Two dimensional variational problems with linear growth

Abstract.

 Suppose that f: ℝ^nN→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 u ₀ W ¹ ₁ (ω;ℝ^N) are additionally assumed to be of class L ^∞(ω;ℝ^N). Moreover, if μ<3, then the boundedness of u ₀ yields local C ^1,α-regularity (and uniqueness up to a constant) of generalized minimizers of the problem

In our paper we show that the restriction u ₀L ^∞(ω;ℝ^N) is superfluous in the two dimensional case n=2, hence we may prescribe boundary values from the energy class W ¹ ₁ (ω;ℝ^N) and still obtain the above results.