example of a BV function which is not W1,1

The following example presents a function uBV(Ω)W1,1(Ω).

Example 1.

Let Ω:=(-1,1)×(-1,1)2. We will show that the function

u(x,y)={1,if x00,if x<0

belongs to BV(Ω). Given ϕCc1(Ω,2), ϕ=(ϕ1,ϕ2), one has

Ωu(x,y)divϕ(x,y)𝑑x𝑑y =-11[01ϕx1(x,y)𝑑x]𝑑y+01[-11ϕy2(x,y)𝑑y]𝑑x

if we choose μ:=(μ1,μ2):=(1({0}×(-1,1)),0). So we notice that uBV(Ω) and Du=μ is singularPlanetmathPlanetmath with respect to the Lebesgue measureMathworldPlanetmath .

Title example of a BV function which is not W1,1
Canonical name ExampleOfABVFunctionWhichIsNotW11
Date of creation 2013-03-22 15:12:59
Last modified on 2013-03-22 15:12:59
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 5
Author paolini (1187)
Entry type Example
Classification msc 26B30