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W 1,p-REDUCTION PRINCIPLE AND EXISTENCE FOR PARTIAL DIFFERENTIAL RELATIONS

We study the solvability of special partial differential relations of the form Du(x) ∈ K for u ∈ W (Ω;R), where Ω ⊂ R is a bounded domain, Du(x) is the Jacobi matrix of u, and K is a given set of m × n matrices. In this note, we present some results obtained in the paper [16] on the W -reduction principles and a general existence theorem that generalize the similar reduction principles for Lipschitz solutions in [11]. Our methods are different and rely on a new Baire’s category argument concerning the residual continuity of a Baireone function. We also present some new results related to the special weakly quasiconformal mappings.