Domain In R3 Research Articles

Ginzburg-Landau equation and Meissner states of multiply-connected superconductors

This paper concerns the Ginzburg-Landau equation and the Meissner equations on a bounded multiply-connected domain in R3. The novelty of these equations is that both of them contain an unknown potential and an unknown Neumann field as the Lagrange multipliers. The minimal and non-minimal solutions for each equation are obtained by variational methods, and the effect of the applied magnetic field on the Meissner states is examined. The Meissner A-system with potential is also examined and a solution is obtained by using the Mountain Pass Lemma of Ambrosetti and Rabinowitz. In our analysis the estimates of the natural boundary value problem of the Maxwell-Stokes system play an important role.

A necessary condition for Sobolev extension domains in higher dimensions

We give a necessary condition for a domain to have a bounded extension operator from L1,p(Ω) to L1,p(Rn) for the range 1<p<2. The condition is given in terms of a power of the distance to the boundary of Ω integrated along the measure theoretic boundary of a set of locally finite perimeter and its extension. This generalizes a characterizing curve condition for planar simply connected domains, and a condition for W1,1-extensions. We use the necessary condition to give a quantitative version of the curve condition. We also construct an example of an extension domain in R3 that is homeomorphic to a ball and has 3-dimensional boundary.

On Nonlocal Elliptic Problems of the Kirchhoff Type Involving the Hardy Potential and Critical Nonlinearity

In this article, we deal with the nonlocal elliptic problems of the Kirchhoff type involving the Hardy potential and critical nonlinearity on a bounded domain in R 3 . Under an appropriate condition on the nonhomogeneous term and using variational methods, we obtain two distinct solutions.

Asymptotic linking of volume-preserving actions of [formula omitted

We extend V. Arnold's work on asymptotic linking for two volume preserving flows on a domain in R3 and S3 to volume preserving actions of Rk and Rℓ on certain domains in Rn and also to linking of a volume preserving action of Rk with a closed oriented singular ℓ-dimensional submanifold in Rn, where n=k+ℓ+1.

Completeness of products of homogeneous harmonic polynomials and uniqueness of the solution to an inverse wave sounding problem

We prove that pairwise products of elements from two classes of homogeneous harmonic polynomials (HHPs) are complete in L2(D), where D is a bounded domain in R3. One of the classes is formed by all HHPs, the second class contains one HHP of each degree l=0,1,…. The result reinforces a long–standing theorem by A. Calderon (1980), in which both classes consisted of all HHPs. The strengthened Calderon's theorem is used to substantiate uniqueness of the solution to an inverse wave sounding problem in the spatially nonoverdetermined setting.

Sub-elliptic boundary value problems in flag domains

Abstract A flag domain in ℝ 3 {\mathbb{R}^{3}} is a subset of ℝ 3 {\mathbb{R}^{3}} of the form { ( x , y , t ) : y &lt; A ⁢ ( x ) } {\{(x,y,t):y&lt;A(x)\}} , where A : ℝ → ℝ {A\colon\mathbb{R}\to\mathbb{R}} is a Lipschitz function. We solve the Dirichlet and Neumann problems for the sub-elliptic Kohn–Laplacian △ ♭ = X 2 + Y 2 {\bigtriangleup^{\flat}=X^{2}+Y^{2}} in flag domains Ω ⊂ ℝ 3 {\Omega\subset\mathbb{R}^{3}} , with L 2 {L^{2}} -boundary values. We also obtain improved regularity for solutions to the Dirichlet problem if the boundary values have first order L 2 {L^{2}} -Sobolev regularity. Our solutions are obtained as sub-elliptic single and double layer potentials, which are best viewed as integral operators on the first Heisenberg group. We develop the theory of these operators on flag domains, and their boundaries.

