In this paper, we consider a convex function defined as a 1D-regularized total variation with nonhomogeneous coefficients, and prove the Main Theorem concerned with the decomposition of the subdifferential of this convex function to a weighted singular diffusion and a linear regular diffusion. The Main Theorem will be to enhance the previous regularity result for quasilinear equation with singularity, and moreover, it will be to provide some useful information in the advanced mathematical studies of grain boundary motion, based on KWC type energy.

Shodai Kubota, PhD Student, Department of Mathematics and Informatics, Graduate School of Science and Engineering, Chiba University, 1–33, Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan, tel: +81 (0) 43-290-2665, email: skubota@chiba-u.jp

Kubota S. Subdifferential Decomposition of 1D-regularized Total Variation with Nonhomogeneous Coefficients. The Bulletin of Irkutsk State University. Series Mathematics, 2021, vol. 36, pp. 69-83. https://doi.org/10.26516/1997-7670.2021.36.69

1. Ambrosio L., Fusco N., Pallara D. Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. New York, The Clarendon Press, Oxford University Press, 2000. http://doi.org/10.1017/S0024609301309281
2. Attouch H. Variational Convergence for Functions and Operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA, 1984. http://doi.org/10.1112/blms/18.2.222
3. Barbu V. Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics. New York, Springer, 2010. http://doi.org/10.1007/978-1-4419-5542-5
4. Bellettini G., Bouchitt´e G., Fragal`a I. BV functions with respect to a measure and relaxation of metric integral functionals. J. Convex Anal., 1999, vol. 6, no. 2, pp. 349-366.
5. Br´ezis H. Op´erateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland Publishing Co., Amsterdam-London, American Elsevier Publishing Co., Inc., New York, 1973, North-Holland Mathematics Studies, no. 5. Notas de Matem´atica (50).
6. Colli P., Gilardi G., Nakayashiki R., Shirakawa K. A class of quasi-linear Allen–Cahn type equations with dynamic boundary conditions. Nonlinear Anal., 2017, vol. 158, pp. 32-59. http://doi.org/10.1016/j.na.2017.03.020
7. Ekeland I., T´emam R. Convex analysis and variational problems, 1999, vol. 28 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, english edition. Translated from the French. http://doi.org/10.1137/1.9781611971088
8. Giga Y., Kashima Y., Yamazaki N. Local solvability of a constrained gradient system of total variation. Abstr. Appl. Anal., 2004, vol. 8, pp. 651-682. http://doi.org/10.1155/S1085337504311048
9. Kobayashi R., Warren J.A., Carter W.C. A continuum model of grain boundaries. Phys. D, 2000, vol. 140, no. 1-2, pp. 141-150. http://doi.org/10.1016/S0167-2789(00)00023-3
10. Ladyˇzenskaja O.A., Solonnikov V.A., Ural’ceva N.N. Linear and Quasilinear Equations of Parabolic Type, 1968, vol. 23 of Translations of Mathematical Monographs. American Mathematical Society, Providence, R.I. http://doi.org/10.1090/mmono/023
11. Mosco U. Convergence of convex sets and of solutions of variational inequalities. Advances in Math., 1969, vol. 3, pp. 510-585, http://doi.org/10.1016/0001-8708(69)90009-7
12. Mucha P.B., Rybka P. Well posedness of sudden directional diffusion equations. Math. Methods Appl. Sci., 2013, vol. 36, no. 17, pp. 2359–2370. http://doi.org/10.1002/mma.2759
13. Shirakawa K., Watanabe H., Yamazaki N. Phase-field systems for grain boundary motions under isothermal solidifications. Adv. Math. Sci. Appl., 2014, vol. 24, no. 2, pp. 353-400.
14. Watanabe H., Shirakawa K. Energy-dissipation in a coupled system of Allen-Cahn-type equation and Kobayashi-Warren-Carter-type model of grain boundary motion. Math. Methods Appl. Sci., 2020, vol. 43, no. 17, pp. 10138-10167. http://doi.org/10.1002/mma.6684