On the convexity of level lines of the fundamental mode in the clamped membrane problem,and the existence of convex solutions in a related free boundary problem

It follows from a result of Brascamp and Lieb that the level curves of the first eigenfunction in the clamped membrane problem for a convex region are convex. This paper gives an alternate proof of the above convexity theorem which makes use of the maximum principle — a method which also yields pointwise bounds for the curvature of the level curve through an arbitrary point in. The convexity theorem is then used to establish the existence of convex solutions to a related free boundary problem.

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Zusammenfassung Aus einem Ergebnis von Brascamp und Lieb folgt die Konvexität der Niveaulinien der ersten Eigenfunktion einer eingespannten Membran auf einem konvexen ebenen Gebiet. Hier wird ein anderer Beweis desselben Konvexitätssatzes gegeben, welcher auf dem Maximum-prinzip beruht; diese Methode liefert auch punktweise Schranken für die Krümmung der Niveaulinie durch einen beliebigen Punkt in. Mit Hilfe des Konvexitätssatzes wird dann die Existenz von konvexen Lösungen einer verwandten freien Randwertaufgabe bewiesen.

     

On the directional derivative problem for an elliptic equation

Siberian Mathematical Journal -     

On the approximability of the minimum strictly fundamental cycle basis problem

We consider the problem of finding a strictly fundamental cycle basis of minimum weight in the cycle space associated with an undirected connected graph G, where a nonnegative weight is assigned to each edge of G and the total weight of a basis is defined as the sum of the weights of all the cycles in the basis. Several heuristics have been proposed to tackle this NP-hard problem, which has some interesting applications. In this paper we show that this problem is APX-hard, even when restricted to unweighted graphs, and hence does not admit a polynomial-time approximation scheme, unless P=NP. Using a recent result on the approximability of lower-stretch spanning trees (Elkin et al. (2005) [7]), we obtain that the problem is approximable within O(log²nloglogn) for arbitrary graphs. We obtain tighter approximability bounds for dense graphs. In particular, the problem restricted to complete graphs admits a polynomial-time approximation scheme.     

Inequalities for eigenvalues of a clamped plate problem

Let be a connected bounded domain in an -dimensional Euclidean space . Assume that

are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator:

Then, we give an upper bound of the -th eigenvalue in terms of the first eigenvalues, which is independent of the domain , that is, we prove the following:

Further, a more explicit inequality of eigenvalues is also obtained.

     

On two conjectures in the fixed membrane eigenvalue problem

In this paper we give conditions sufficient to insure that the following two results hold for eigenfunctions of the fixed membrane problem defined over a bounded region of the plane:

1. The first eigenfunction has only one interior critical point.
2. The second eigenfunction has no closed interior nodal curve.

     

Inequalities for eigenvalues of a clamped plate problem

In this paper we study eigenvalues of a clamped plate problem on compact domains in complete manifolds. For complete manifolds admitting special functions, we prove universal inequalities for eigenvalues of clamped plate problem independent of the domains of Payne?CPólya?CWeinberger?CYang type. These manifolds include Hadamard manifolds with Ricci curvature bounded below, a class of warped product manifolds, the product of Euclidean spaces with any complete manifolds and manifolds admitting eigenmaps to a sphere. In the case of warped product manifolds, our result implies a universal inequality on hyperbolic space proved by Cheng?CYang. We also strengthen an inequality for eigenvalues of clamped plate problem on submanifolds in a Euclidean space obtained recently by Cheng, Ichikawa and Mametsuka.     

On the first eigenvalue of the clamped plate

Summary We find a lower bound for the ratio between the first eigenvalue of any homogeneous thin plate G, which is clamped on its boundary, and the first eigenvalue of the spherical clamped plate having the same measure as G. In two dimensions, our bound is about 0.98.     

On a variational problem with lack of compactness related to the Strichartz inequality

We study a variational problem from nonlinear fiber optics which strongly lacks compactness, due to the absence of a priori bounds in spaces different from . A method is established how to restore this missing compactness by means of the dispersive properties inherent in the problem.Received: 30 October 2002, Accepted: 15 April 2003, Published online: 1 July 2003     

On fundamental solutions of the Cauchy problem for a class of degenerate parabolic equations

We present the results of an investigation and some applications of fundamental solutions of the Cauchy problem for a new class of parabolic equations. In these equations: (i) there exist three groups of spatial variables, one basic and two auxiliary, (ii) different weights of spatial variables from the basic group with respect to the time variable are admitted, (iii) degeneracies in variables from the auxiliary groups are present, (iv) a degeneracy on the initial hyperplane is present. Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv; Ternopil' Academy of Economics, Ternopil'. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 13–19, April–June, 1998.     

带裂纹的弹性狭长体基本问题

利用复变方法和积分方程理论，讨论带任意裂纹的各向同性弹性狭长体的基本问题。通过适当的函数分解和积分变换，将问题简化为一正则型奇异积分方程。对方程解的情况和求解方法进行了研究，并导出裂纹尖端的应力强度因子。     

On the fundamental group-scheme

We show that the fundamental group-scheme of a separably rationally connected variety defined over an algebraically closed field is trivial. Let X be a geometrically irreducible smooth projective variety defined over a finite field k admitting a k-rational point. Let {En,σn}_n?0 be a flat principal G-bundle over X, where G is a reductive linear algebraic group defined over k. We show that there is a positive integer a such that the principal G-bundle is isomorphic to E₀, where FX is the absolute Frobenius morphism of X. From this it follows that E₀ is given by a representation of the fundamental group-scheme of X in G.     

A lower bound for eigenvalues of a clamped plate problem

In this paper, we study eigenvalues of a clamped plate problem. We obtain a lower bound for eigenvalues, which gives an important improvement of results due to Levine and Protter.     

A fundamental problem in linear inequalities with applications to the travelling salesman problem

We consider a system ofm linearly independent equality constraints inn nonnegative variables:Ax = b, x 0. The fundamental problem that we discuss is the following: suppose we are given a set ofr linearly independent column vectors ofA, known asthe special column vectors. The problem is to develop an efficient algorithm to determine whether there exists a feasible basis which contains all the special column vectors as basic column vectors and to find such a basis if one exists. Such an algorithm has several applications in the area of mathematical programming. As an illustration, we show that the famous travelling salesman problem can be solved efficiently using this algorithm. Recent published work indicates that this algorithm has applications in integer linear programming. An algorithm for this problem using a set covering approach is described.This research has been partially supported by the ISDOS research project and the National Science Foundation under Grant GK-27872 with the University of Michigan.     

Biorthogonal expansions in the first fundamental problem of elasticity theory

The first fundamental boundary-value problem of elasticity theory is considered for a rectangular semi-infinite strip whose long sides are free of stress. Separation of variables is used to reduce the solution to a series expansion of two functions defined in a closed interval (the “end” of the half-strip), in terms of homogeneous solutions. The system of homogeneous solutions over an interval of the real axis is proved to be complete in L₂. Systems of functions biorthogonal to the systems of homogeneous solutions are constructed on a certain contour on the Riemann surface of the logarithm. This biorthogonality concept is a natural generalization of biorthogonality over a closed interval. The biorthogonal systems constructed are used to find explicit expressions for the expansion coefficients.     

Estimates for lower order eigenvalues of a clamped plate problem

For a bounded domain Ω in a complete Riemannian manifold M ⁿ , we study estimates for lower order eigenvalues of a clamped plate problem. We obtain universal inequalities for lower order eigenvalues. We would like to remark that our results are sharp.