Wrinkle-free solutions in the theory of annular elastic membranes

Rotationally symmetric deformations of a flat annular elastic membrane under a gravitational force are studied, with prescribed radial stresses or horizontal displacements at the edges. The small-finitedeflection theory of Föppl-Hencky as well as a simplified version of Reissner's static first approximation theory of thin shells of revolution are applied which lead to consider a single, second-order, ordinary differential equation for the derivation of the principal stresses in the membrane. Using analytical methods, the range of those boundary data is determined for which the solutions of the differential equation are wrinkle free in the sense that both the radial and the circumferential stress components are nonnegative everywhere.     

Boundary layer solutions for some nonlinear elastic membrane problems

Summary The method of matched asymptotic expansions is used to describe finite axisymmetric deformations of two thin elastic membrane problems: a circular membrane with a small radial traction at the edge, an annular membrane with a small circular hole at the center. In both problems, the surface load is assumed to be an axial pressure. There is a boundary layer at the outer edge in the first problem, and at the hole in the second problem.

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Zusammenfassung Mit Hilfe der in der singulären Störungsrechnung benutzten Methode einer inneren und äußeren asymptotischen Entwicklung werden endliche, axialsymmetrische Deformationen bei zwei Membranproblemen beschrieben: eine dünne elastische Kreismembran mit kleiner radialer Zugspannung am Rande, und eine Kreisringmembran mit einem kleinen konzentrischen Loch. In beiden Problemen wird die Belastung als ein axial gerichteter Oberflächendruck angenommen. Im ersten Problem zeigt die Lösung am äußeren Rand Grenzschichtverhalten, welches im zweiten Problem am Lochrand vorliegt.

     

Some extremal problems for circular polygons

The main result of this paper is the solution of the following problem posed by J. Hersch: find the maximal conformal radius on the family of all hyperbolic polygons with n sides (n ≥ 3). It is proved that the maximum is attained on a regular polygon. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 206, 1993, pp. 127–136. Translated by A. Yu. Solynin.     

Two Saint-Venant problems for anisotropic circular cylinder

In the paper the method of homogeneous solutions is used to reduce the torsion and tension Saint-Venant problems for a circular cylinder with rhombohedral cylindrical anisotropy to construction of solutions of two boundary value problems for a system of ordinary differential equations. By integrating it is found that if the twisting moment is nonzero, then a longitudinal strain, which is proportional to the twisting moment, is brought about in the cylinder, provided that the boundary conditions on at least one of the ends of the cylinder do not impede longitudinal displacements. Otherwise, an axial force arise in the cylinder under torsion.     

Nonuniqueness of solutions of Riemann problems

We investigate a general mechanism, utilizing nonclassical shock waves, for nonuniqueness of solutions of Riemann initial-value problems for systems of two conservation laws. This nonuniqueness occurs whenever there exists a pair of viscous shock waves forming a 2-cycle, i.e., two statesU ₁ andU ₂ such that a traveling wave leads fromU ₁ toU ₂ and another leads fromU ₂ toU ₁. We prove that a 2-cycle gives rise to an open region of Riemann data for which there exist multiple solutions of the Riemann problem, and we determine all solutions within a certain class. We also present results from numerical experiments that illustrate how these solutions arise in the time-asymptotic limit of solutions of the conservation laws, as augmented by viscosity terms.     

Structured solutions in scheduling problems

A generalization of the well-known optimal scheduling problem for jobs with lower and upper deadlines is considered. The existence of structured solutions is proved for a wide class of these problems.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 80, pp. 239–248, 1978.     

Polynomial solutions to Dirichlet problems

The Dirichlet problem

is considered where the functions and are polynomials. The authors study the problem of determining which functions will admit polynomial solutions for any polynomial . When one additionally requires the classical condition , this forces , and a complete classification is obtained. Some necessary conditions are obtained for the case 2$">.

     

Viscosity solutions for elliptic-parabolic problems

We study an elliptic-parabolic problem appearing in the theory of partially saturated flows in the framework of viscosity solutions. This is part of current investigation to understand the theory of viscosity solutions for PDE problems involving free boundaries. We prove that the problem is well posed in the viscosity setting and compare the results with the weak theory. Dirichlet or Neumann boundary conditions are considered.     