Bounded solutions to the axially symmetric Navier Stokes equation in a cusp region

A domain in R3 that touches the x3 axis at one point is found with the following property. For any initial value in a C2 class, the axially symmetric Navier Stokes equations with Navier slip boundary condition have a finite energy solution that stays bounded for any given time, i.e. no finite time blow up of the fluid velocity occurs. The result seems to be the first case where the Navier-Stokes regularity problem is solved beyond dimension 2.

A new sufficient condition for local regularity of a suitable weak solution to the MHD equations

We assume that Ω is either the whole space R3 or a half-space or a smooth bounded or exterior domain in R3, T>0 and (u,b,p) is a suitable weak solution of the MHD equations in Ω×(0,T). We show that (x0,t0)∈Ω×(0,T) is a regular point of the solution (u,b,p) if the limit inferior (for t→t0−) of the sum of the L3–norms of u and b over an arbitrarily small ball Bρ(x0) is less than infinity.

Number of synchronized solutions for linearly coupled elliptic systems

In this paper, we consider the following linearly coupled Schrödinger system: (Pε)−ε2Δu+u=u3+λv in Ω,−ε2Δv+v=v3+λu in Ω,u>0,v>0 in Ω,∂u∂n=∂v∂n=0 on ∂Ω,where 0<ε<1 is a small parameter, 0<λ<1 is a coupling parameter, Ω is a smooth and bounded domain in R3, and n is the outer normal vector defined on ∂Ω, the boundary of Ω. Motivated by the works of Ao and Wei (2014) and Ao et al. (2013), we use the Lyapunov–Schmidt reduction method to construct a positive synchronized solution of the problem (Pε) with O(ε−3) interior spikes for sufficiently small ε and some λ near 1. In particular, we also show that the problem (Pε) has exactly O(ε−3) many positive synchronized solutions.

On the role of pressure in the theory of MHD equations

We consider the system of MHD equations in Ω×(0,T), where Ω is a domain in R3 and T>0, with the no slip boundary condition for the velocity u and the Navier-type boundary condition for the magnetic induction b. We show that an associated pressure p, as a distribution with a certain structure, can be always assigned to a weak solution (u,b). The pressure is a function with some rate of integrability if the domain Ω is “smooth”, see section 3. In section 4, we study the regularity of p in a sub-domain Ω1×(t1,t2) of Ω×(0,T), where u (or, alternatively, both u and b) satisfies Serrin’s integrability conditions. Regularity criteria for weak solutions to the MHD equations in terms of π≔p+12|b|2 are studied in section 5. Finally, section 6 contains remarks on analogous results in the case of Navier’s or Navier-type boundary conditions for the velocity u.

The general magneto-static model and Maxwell-Stokes system with topological parameters

In this paper we study a nonlinear magneto-static model on a general domain in R3 which is multiply-connected and has holes, and under a nonlinear relation between the magnetic induction B and the magnetic field H. The equation of the model contains a Neumann field h1∈H1(Ω) and a Dirichlet field h2∈H2(Ω), which represent the effects of domain topology. In the case of a general electric current, the equation contains also an unknown gradient ∇p(x), which represents the electric field. Existence results of solutions of the boundary value problems of this model and of a more general nonlinear Maxwell-Stokes system with topological parameters h1 and h2 are proved, which exhibit the effects of domain topology on the electromagnetic fields and on the nonlinear systems involving curl.

Global solutions to compressible Navier-Stokes-Poisson and Euler-Poisson equations of plasma on exterior domains

The initial boundary value problems for compressible Navier-Stokes-Poisson and Euler-Poisson equations of plasma are considered on exterior domains in this paper. With the radial symmetry assumption, the global existence of solutions to compressible Navier-Stokes-Poisson equations with the large initial data on a domain exterior to a ball in Rn(n≥1) is proved. Moreover, without any symmetry assumption, the global existence of smooth solutions near a given constant steady state for both compressible Navier-Stokes-Poisson and Euler-Poisson equations on an exterior domain in R3 with physical boundary conditions is also established with the exponential stability. A key issue addressed in this paper is on the global-in-time regularity of solutions near physical boundaries. This is in particular so for the 3-D compressible Navier-Stokes-Poisson equations to which global smooth solutions of initial boundary value problems are seldom found in literature to the best of knowledge.

