For corresponding parabolic-parabolic Neumann-type boundary-value problems in bounded domains , where for some and , it is known that the system does not emit blow-up solutions if the quantities and , for some , are all bounded by some small enough.We will show that whenever and in , a measure-valued global-in-time weak solution to the system above can be constructed which blows up immediately. Since these conditions are independent of and c0 > 0, we obtain a strong indication that in fact is critical for the existence of global bounded solutions under a smallness conditions as described above.

A convergence result for the ergodic problem for Hamilton–Jacobi equations with Neumann-type boundary conditions

We consider the ergodic (or additive eigenvalue) problem for the Neumann-type boundary-value problem for Hamilton–Jacobi equations and the corresponding discounted problems. Denoting by uλ the solution of the discounted problem with discount factor λ > 0, we establish the convergence of the whole family to a solution of the ergodic problem as λ → 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting by-product, we introduce Mather measures associated with Hamilton–Jacobi equations with the Neumann-type boundary conditions. These results are variants of the main results in a recent paper by Davini et al., who study the same convergence problem on smooth compact manifolds without boundary.

Boundary value problems of stationary oscillation of thermoelasticity of microstretch materials with microtemperatures

Abstract We consider the stationary oscillation case of the theory of linear thermoelasticity with microtemperatures of microstretch materials. A representation formula of a general solution of a homogeneous system of differential equations is written in terms of eight metaharmonic functions. Such formulas are very convenient and useful in many specific problems of concrete geometry. We demonstrate an application of these formulas to Dirichlet and Neumann type boundary value problems in a ball. Explicit solutions in the form of absolutely and uniformly convergent series are constructed.

On Neumann type problems for nonlocal equations set in a half space

We study Neumann type boundary value problems for nonlocal equations related to Levy processes. Since these equations are nonlocal, Neumann type problems can be obtained in many ways, depending on the kind of reflection we impose on the outside jumps. To focus on the new phenomenas and ideas, we consider different models of reflection and rather general non-symmetric Levy measures, but only simple linear equations in half-space domains. We derive the Neumann/reflection problems through a truncation procedure on the Levy measure, and then we develop a viscosity solution theory which includes comparison, existence, and some regularity results. For problems involving fractional Laplacian type operators like e.g.$(-\Delta)^{\alpha/2}$, we prove that solutions of all our nonlocal Neumann problems converge as alpha goes to 2 to the solution of a classical Neumann problem. The reflection models we consider include cases where the underlying Levy processes are reflected, projected, and/or censored upon exiting the domain.

Computation of aeroelastic transfer functions of aircraft elements under vibrations in a compressible flow

We consider the problem on the computation of aeroelastic transfer functions of an airfoil under vibrations in a compressible flow with the actual natural frequencies of the airfoil taken into account. The problem is reduced to a Neumann type boundary value problem for the scalar Helmholtz equation.We estimate the reliability of the computation results.

Ergodic BSDEs and related PDEs with Neumann boundary conditions

We study a new class of ergodic backward stochastic differential equations (EBSDEs for short) which is linked with semi-linear Neumann type boundary value problems related to ergodic phenomena. The particularity of these problems is that the ergodic constant appears in Neumann boundary conditions. We study the existence and uniqueness of solutions to EBSDEs and the link with partial differential equations. Then we apply these results to optimal ergodic control problems.

An Existence Result for Neumann Type Boundary Value Problems for Second Order Nonlinear Functional Differential Equation

Abstract. New sufficient conditions for the existence of at least one solution of Neumanntype boundary value problems for second order nonlinear differential equationsˆ(p(t)φ(x 0 (t))) 0 = f(t,x(t),x(τ 1 (t)),··· ,x(τ m (t))), t ∈ [0,T],x 0 (0) = 0, x 0 (T) = 0,are established. 1. IntroductionRecently, there have been many papers discussed the solvability of two-point,or multi-point boundary value problems for second order or higher order ordinaryor functional differential equations, we refer the readers to the text books [1], [2]and papers [6]-[9], [13]-[15] and the references therein.Xuan and Chen in [3] studied the solvability of singular one dimensionalp−Laplacian-like equation with Neumann boundary conditions(1)ˆ(A(|u 0 |)u 0 ) 0 −g(u(t)) = h(t), 0 0, h(t) ∈ L 1 [0,1], g is a continuous function definedon (−∞,0) ∪ (0,+∞) such that g(ξ) → 0 as |ξ| → 0, g(ξ) → +∞ as ξ → 0 + ,g(ξ) → −∞ as ξ → 0 − , and g(ξ)ξ > 0 for ξ 6= 0. Suppose(i) H(ξ) = ξA(|ξ|) is strictly increasing homeomorphism of (0,+∞) withH(0) = 0.(ii) lim

Continuous dependence results for non-linear Neumann type boundary value problems

We obtain estimates on the continuous dependence on the coefficient for second-order non-linear degenerate Neumann type boundary value problems. Our results extend previous work of Cockburn et al., Jakobsen and Karlsen, and Gripenberg to problems with more general boundary conditions and domains. A new feature here is that we account for the dependence on the boundary conditions. As one application of our continuous dependence results, we derive for the first time the rate of convergence for the vanishing viscosity method for such problems. We also derive new explicit continuous dependence on the coefficients results for problems involving Bellman–Isaacs equations and certain quasilinear equation.