Abstract
AbstractWe study the set of possible traces of anisotropic least gradient functions. We show that even on the unit disk it changes with the anisotropic norm: for two sufficiently regular strictly convex norms the trace spaces coincide if and only if the norms coincide. The example of a function in exactly one of the trace spaces is given by a characteristic function of a suitably chosen Cantor set.
Publisher
Springer Science and Business Media LLC
Reference19 articles.
1. Amar, M., Bellettini, G.: A notion of total variation depending on a metric with discontinuous coefficients. Ann. Inst. Henri Poincaré Analyse Non Linéaire 11, 91–133 (1994)
2. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs (2000)
3. Bombieri, E., De Giorgi, E., Giusti, E.: Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969)
4. Dos Santos, M.: Characteristic functions on the boundary of a planar domain need not be traces of least gradient functions. Conflu. Math. 9(1), 65–93 (2017)
5. Dweik, S., Santambrogio, F.: $$L^p$$ bounds for boundary-to-boundary transport densities, and $$W^{1,p}$$ bounds for the BV least gradient problem in 2D. Calc. Var. Partial Differ. Equ. 58(1), Art. 31 (2019)