Convex integration for Lipschitz mappings and counterexamples to regularity

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We study Lispchitz solutions of partial differential relations ∇ u∈ K, where u is a vector-valued function in an open subset of Rⁿ. In some cases the set of solutions turns out to be surprisingly large. The general theory is then used to construct counter-examples to regularity of solutions of Euler-Lagrange systems satisfying classical ellipticity conditions.