Poincaré Trace Inequalities in BV(B^n) with Non-standard Normalization

Extremal functions are exhibited in Poincaré trace inequalities for functions of bounded variation in the unit ball B^n of the n-dimensional Euclidean space ℝ^n. Trial functions are subject to either a vanishing mean value condition, or a vanishing median condition in the whole of B^n, instead of just on ∂B^n, as customary. The extremals in question take a different form, depending on the constraint imposed. In particular, under the latter constraint, unusually shaped extremal functions appear. A key step in our approach is a characterization of the sharp constant in the relevant trace inequalities in any admissible domain Ω⊂ℝ^n, in terms of an isoperimetric inequality for subsets of Ω.