Stability of the solutions of one-sided variational problems with applications to the theory of plasticity

Abstract

In the paper one investigates the one-sided variational problem of the form $$/u - u_0 / = min, u \in M,$$ where ¦·¦ is the norm in some Hilbert space H0, ℳ is a nonempty convex set, closed in the metric of H0, and u0 is a given element of this space. The fundamental results are: 1) The solution of the problem (1) is stable relative to small perturbations of the data of this problem: the element u0, the norm ¦·¦, and the set ℳ the concept of small perturbation is precisely formulated. 2) Assume that the set ℳ is defined by the formula where g is an element of some Hilbert space containing the space H0, ||| · ||| is some seminorm, and a is a positive constant. Let H(n) be a subspace of the space H0 on all of whose elements the seminorm ||| · ||| is finite. If un is the approximate solution of the problem (1), obtained as the solution of the problem ¦u−u0¦=min, un ∈ M ∩ H(n), then ¦u*−un¦=0 (en(u*)), where u* is the exact solution of the problem (1), while en(u*) is the best approximation of u* by the elements of the subspace H(n). The given results are used in a series of problems regarding the elastoplastic state according to the Saint-Venant-Mises theory; one assume that for these problems the Haar-Karman variational principle holds.