Magnetic bottles for the Neumann problem: The case of dimension 3

The main object of this paper is to analyze the recent results obtained on the Neumann realization of the Schrödinger operator in the case of dimension 3 by Lu and Pan. After presenting a short treatment of their spectral analysis of keymodels, we show briefly how to implement the techniques of Helffer-Morame in order to give some localization of the ground state. This leaves open the question of the localization by curvature effect which was solved in the case of dimension 2 in our previous work and will be analysed in the case of dimension 3 in a future paper.     

Neumann boundary value problem for general curvature flow with forcing term

Geometriae Dedicata - We consider the evolution of a strictly convex hypersurface by a class of general curvature. We prove that given some Neumann boundary condition, the flow exists for all time...     

The Neumann problem for the Laplace equation on general domains

The solution of the weak Neumann problem for the Laplace equation with a distribution as a boundary condition is studied on a general open set G in the Euclidean space. It is shown that the solution of the problem is the sum of a constant and the Newtonian potential corresponding to a distribution with finite energy supported on ∂G. If we look for a solution of the problem in this form we get a bounded linear operator. Under mild assumptions on G a necessary and sufficient condition for the solvability of the problem is given and the solution is constructed. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.     

Existence and multiplicity of solutions for a general p(x)-Laplacian Neumann problem

In this paper, we verify that a general p(x)$p\left(x\right)$-Laplacian Neumann problem has at least three weak solutions, which generalizes the corresponding result of the reference [Mihai Mihilescu, Existence and multiplicity of solutions for a Neumann problem involving the p(x)$p\left(x\right)$-Laplace operator, Nonlinear Anal. 67 (2007) 1419–1425].     

The question of interior blow-up points for an elliptic Neumann problem: the critical case

In contrast with the subcritical case, we prove that for any bounded domain $\text{Ω}$ in $\text{R}{}^3$, the Neumann elliptic problem with critical nonlinearity $\text{−Δu+μu=u}{}^5\text{,}\text{u>0}\text{in}\text{Ω;}\text{∂u/∂ν=0}\text{on}\text{∂Ω}$ has no solution blowing up at only interior points as μ goes to infinity.     

On the Dirichlet problem for the prescribed mean curvature equation over general domains

We study and solve the Dirichlet problem for graphs of prescribed mean curvature in Rn+1 over general domains Ω without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result by Schulz and Williams we can then also solve the Dirichlet problem for boundary values satisfying a Lipschitz condition.     

A variable dimension solution approach for the general spatial price equilibrium problem

Algorithms are presented that are specifically designed for solving general nonlinear multicommodity spatial price equilibrium problems, i.e., problems with nonlinear transportation cost functions, nonlinear supply and demand functions, inter-commodity congestion effects, intercommodity substitution and complementarity effects and interactions among transportation links and among spatially separated markets. The algorithms are specializations of an iterative method for solving nonlinear complementarity problems that requires solving a system of nonlinear equations at each iteration. The algorithms exploit the network structure of the problems to reduce the size of the system of equations to be solved at each iteration. The decision rules for determining which equations are to be included in the system at each iteration are extremely simple, and the remainder of the computational work is carried out by the nonlinear equation solver. Because of this, the algorithms are very easy to implement with readily available software. In addition, since the decision rules only require sign information, only the final system needs to be solved with precision.     

Torsion of SU(2)-structures and Ricci curvature in dimension 5

Following the approach of Bryant [R. Bryant, Some remarks on G₂-structures, in: S. Akbulut, T. Önder, R.J. Stern (Eds.), Proceeding of Gökova Geometry-Topology Conference 2005, International Press, 2006], we study the intrinsic torsion of an SU(2)-structure on a 5-dimensional manifold deriving an explicit expression for the Ricci and the scalar curvature in terms of torsion forms and its derivative. As a consequence of this formula we prove that the α-Einstein condition forces some special SU(2)-structures to be Sasaki-Einstein.     

Neumann problem for the operator of elasticity theory

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 9, pp. 1279–1281, September, 1989.     

