Journal of Convex Analysis 25 (2018), No. 1, 219--223
Copyright 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
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