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Determination of the potential form of operators

100%

Telega J. J.

Mathematica Applicanda

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1982

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tom 10

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nr 18

PL

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EN

The author extends ideas of duality [see, for example, B. Noble and M. J. Sewell, J. Inst. Math. Appl. 9 (1972), 123–193; MR0307012] to a class of nonlinear operators on Banach spaces. Let U, V be Banach spaces and a(u,v) a bilinear form on U×V. Let N be a (nonlinear) operator N:U→V. GN(u)h denotes the Gâteaux derivative of N in the direction of h, computed at the point u∈U. Let us assume that a separates points in U×V (as defined by Marshall Stone). If there is v∈V such that a(h,v)=⟨h,Gf(u)⟩ for a functional f:U→R then v is called the gradient of f(u). The operator N is called potential if a suitable functional f satisfying this condition exists. The problem of symmetrizing N involves a suitable choice of the bilinear form a. For example, the operator N(u(t))=[(du/dt)2−g(t)] is not potential with respect to the usual L2 product. The author formulates a number of variational principles and discusses specific examples. This is an interesting article, supplementing the ideas of E. Tonti and of R. W. Atherton and G. M. Homsy [Studies in Appl. Math. 54 (1975), no. 1, 31–60; MR0458271].

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Two-dimensional examples of rank-one convex functions that are not quasiconvex

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Benaouda M. K., Telega J. J.

Annales Polonici Mathematici

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2000

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tom 73

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nr 3

291-295

EN

The aim of this note is to provide two-dimensional examples of rank-one convex functions which are not quasiconvex.

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Young measures and their applications in micromechanics and optimization. I. Mathematical principles.

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Bielski W., Kruglenko E., Telega J. J.

Mathematica Applicanda

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2003

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tom 31

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nr 45/04

PL

Artykuł nie zawiera streszczenia

EN

The paper is a review of modern mathematical methods for the analysis of continuous problems of nonlinear mechanics and magnetism. The bibliography contains 175 items. The second part of the paper deals with mechanical problems described by nonconvex density energy functions and with numerical methods to solve such problems.

Ograniczanie wyników

2 Mathematica Applicanda

1 Annales Polonici Mathematici

3 Telega J. J.

1 Benaouda M. K.

1 Bielski W.

1 Kruglenko E.

1 2003

1 2000

1 1982

Determination of the potential form of operators

Two-dimensional examples of rank-one convex functions that are not quasiconvex

Young measures and their applications in micromechanics and optimization. I. Mathematical principles.