Existence of parabolic minimizers to the total variation flow on metric measure spaces
Existence of parabolic minimizers to the total variation flow on metric measure spaces
Abstract We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $$({\mathcal {X}}, d, \mu )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mo>,</mml:mo><mml:mi>μ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide …