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Analysis of total variation flow and its finite element approximations

Xiaobing Feng, Andreas Prohl (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ε, see (1.7)) and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since...

Asymmetric heteroclinic double layers

Michelle Schatzman (2002)

ESAIM: Control, Optimisation and Calculus of Variations

Let W be a non-negative function of class C 3 from ℝ 2 to ℝ , which vanishes exactly at two points 𝐚 and 𝐛 . Let S 1 ( 𝐚 , 𝐛 ) be the set of functions of a real variable which tend to 𝐚 at - ∞ and to 𝐛 at + ∞ and whose one dimensional energy E 1 ( v ) = ∫ ℝ W ( v ) + | v ' | 2 / 2 d x is finite. Assume that there exist two isolated minimizers z + and z - of the energy E 1 over S 1 ( 𝐚 , 𝐛 ) . Under a mild coercivity condition on the potential W and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator at z + and z - , it is possible to prove...

Asymmetric heteroclinic double layers

Michelle Schatzman (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Let W be a non-negative function of class C3 from 2 to , which vanishes exactly at two points a and b. Let S1(a, b) be the set of functions of a real variable which tend to a at -∞ and to b at +∞ and whose one dimensional energy E 1 ( v ) = ∫ W ( v ) + | v ' | 2 / 2 x is finite. Assume that there exist two isolated minimizers z+ and z- of the energy E1 over S1(a, b). Under a mild coercivity condition on the potential W and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator at z+ and...

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