Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 25 (2018), No. 1, 219--223Copyright Heldermann Verlag 2018 Some Remarks on the (Non-) Attainment of the Boundary Data for Variational Problems in the Space BV Michael Bildhauer Fachbereich Mathematik, Universität des Saarlandes, Postfach 15 11 50, 66041 Saarbrücken, Germany bibi@math.uni-sb.de Martin Fuchs Fachbereich Mathematik, Universität des Saarlandes, Postfach 15 11 50, 66041 Saarbrücken, Germany fuchs@math.uni-sb.de [Abstract-pdf] We discuss the standard relaxed version of a minimization problem for variational integrals of linear growth together with prescribed Dirichlet boundary data $u_0$ and give estimates for the size of the set $\{x \in \partial \Omega : u (x) \not= u_0 (x)\}$ for BV-minimizers $u$ which imply $${\cal{H}}^{n -1} \left(\left\{x \in \partial \Omega : u (x) < u_0 (x)\right\}\right) = {\cal{H}}^{n - 1} \left(\left\{x \in \partial \Omega : u (x) > u_0 (x) \right\}\right)$$ in the case of minimal surfaces $u$ not attaining the boundary values $u_0$ on a subset of $\partial \Omega$ with positive measure. Keywords: Variational problems of linear growth, boundary behaviour, minimal surfaces. MSC: 49J40, 49J45, 49Q05 [ Fulltext-pdf  (75  KB)] for subscribers only.