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

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, ? 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 ? the concept of small perturbation is precisely formulated. 2) Assume that the set ? 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 u_n is the approximate solution of the problem (1), obtained as the solution of the problem ¦u?u₀¦=min, u_n ∈ M ∩ H⁽ⁿ⁾, then ¦u_*?u_n¦=0 (e_n(u_*)), where u_* is the exact solution of the problem (1), while e_n(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.