BV Functions and Nonlocal Functionals in Metric Measure Spaces

Abstract We study the asymptotic behavior of three classes of nonlocal functionals in complete metric spaces equipped with a doubling measure and supporting a Poincaré inequality. We show that the limits of these nonlocal functionals are comparable to the total variation $$\Vert Df\Vert (\Omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>‖</mml:mo> <mml:mi>D</mml:mi> <mml:mi>f</mml:mi> <mml:mo>‖</mml:mo> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> or the Sobolev semi-norm $$\int _\Omega g_f^p\, d\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>∫</mml:mo> <mml:mi>Ω</mml:mi> </mml:msub> <mml:msubsup> <mml:mi>g</mml:mi> <mml:mi>f</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> <mml:mspace/> <mml:mi>d</mml:mi> <mml:mi>μ</mml:mi> </mml:mrow> </mml:math> , which extends Euclidean results to metric measure spaces. In contrast to the classical setting, we also give an example to show that the limits are not always equal to the corresponding total variation even for Lipschitz functions.