Characteristic functions on the boundary of a planar domain need not be traces of least gradient functions
Confluentes Mathematici, Tome 9 (2017) no. 1, pp. 65-93.

Given a smooth bounded planar domain Ø, we construct a compact set on the boundary such that its characteristic function is not the trace of a least gradient function. This generalizes the construction of Spradlin and Tamasan [3] when Ø is a disc.

Reçu le : 2016-10-15
Révisé le : 2017-02-27
Accepté le : 2017-02-28
Publié le : 2017-09-14
DOI : https://doi.org/10.5802/cml.36
Classification : 26B30,  35J56
Mots clés: traces of functions of bounded variation, least gradient problem
@article{CML_2017__9_1_65_0,
     author = {Micka\"el Dos Santos},
     title = {Characteristic functions on the boundary of a planar domain need not be traces of least gradient functions},
     journal = {Confluentes Mathematici},
     publisher = {Institut Camille Jordan},
     volume = {9},
     number = {1},
     year = {2017},
     pages = {65-93},
     doi = {10.5802/cml.36},
     language = {en},
     url = {cml.centre-mersenne.org/item/CML_2017__9_1_65_0/}
}
Dos Santos, Mickaël. Characteristic functions on the boundary of a planar domain need not be traces of least gradient functions. Confluentes Mathematici, Tome 9 (2017) no. 1, pp. 65-93. doi : 10.5802/cml.36. https://cml.centre-mersenne.org/item/CML_2017__9_1_65_0/

[1] G. Anzellotti; M. Giaquinta Funzioni BV e tracce, Rend. Sem. Mat. Univ. Padova, Tome 60 (1978), pp. 1-21

[2] E. Giusti Minimal surfaces and functions of bounded variation, Springer Science & Business Media, 1984 no. 80

[3] G. Spradlin; A. Tamasan Not All Traces on the Circle Come from Functions of Least Gradient in the Disk, Indiana Univ. Math. J., Tome 63 (2014) no. 3, pp. 1819-1837 | Article

[4] P. Sternberg; G. Williams; W. Ziemer Existence, uniqueness, and regularity for functions of least gradient., J. reine angew. Math., Tome 430 (1992), pp. 35-60