The Total Variation Flow in RN

The purpose of this chapter is to prove existence and uniqueness of the minimizing total variation flow in ℝ N $$ \frac{{\partial u}} {{\partial t}} = div\left( {\frac{{Du}} {{\left| {Du} \right|}}} \right) in\left] {0,\infty } \right[ \times \mathbb{R}^N , $$ (3.1) coupled with the initial condition $$ u(0,x) = {u_0}(x){\text{ }}x \in {\mathbb{R}^N}, $$ (3.2) when u0 ∈ L loc 1 (L N ).

[1]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

[2]  Michael G. Crandall,et al.  GENERATION OF SEMI-GROUPS OF NONLINEAR TRANSFORMATIONS ON GENERAL BANACH SPACES, , 1971 .

[3]  E. Giusti.  On the equation of surfaces of prescribed mean curvature , 1978 .

[4]  G. Anzellotti,et al.  Pairings between measures and bounded functions and compensated compactness , 1983 .

[5]  Gian-Carlo Rota.  Opérateurs maximaux monotones: H. Brézis, North-Holland, 1983, 183 pp. , 1985 .

[6]  W. Ziemer.  Weakly differentiable functions , 1989 .

[7]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[8]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[9]  R. DeVore,et al.  Fast wavelet techniques for near-optimal image processing , 1992, MILCOM 92 Conference Record.

[10]  J. Carrillo.  On the uniqueness of the solution of the evolution dam problem , 1994 .

[11]  L. Hörmander.  Notions of Convexity , 1994 .

[12]  D. Donoho.  Nonlinear Solution of Linear Inverse Problems by Wavelet–Vaguelette Decomposition , 1995 .

[13]  I. Johnstone,et al.  Wavelet Shrinkage: Asymptopia? , 1995 .

[14]  David L. Donoho,et al.  De-noising by soft-thresholding , 1995, IEEE Trans. Inf. Theory.

[15]  J. Vázquez,et al.  An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations , 1995 .

[16]  G. Bellettini,et al.  Anisotropic motion by mean curvature in the context of Finsler geometry , 1996 .

[17]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .

[18]  J. M. Mazón,et al.  Existence and uniqueness for a degenerate parabolic equation with L1-data , 1999 .

[19]  R. DeVore,et al.  Nonlinear Approximation and the Space BV(R2) , 1999 .

[20]  Gene H. Golub,et al.  A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration , 1999, SIAM J. Sci. Comput..

[21]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[22]  S. Durand,et al.  IMAGE DEBLURRING, SPECTRUM INTERPOLATION AND APPLICATION TO SATELLITE IMAGING , 2000 .

[23]  Mila Nikolova,et al.  Local Strong Homogeneity of a Regularized Estimator , 2000, SIAM J. Appl. Math..

[24]  J. Morel,et al.  Connected components of sets of finite perimeter and applications to image processing , 2001 .

[25]  C. Ballester,et al.  The Dirichlet Problem for the Total Variation Flow , 2001 .

[26]  M. Novaga,et al.  On a Crystalline Variational Problem, Part I:¶First Variation and Global L∞ Regularity , 2001 .

[27]  Maurizio Paolini,et al.  Characterization of facet-breaking for nonsmooth mean curvature flow in the convex case , 2001 .

[28]  M. Novaga,et al.  On a Crystalline Variational Problem, Part II:¶BV Regularity and Structure of Minimizers on Facets , 2001 .