Counting solutions in bifurcation problems

In this paper, we use the topological degree to obtain some sharp lower bounds for the number of solutions of the parameter slices of the semi-bounded components of the set of nontrivial solutions of an abstract nonlinear equation with a trivial state. By a semi-bounded component one means a component that is bounded in one direction of the parameter. The spectrum of linearization of the equation at the trivial state is not assumed to be discrete. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 149–167, 2006.     

Generalized solutions to Cauchy problems

This paper continues previous attempts to find a convenient mathematical setting in which linear and nonlinear Cauchy problems have a unique global solution, that reduces to a classical solution when the latter exists.With 1 Figure     

Convex solutions of boundary value problems

We establish two criteria for the existence of convex solutions for a boundary value problem arising from the study of the existence of convex radial solutions for the Monge-Ampère equations. We shall use fixed point theorems in a cone.     

Approximation of solutions for location problems

This paper is devoted to a class of location problems with polyhedral norms. The objective function is shown to be a piecewise convex function which has to be maximized. We prove that the optimal locations belong to a finite set of intersection points, and we present an efficient method operating upon this finite set and providing a strict local maximum with few computational effort.     

Proper solutions of vector optimization problems

We define proper solutions in the Kuhn-Tucker sense for multiobjective mathematical programming problems with parameters in infinite-dimensional spaces and compare them with other definitions via suitable representatives: the Benson, Geoffrion, and Hurwicz properness. Necessary and/or sufficient conditions for proper solutions are proved. Problems with and without constraint qualifications are considered under relaxed convexity and differentiability assumptions.The author is grateful to Prof. W. Stadler and two referees for valuable remarks and suggestions concerning a previous draft of this paper.     

Renormalized solutions of degenerate elliptic problems

We consider a general class of degenerate elliptic problems of the form Au+g(x,u,Du)=f, where A is a Leray-Lions operator from a weighted Sobolev space into its dual. We assume that g(x,s,ξ) is a Caratheodory function verifying a sign condition and a growth condition on ξ. Existence of renormalized solutions is established in the L¹-setting.     

Test solutions of porous media problems

Laminar isothermal fluid flow of two immiscible compressible fluid phases in a porous medium is formulated in terms of four unknown functions ?₁, ?₂, S₁ and S₂ in a pair of partial differential equations, $\text{(}\text{?}\text{?t}\text{)[φ(x, t)S}{}_{\mathrm{i}}\text{?}{}_{\mathrm{i}}\text{] = (}\text{?}\text{?x}\text{){κ(x, t) σ}{}_{\mathrm{i}}\text{(S}{}_{\mathrm{i}}\text{)(}\text{?}\text{?x}\text{)[Φ}{}_{\mathrm{i}}\text{(?}{}_{\mathrm{i}}\text{)]}}$, and a pair of auxiliary relations, S₁ = Γ₁(?₁, ?₂) and S₂ = 1 ? S₁. This system is studied in terms of solutions which are constant along curves, viz., horizontal lines, vertical lines, and parabolas relevant to the parabolic nature of each of the partial differential equations in the system. Such solutions are described, although subject to solution of ordinary differential equations, for use in testing either numerical methods or proposed theorems for the original larger problem.     

Slope-changing solutions of elliptic problems on

We consider the problem on , where F is a smooth function periodic of period 1 in all its variables. We are going to find a non-degeneracy condition on F for which the following holds. If we are given a sequence of positive integers and a sequence of real numbers (the slopes), then we shall find an increasing sequence {Q_i} of integers and a solution u which is entire, periodic in (x₂,…,x_n) and which is close to the plane α₁(x₁−Q_i)+u(Q_i,0,…,0) for .     

Characterizing solutions of rough programming problems

In this paper we define a new kind of mathematical programming problems. This kind, in which the decision set is a rough set, is called a rough programming problem. A rough optimal solution and a rough saddle point will be characterized. Some illustrative examples are presented.