Differential and Integral Equations

Minimizing total variation flow

F. Andreu, C. Ballester, V. Caselles, and J. M. Mazón

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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.

Article information

Differential Integral Equations, Volume 14, Number 3 (2001), 321-360.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions 35D05 35K60: Nonlinear initial value problems for linear parabolic equations 35K65: Degenerate parabolic equations 35K90: Abstract parabolic equations


Andreu, F.; Ballester, C.; Caselles, V.; Mazón, J. M. Minimizing total variation flow. Differential Integral Equations 14 (2001), no. 3, 321--360. https://projecteuclid.org/euclid.die/1356123331

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