A spectral method for elliptic equations: the Neumann problem

Let Ω be an open, simply connected, and bounded region in ℝ ^d , d ≥ 2, and assume its boundary ∂Ω is smooth. Consider solving the elliptic partial differential equation − Δu + γu = f over Ω with a Neumann boundary condition. The problem is converted to an equivalent elliptic problem over the unit ball B, and then a spectral method is given that uses a special polynomial basis. In the case the Neumann problem is uniquely solvable, and with sufficiently smooth problem parameters, the method is shown to have very rapid convergence. Numerical examples illustrate exponential convergence.     

On the prescribed scalar curvature problem on Sn: The degree zero case

In this Note, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere $S^n$, $n?3$. We give new existence and multiplicity results based on a new Euler–Hopf formula type. Our argument also has the advantage of extending the well known results due to Y. Li (1995) [10].     

Solution of the Neumann problem for the Laplace equation

For fairly general open sets it is shown that we can express a solution of the Neumann problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series. If the open set is simply connected and bounded then the solution of the Dirichlet problem is the double layer potential with a density given by a similar series.     

A saint-venant principle for the gradient in the Neumann problem

The maximum principle for subharmonic functions is used to obtain upper bounds for the gradient in the Neumann problem of potential theory. These bounds, which concern a curvilinear strip domain having nonzero boundary data only on an end, entail an exponential decay of the gradient magnitude with distance from that end.

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Zusammenfassung Unter Benutzung des Maximumprinzips für subharmonische Funktionen werden obere Schranken angegeben für den Gradienten des Neumannschen Problems der Potential-Theorie. Diese Schranken betreffen einem gekrümmten Streifenbereich mit von Null verschiedenen Grenzdaten an einem Ende und haben zur Folge, dass der Betrag des Gradienten exponentiell mit dem Abstand von diesem Ende abfällt.

     

The packing problem for projective geometries over GF(3) with dimension greater than five

An (n, d) set in the projective geometry PG(r, q) is a set of n points, no d of which are dependent. The packing problem is that of finding n(r, q, d), the largest size of an (n, d) set in PG(r, q). The packing problem for PG(r, 3) is considered. All of the values of n(r, 3, d) for r ? 5 are known. New results for r = 6 are n(6, 3, 5) = 14 and 20 ? n(6, 3, 4) ? 31. In general, upper bounds on n(r, q, d) are determined using a slightly improved sphere-packing bound, the linear programming approach of coding theory, and an orthogonal (n, d) set with the known extremal values of n(r, q, d)—values when r and d are close to each other. The BCH constructions and computer searches are used to give lower bounds. The current situation for the packing problem for PG(r, 3) with r ? 15 is summarized in a final table.     

An efficient alorithm for the general multiple-choice knapsack problem (GMKP)

A common problem frequently faced by business firms and individual investors is to select a few investment opportunities from many available possibilities. This problem, in its simplest form, can be modeled as a 0–1 knapsack problem. In a more general investment scenario, however, we obtain a model which is a general knapsack problem with a multiple-choice constraint. To solve this problem, an efficient enumerative algorithm is developed. The algorithm includes an efficient procedure to solve the LP-relaxed problem, a reduction algorithm which may allow the initial fixing of some of the variables, and various other implicit enumeration criteria derived from the group problem. Extensive computational experience illustrates the efficiency of the algorithm and related results.     

Reflections on the U(1) problem in general relativity

After reviewing the known stability results for vacuum, U(1) symmetric solutions to Einstein’s field equations, we shall describe some mathematical ideas and techniques that were unavailable during the earlier research in this area and which might conceivably be exploited to shed new light on the large data global Cauchy problem for this interesting class of spacetimes.     

Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac equations in one space dimension

The existence of global solutions of the Cauchy problem is proved for the Maxwell-Dirac equations coupled through the standard electromagnetic interaction. The proof depends on the conservation of charge and an a priori estimate on the electromagnetic potential. The technique also applies to the Dirac-Klein-Gordon equations with Yukawa coupling.