Publication: Some qualitative properties for the total variation flow
Full text at PDC
Advisors (or tutors)
We prove the existence of a finite extinction time for the solutions of the Dirichlet problem for the total variation flow. For the Neumann problem, we prove that the solutions reach the average of its initial datum in finite time. The asymptotic profile of the solutions of the Dirichlet problem is also studied. It is shown that the profiles are nonzero solutions of an eigenvalue-type problem that seems to be unexplored in the previous literature. The propagation of the support is analyzed in the radial case showing a behaviour entirely different to the case of the problem associated with the p-Laplacian operator. Finally. the study of the radially symmetric case allows us to point out other qualitative properties that are peculiar of this special class of quasilinear equations.
L. Ambrosio, V. Caselles, S. Masnou, and J. M. Morel, Connected components of sets of finite perimeter and applications to image processing, European J. Appl. Math. 3 (2001), 39–92. L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, 2000. F. Andreu, C. Ballester, V. Caselles, and J. M. Mazón, Minimizing total variation flow, Differential Integral Equations 14 (2001), 321–360. F. Andreu, C. Ballester, V. Caselles, and J. M. Mazón, The Dirichlet problem for the total variation flow, J. Funct. Anal. 180 (2001), 347–403. S. N. Antonsev and J. I. Díaz, New results on space and time localization of solutions of nonlinear elliptic or parabolic equations via energy methods, Soviet Math. Dokl. 303 (1988), 524–528. S. N. Antonsev, J. I. Díaz, and S. I. Shmarev, "Energy Methods for Free Boundary Problems," Birkhäuser, Boston, 2001. G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 (1983), 293–318. A. Bamberger, Etude d'une équation doublement non linéaire, J. Funct. Anal. 24 (1977), 148–155. Ph. Benilan and M. G. Crandall, Regularizing effects in homogeneous equations, in "Contributions to Analysis Geometry" (D. N. Clark et al., Eds.), pp. 23–39, John Hopkins Univ. Press, Baltimore, 1981. Ph. Benilan and M. G. Crandall, Completely accretive operators, in "Semigroups Theory and Evolution Equations" (Ph. Clement et al., Eds.), pp. 41–76, Marcel Dekker, New York, 1991. J. G. Berryman and C. J. Holland, Stability of the separable solution for fast diffusion, Arch. Rational. Mech. Anal. 74 (1980), 279–288. H. Brezis, "Opérateurs Maximaux Monotones," North-Holland, Amsterdam, 1973. M. G. Crandall and T. M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265–298. J. I. Díaz and M. A. Herrero, Propriétés de support compact pour certaines équations elliptiques et parabolique non linéaires, C. R. Acad. Sci. Paris 286 (1978), 815–817. J. I. Díaz and M. A. Herrero, Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems, Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), 249–258. J. I. Díaz and A. Li~nán, Movimiento de descarga de gases en conductos largos: modelización y estudio de una ecuación doblemente no lineal, in "Reunión Matemática en Honor de A. Dou" (J. I. Díaz and J. M. Vegas, Eds.), pp. 95–119, Universidad Complutense de Madrid, Madrid, 1989. L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Math., CRC Press, Boca Raton, FL, 1992. E. Giusti, Boundary value problems for non-parametric surfaces of prescribed mean curvature, Ann. Scuola Norm. Sup. Pisa Sci. (4) 3 (1976), 501–548. R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys. 95 (1999), 1187–1220. R. Hardt and X. Zhou, An evolution problem for linear growth functionals, Comm. Partial Differential Equations 19 (1984), 1879–1907. M. Herrero and J. L. Vazquez, Asymptotic behaviour of the solutions of a strongly nonlinear parabolic problem, Ann. Fac. Sci. Toulouse 3 (1981), 113–127. M. Herrero and J. L. Vazquez, On the propagation properties of a nonlinear degenerate parabolic equation, Comm. Partial Differential Equations 7 (1982), 1381–1402. R. Kohn and R. Temam, Dual space of stress and strains with application to Hencky plasticity, Appl. Math Optm. 10 (1983), 1–35. L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), 259–268. L. Veron, Effects régularisantes des semi-groupes non-linéaires dans les espaces de Banach, Ann. Fac. Sci. Toulouse 1 (1979), 171–200. W. P. Ziemer, "Weakly Differentiable Functions," GTM 120, Springer-Verlag, Berlin, 1989.