UDC 517.97 V. V. ZHIKOV ABSTRACT. Duality methods in the theory of averaging of nonlinear variational problems are developed. The questions of a general nature that are discussed include a detailed analysis of the concept of regularity, an example of a nonregular Lagrangian, and the derivation of duality formulas that take account of the regularity problem.The main content is concerned with the averaging of variational problems with stochastic Lagrangians. Three groups of questions are investigated: 1) averaging of Lagrangians of a general form; 2) averaging of the Lagrangians of plasticity (the theory of the limit load); and 3) averaging of degenerate Lagrangians (problems with random soft or rigid inclusions).Bibliography: 13 titles.f f{x,\ni)dx = / f(x,vu)dx + ( \νΰ + Vzf 1 dx.

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