6533b7dcfe1ef96bd12729c3

RESEARCH PRODUCT

Minimizing total variation flow

Vicent CasellesFuensanta AndreuColoma BallesterJosé M. Mazón

subject

Dirichlet problem35K90Partial differential equationMeasurable functionApplied MathematicsMathematical analysis35B40Existence theorem35K65General Medicine35D0535K60Maxima and minimaUniqueness theorem for Poisson's equation35K55Neumann boundary conditionUniquenessAnalysisMathematics

description

We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in $L^1$. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect, and the solution converges to the spatial average of the initial datum as $t \to \infty$. We also prove that local maxima strictly decrease with time; in particular, flat zones immediately decrease their level. We display some numerical experiments illustrating these facts.

http://projecteuclid.org/euclid.die/1356123331