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Two dimensional variational problems with linear growth

Abstract. Suppose that f: ℝnN→ℝ is a strictly convex energy density of linear growth, f(Z)=g(|Z|2) if N>1. If f satisfies an ellipticity condition of the form then, following [Bi3], there exists a unique (up to a constant) solution of the variational problem provided that the given boundary data u0W11(ω;ℝN) are additionally assumed to be of class L∞(ω;ℝN). Moreover, if μ<3, then the boundedness of u0 yields local C1,α-regularity (and uniqueness up to a constant) of generalized minimizers of the problem In our paper we show that the restriction u0L∞(ω;ℝN) is superfluous in the two dimensional case n=2, hence we may prescribe boundary values from the energy class W11(ω;ℝN) and still obtain the above results.