On the Dirichlet problem for variational integrals in BV

Lisa Beck, Thomas Schmidt
2013 Journal für die Reine und Angewandte Mathematik  
We investigate the Dirichlet problem for multidimensional variational integrals with linear growth which is formulated in a generalized way in the space of functions of bounded variation. We prove uniqueness of minimizers up to additive constants and deduce additional assertions about these constants and the possible (non-)attainment of the boundary values. Moreover, we provide several related examples. In the case of the model integral dx for w : R n I W ! R N our results extend classical
more » ... ts from the scalar case N ¼ 1-where the problem coincides with the non-parametric least area problem-to the general vectorial setting N A N. 11) In the proof of Lemma 2.10-as in the whole paper-we have suppressed an explicit notation for the continuous linear trace operator T. However, it should be noted that working withũ u on qW we are implicitly using T maxfu; vg ¼ maxfTu; Tvg on qW. To establish this equality one first proves Tju À vj ¼ jTðu À vÞj (approximating u À v with continuous functions on W) and then writes 2 maxfu; vg ¼ u þ v þ ju À vj.
doi:10.1515/crelle.2011.188 fatcat:v6b2bcnrw5bsrnnkmzhrri7cou