Abstract
The paper examines the one-dimensional total variation flow equation with Dirichlet boundary conditions. Thanks to a new concept of “almost classical” solutions we are able to determine evolution of facets – flat regions of solutions. A key element of our approach is the natural regularity determined by the nonlinear elliptic operator, for which x 2 is an example of an irregular function. Such a point of view allows us to construct solutions. We apply this idea to numerical simulations for typical initial data. Due to the nature of Dirichlet data, any monotone function is an equilibrium. We prove that each solution reaches such a steady state in finite time.
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References
W.K. Allard, Total Variation Regularization for Image Denoising, I. Geometric theory, SIAM J. Math. Anal. 39 (2007/08), (4), 1150–1190.
Andreu F., Ballester C., Caselles V., Mazón J.M.: The Dirichlet problem for the total variation flow. J. Funct. Anal. 180, 347–403 (2001)
Andreu F., Ballester C., Caselles V., Mazón J.M.: Minimizing total variation flow. Diff. Int. Equations 14, 321–360 (2001)
Angenent S., Gurtin M.E.: Multiphase thermomechanics with interfacial structure. II. Evolution of a isothermal interface. Arch. Rational Mech. Anal. 108(4), 323–391 (1989)
F. Andreu-Vaillo, V. Caselles, V., J.M. Mazón, “Parabolic quasilinear equations minimizing linear growth functionals”. Birkhäuser, Basel, (2004).
Andreu F., Caselles V., Mazón J.M., Moll S.: The Dirichlet problem associated to the relativistic heat equation. Math. Ann. 347, 135–199 (2010)
Andreu F., Caselles V., Díaz J.I., Mazón J.M.: Some qualitative properties for the total variation flow. J. Funct. Anal. 188, 516–547 (2002)
G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 (1983), 293–318.
H. Attouch, G. Buttazzo, G. Michaille, “Variational analysis in Sobolev and BV spaces. Applications to PDEs and optimization”. MPS/SIAM Series on Optimization 6. Philadelphia, PA, SIAM. Philadelphia, 2006.
V. Barbu, “Nonlinear semigroups and differential equations in Banach spaces”, Leyden, The Netherlands, Noordhoff International Publishing, 1976.
Belik P., Luskin M.: A total-variation surface energy model for thin films of martensitic crystals. Interfaces Free Bound. 4, 71–88 (2002)
Bellettini G., Caselles V., Novaga M.: The total variation flow in \({\mathbb{R}^N}\) . J. Diff. Eqns. 184, 475–525 (2002)
Bonforte M., Figalli A.: Total Variation Flow and Sign Fast Diffusion in one dimension. J. Differential Equations 252, 4455–4480 (2012)
Briani A., Chambolle A., Novaga M., Orlandi G.: On the gradient flow of a one-homogeneous functional. Confluentes Math. 3, 617–635 (2011)
H. Brézis, “Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert”, Amsterdam-London, North-Holland Publishing, 1973.
Chang K.-Ch.: Variational methods for non-differentiable functionals and their applications to Partial Differential Equations. J. Math. Anal. Appl. 80, 102–129 (1981)
Crandall M.G., Liggett T.M.: Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math. 93, 265–298 (1971)
diBlasio G.: Differtiability of Spatially Homogeneous Solution of the Boltzmann Equation in the Non Maxwellian Case. Commun. Math. Phys. 38, 331–340 (1974)
Feng X., Oehsen M., Prohl A.: Rate of convergence of regularization procedures and finite element approximations for the total variation flow. Numer. Math. 100(3), 441–456 (2005)
T. Fukui, Y. Giga, Motion of a graph by nonsmooth weighted curvature, in: World Congress of Nonlinear Analysts ’92, Vol. I–IV (Tampa, FL, 1992), 47–56, de Gruyter, Berlin, 1996.
Giga M.-H., Giga Y.: Very singular diffusion equations: second and fourth order problems, Japan J. Indust. Appl. Math. 27, 323–345 (2010)
Giga M.-H., Giga Y.: Evolving graphs by singular weighted curvature. Arch. Rational Mech. Anal. 141(2), 117–198 (1998)
Giga M.-H., Giga Y.: Stability for evolving graphs by nonlocal weighted curvature. Comm. Partial Differential Equations 24(1–2), 109–184 (1999)
M.-H. Giga, Y. Giga, Yoshikazu, H.Hontani, Self-similar expanding solutions in a sector for a crystalline flow, SIAM J. Math. Anal., 37 (4), (2006), 1207–1226.
M.-H. Giga, Y. Giga, R. Kobayashi, Very Singular Diffusion Equations, Advanced Studies in Pure Mathematics 31, 2001 Taniguchi Conference on Mathematics Nara ’98 93–125.
M.-H. Giga, Y. Giga, P. Rybka, A comparison principle for singular diffusion equations with spatially inhomogeneous driving force, preprint.
Kobayashi R., Giga Y.: Equations with Singular Diffusivity. Journal of Statistical Physics 95(5/6), 1197–1220 (1999)
Mucha P.B.: Global existence for the Einstein-Boltzmann equation in the flat Robertson-Walker spacetime. Comm. Math. Phys. 203(1), 107–118 (1999)
Mucha P.B., Rybka P.: A new look at equilibria in Stefan type problems in the plane. SIAM J. Math. Anal. 39(4), 1120–1134 (2007)
Mucha P.B., Rybka P.: A caricature of a singular curvature flow in the plane. Nonlinearity 21, 2281–2316 (2008)
P.B. Mucha, P. Rybka, Almost classical solutions of static Stefan type problems involving crystalline curvature, in: “Nonlocal and Abstract Parabolic Equations and their Applications”, Banach Center Publ. 86, IMPAN, Warszawa, 2009, 223–234.
J. Quah, D. Margetis, Anisotropic diffusion in continuum relaxation of stepped crystal surfaces, J. Phys. A: Math. Theor. 41 (2008) 235004 (18pp).
Ring W.: Structural properties of solutions to total variation regularization problems, M2AN Math. Model. Numer. Anal. 34(4), 799–810 (2000)
Rudin L.I., Osher S., Fatemi E.: Nonlinear total variation based noise removal argorithms. Physica D 60, 259–268 (1992)
Struwe M.: “Variational Methods”. Springer, New York (1990)
J. Taylor, Motion of curves by crystalline curvature, including triple junctions and boundary points, In: Differential Geometry: Partial Differential Equations on Manifolds (eds. R. Greene and S. T. Yau), Proc. Symp. Pure Math., 54 (1993) Part I, pp.417–438, Amer. Math. Society Providence, RI.
Tsai Y.H.R., Giga Y., Osher S.: A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations. Math. Comp. 72(241), 159–181 (2003)
Ziemer W.P.: “Weakly differentiable functions”. Springer, New York (1989)
Acknowledgments
PR thanks Professor José Mazón for inspiring conversations on the topic of this paper. Both authors thank referees for precious comments improving the final version of the paper. In particular, one of the referees suggested the presented proof of Proposition 2.1. Special thanks go to the Iberia airline for creating extra opportunities to work on this paper and on related topics. After the submission of the paper, PR and PBM were informed by authors of [13] about their results concerning qualitative analysis around facets in 1d TVF. The present work has been partly supported by MN grant No. N N201 268935.
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Kielak, K., Mucha, P.B. & Rybka, P. Almost classical solutions to the total variation flow. J. Evol. Equ. 13, 21–49 (2013). https://doi.org/10.1007/s00028-012-0167-x
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DOI: https://doi.org/10.1007/s00028-012-0167-x