The incompressible limit of compressible finitely extensible nonlinear bead-spring chain models for dilute polymeric fluids

We explore the behaviour of global-in-time weak solutions to a class of bead-spring chain models, with finitely extensible nonlinear elastic (FENE) spring potentials, for dilute polymeric fluids. In the models under consideration the solvent is assumed to be a compressible, isentropic, viscous, isothermal Newtonian fluid, confined to a bounded open domain in R3, and the velocity field is assumed to satisfy a complete slip boundary condition. We show that as the Mach number tends to zero the system is driven to its incompressible counterpart.

Sign changing bubbling solutions for a critical Neumann problem in 3D

Let Ω be a smooth bounded domain in R3. We consider the problem Δu−λu+|u|4u=0inΩ,with zero Neumann boundary conditions, and investigate the existence of sign changing solutions, which concentrate simultaneously around k different points of Ω as λ approaches a special value λ0. We characterize the number λ0 in terms of the Green function of −Δ+λ. In particular, we construct a family of solutions exhibiting bubbling behavior at two different points of an annular domain as λ tends to λ0. In doing so, we also derive a representation formula of the Green function of −Δ with Neumann boundary conditions in the annulus in terms of zonal harmonics.

Global solutions for the compressible quantum hydrodynamic model in a bounded domain

This paper concerns the existence of global smooth solutions to the initial boundary value problem for a three-dimensional compressible quantum hydrodynamic model with damping and heat diffusion in a bounded domain in R3. Based on the continuation argument and the uniform a priori estimates with respect to the time, we obtain the existence of global solutions in a bounded smooth domain provided that the initial perturbation around a constant state is small enough. The key difficulty is to deal with the higher order quantum terms, which do play an essential role in establishing the a priori estimates. The boundary conditions finally adopted are the insulating boundary conditions.

Infinitely many bound state solutions for Kirchhoff type problems

This paper is concerned with the following Kirchhoff type problems (0.1)−(a+b∫Ω|∇u|2dx)Δu=λ|u|q−2u+|u|p−2u,x∈Ω,u=0,x∈∂Ω,where constants a,b>0, the parameter λ≥0, 1<q<2<p<6 and Ω is a smooth bounded domain in R3. We consider two cases: the concave–convex nonlinearities (λ>0) and the superlinear nonlinearities (λ=0). By using the genus type min–max argument, we prove that (0.1) admits infinitely many bound state solutions for the above two cases.

An iterative procedure for finding locally and globally optimal arrangements of particles on the unit sphere

The problem of determination of globally optimal arrangements of N pairwise-interacting particles arises in a variety of biological, physical, and chemical applications. At the same time, the important related question of finding all, or many, local minima of the corresponding energy functions, and the study of structure of these minima, has received relatively little attention.A computational procedure is proposed to compute locally optimal and putative globally optimal arrangements of N particles constrained to a sphere. The procedure is able to handle a wide class of pairwise potentials, and can be generalized to other kinds of surfaces and interactions.As computational examples, locally and globally energy-minimizing arrangements of particles on the unit sphere, interacting via the Coulombic, logarithmic, and inverse square law potentials, are computed. We present new results for the logarithmic potential consisting of 45 new local minima for N≤65 and two new global minima (N=19,46), as well as results for the inverse square law potential which has not previously been studied. We provide comprehensive tables of all minima found, and exclude saddle points. The algorithm can perform computations exceeding N=100 with reasonable execution times. Program summaryProgram Title: EOPS 1.0 - Energy Optimizer for Particles on the SphereProgram Files doi:http://dx.doi.org/10.17632/cbn8jt2ffw.1Licensing provisions: GPLv3Programming language: MATLAB 2015b, C++98, MapleNature of problem: Computation of locally and globally optimal arrangements of N particles on the sphere for different pairwise potentials. This constitutes a constrained local optimization problem with 2N−3 degrees of freedom.Solution method: For N particles, the pairwise potential energy is minimized via steepest descent trajectory from a starting configuration generated from known putative (N−1)-particle optimal configurations.Restrictions: Spherical domain in R3 and pairwise potentials. The number of particles is limited by the computing power and memory of the machine.Unusual features: The programs are executed from MATLAB scripts which call C++ and Maple procedures which perform the bulk of the computations.

