We consider the gradient flow of a one-homogeneous functional, whose dual involves the derivative of a constrained scalar function. We show in this case that the gradient flow is related to a weak, generalized formulation of a Hele–Shaw flow. The equivalence follows from a variational representation, which is a variant of well-known variational representations for the Hele–Shaw problem. As a consequence we get existence and uniqueness of a weak solution to the Hele–Shaw flow. We also obtain an explicit representation for the Total Variation flow in dimension 1, and easily deduce basic qualitative properties, concerning in particular the "staircasing effect".
@article{CML_2011__3_4_617_0,
author = {Briani, Ariela and Chambolle, Antonin and Novaga, Matteo and Orlandi, Giandomenico},
title = {On the gradient flow of a one-homogeneous functional},
journal = {Confluentes Mathematici},
pages = {617--635},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {3},
number = {4},
year = {2011},
doi = {10.1142/S1793744211000461},
language = {en},
url = {http://www.numdam.org/articles/10.1142/S1793744211000461/}
}
TY - JOUR AU - Briani, Ariela AU - Chambolle, Antonin AU - Novaga, Matteo AU - Orlandi, Giandomenico TI - On the gradient flow of a one-homogeneous functional JO - Confluentes Mathematici PY - 2011 SP - 617 EP - 635 VL - 3 IS - 4 PB - World Scientific Publishing Co Pte Ltd UR - http://www.numdam.org/articles/10.1142/S1793744211000461/ DO - 10.1142/S1793744211000461 LA - en ID - CML_2011__3_4_617_0 ER -
%0 Journal Article %A Briani, Ariela %A Chambolle, Antonin %A Novaga, Matteo %A Orlandi, Giandomenico %T On the gradient flow of a one-homogeneous functional %J Confluentes Mathematici %D 2011 %P 617-635 %V 3 %N 4 %I World Scientific Publishing Co Pte Ltd %U http://www.numdam.org/articles/10.1142/S1793744211000461/ %R 10.1142/S1793744211000461 %G en %F CML_2011__3_4_617_0
Briani, Ariela; Chambolle, Antonin; Novaga, Matteo; Orlandi, Giandomenico. On the gradient flow of a one-homogeneous functional. Confluentes Mathematici, Tome 3 (2011) no. 4, pp. 617-635. doi : 10.1142/S1793744211000461. http://www.numdam.org/articles/10.1142/S1793744211000461/
[1] D. R. Adams and L. I. Hedberg, Functions Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften, Vol. 314 (Springer, 1996).
[2] W. K. Allard, Total variation regularization for image denoising. III. Examples, SIAM J. Imaging Sci. 2 (2009) 532–568.
[3] F. Alter, V. Caselles and A. Chambolle, A characterization of convex calibrable sets in RN , Math. Ann. 332 (2005) 329–366.
[4] S. Baldo, R. Jerrard, G. Orlandi and H. M. Soner, Convergence of Ginzburg–Landau functionals in 3-d superconductivity, arXiv:1102.4650.
[5] G. Bellettini, V. Caselles and M. Novaga, The total variation flow in RN , J. Differ- ential Equations 184 (2002) 475–525.
[6] Yu. K. Belyaev, Continuity and Hölder’s conditions for sample functions of stationary Gaussian processes, in Proc. 4th Berkeley Symp. Math. Statist. and Prob., Vol. II (University of California Press, 1961), pp. 22–33.
[7] M. Bonforte and A. Figalli, Total variation flow and sign fast diffusion in one dimen- sion, arXiv:1107.2153v2.
[8] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (North-Holland, 1973).
[9] L. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl. 4 (1998) 383–402.
[10] L. Caffarelli and A. Friedman, Continuity of the temperature in the Stefan problem, Indiana Univ. Math. J. 28 (1979) 53–70.
[11] C. M. Elliott and V. Janovsk´y, A variational inequality approach to Hele–Shaw flow with a moving boundary, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981) 93–107.
[12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, 1983).
[13] B. Gustafsson, Applications of variational inequalities to a moving boundary problem for Hele–Shaw flows, SIAM J. Math. Anal. 16 (1985) 279–300.
[14] K. Kielak, P. B. Mucha and P. Rybka, Almost classical solutions to the total variation flow, arXiv:1106.5369v1.
[15] C. I. Kim and A. Mellet, Homogenization of a Hele–Shaw problem in periodic and random media, Arch. Rat. Mech. Anal. 194 (2009) 507–530.
[16] D. Kinderlehrer and L. Nirenberg, The smoothness of the free boundary in the one phase Stefan problem, Comm. Pure Appl. Math. 31 (1978) 257–282.
[17] Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equa- tions, University Lecture Series, Vol. 22 (Amer. Math. Soc., 2001).
[18] W. Ring, Structural properties of solutions to total variation regularization problems, M2AN Math. Model. Numer. Anal. 34 (2000) 799–810.
[19] J. F. Rodrigues, Variational Methods in the Stefan Problem, Lecture Notes in Math- ematics, Vol. 1584 (Springer, 1994), pp. 147–212.
[20] E. Sandier and S. Serfaty, Vortices in the Magnetic Ginzburg–Landau Model, Progress in Nonlinear Differential Equations and their Applications, Vol. 70 (Birkhäuser, 2007).
Cité par Sources :
