Relative Extrema Research Articles

Uniqueness in general boundary-value problems for elastic or inelastic solids

The class of solids considered is characterized by a linear relation between the stress-rate and strain-rate tensors. The boundary-value problem is set by prescribed surface velocities or nominal traction-rates, the existing state of stress, anisotropy, etc., being regarded as known. Changes in geometry are unrestricted. Various criteria for uniqueness of the solution ( Hill 1957 a, b, 1959) are re-derived (Sections 2 and 4), together with a useful new transformation (Section 3) and related extremum principles (Section 6). The results are specialized for a homogeneous isotropic elastic solid undergoing infinitesimal strains (Section 5), the thermodynamic restriction of positive strain energy being relaxed. For the displacement boundary-value problem Boggio's (1907) extension of the classical uniqueness theorem of Kirchhoff is recovered by an automatic process, together with related extremum principles of Gurtin and Sternberg (1960). The traction boundary-value problem is re-examined. Plane strain and generalized plane stress are also treated in detail.

Stable Lattices

The consideration of relative extrema to correspond to the absolute extremum which is the critical lattice has been going on for some time. As far back as 1873, Korkine and Zolotareff [6] worked with the ellipsoid in hyperspace (i.e., with quadratic forms), and later Minkowski [8] worked with a general convex body in two or three dimensions. They showed how to find critical lattices by selection from among a finite number of relative extrema. They were aided by the long-recognized premise that only a finite number of lattice points can enter into consideration [1] when one deals with lattices “admissible to convex bodies.”

The Topology of the Level Curves of Harmonic Functions with Critical Points

Introduction. In a previous paper,2 of which this is a continuation, topological properties of curve families which filled the Euclidean plane 7r, or a simply connected domain in r, were investigated. The families were assumed regular (i. e. locally homeomorphic to parallel lines) except at a possibly infinite collection of isolated singularities at each of which the family had the structure of a multiple saddle point; such families were called branched regular curve families. Further investigation of these families, in particular their relation to harmonic functions, is the aim of this paper. In what follows the definitions and theorems in [I] will be assumed, and the same notation will be used. In particular F, G will denote branched regular curve families filling the plane 7r, B will denote the set of singular points, R the domain 7r B in which F is regular, and so on. The Euclidean plane will be taken as a model for all simply connected domains. The principal result of [I] was to prove that any branched regular curve family F filling Xr can be given as the family of level curves of a function f (p) which is continuous on all of 7r and has no relative extrema. This generalizes a portion of [II] in which the same theorem is proved for a curve family without singularities in 7r. In this paper there are two main results: the first, proved in Section 1, is that F is actually homeomorphic to the level curves of a harmonic function; the second, proved in Section 2, asserts the existence of a decomposition of F into a countable collection of subfamilies of curves, each of which has the structure of the parallel lines y=constant of the upper half-plane. Such subfamilies will be called half-parallel, and this decomposition has consequences for the study of harmonic functions and analytic functions which will be mentioned below. These two results generalize

The existence of an extremum in problems of Mayer

one which minimizes a function g [a, b, y(a), y(b) ]. All simple integral problems of the calculus of variations for which a fairly complete theory of the relative extremum problem has been developed may be transformed to a problem of the above type. The theorems we shall prove concerning the existence of an absolute minimum may be translated directly into corresponding existence theorems for the problem of Bolza or for the problem of Lagrange. The problem of Mayer is considered here because it seems notationally simpler than the problem of Bolza. An existence theorem for a complicated problem such as we are considering must naturally impose rather severe restrictions on the involved. In order to treat as wide a variety of cases as possible, the variable functions are divided into groups satisfying different types of conditions. We shall divide them notationally into two groups. The functions yi(x), defined for a ? x < b, will be regarded as independent functions, and the curve they determine in xy-space will be denoted by C. When the functions yi(x) are absolutely continuous, dependent functions z,(x) will be determined by differential equations and initial conditions of the special form

Relative Extrema of Pairs of Quadratic and Hermitian Forms

Relative Extrema of Pairs of Quadratic and Hermitian Forms