Universal meshes for smooth surfaces with no boundary in three dimensions

Summary We introduce a method to mesh the boundary Γ of a smooth, open domain in R3 immersed in a mesh of tetrahedra. The mesh follows by mapping a specific collection of triangular faces in the mesh to Γ. Two types of surface meshes follow: (a) a mesh that exactly meshes Γ, and (b) meshes that approximate Γ to any order, by interpolating the map over the selected faces; i.e., an isoparametric approximation to Γ. The map we use to deform the faces is the closest point projection to Γ. We formulate conditions for the closest point projection to define a homeomorphism between each face and its image. These are conditions on some of the tetrahedra intersected by the boundary, and they essentially state that each such tetrahedron should: (a) have a small enough diameter, and (b) have two of its dihedral angles be acute. We provide explicit upper bounds on the mesh size, and these can be computed on the fly. We showcase the quality of the resulting meshes with several numerical examples. More importantly, all surfaces in these examples were meshed with a single background mesh. This is an important feature for problems in which the geometry evolves or changes, since it could be possible for the background mesh to never change as the geometry does. In this case, the background mesh would be a universal mesh (see [1] ) for all these geometries. We expect the method introduced here to be the basis for the construction of universal meshes for domains in three dimensions. This article is protected by copyright. All rights reserved.

Oseen resolvent estimates with small resolvent parameter

We consider the Oseen system with resolvent term in an exterior domain in R3, supplemented by homogeneous Dirichlet boundary conditions. Under the assumption that the resolvent parameter λ is close to zero and ℜλ≥0, λ≠0, we estimate the Lp-norm of the velocity against the Lp-norm of the right-hand side, times a factor C|λ|−2, with C>0 independent of λ. Such an estimate cannot hold for this range of λ if |λ|−2 is replaced by |λ|−κ with κ<3/2, and there are indications that κ∈[3/2,2) cannot be admitted either. We present various other Lp-estimates of Oseen resolvent flows for the same range of λ. Our article is complementary to the work by T. Kobayashi and Y. Shibata (1998) [20], where Oseen resolvent estimates are derived under the assumption that |λ|≥c0, for some arbitrary but fixed c0>0, with the constant in the resolvent estimate depending on c0.

The spectra of harmonic layer potential operators on domains with rotationally symmetric conical points

We study the adjoint of the double layer potential associated with the Laplacian (the adjoint of the Neumann–Poincaré operator), as a map on the boundary surface Γ of a domain in R3 with conical points. The spectrum of this operator directly reflects the well-posedness of related transmission problems across Γ. In particular, if the domain is understood as an inclusion with complex permittivity ϵ, embedded in a background medium with unit permittivity, then the polarizability tensor of the domain is well-defined when (ϵ+1)/(ϵ−1) belongs to the resolvent set in energy norm. We study surfaces Γ that have a finite number of conical points featuring rotational symmetry. On the energy space, we show that the essential spectrum consists of an interval. On L2(Γ), i.e. for square-integrable boundary data, we show that the essential spectrum consists of a countable union of curves, outside of which the Fredholm index can be computed as a winding number with respect to the essential spectrum. We provide explicit formulas, depending on the opening angles of the conical points. We reinforce our study with very precise numerical experiments, computing the energy space spectrum and the spectral measures of the polarizability tensor in two different examples. Our results indicate that the densities of the spectral measures may approach zero extremely rapidly in the continuous part of the energy space spectrum.