Skip to main content
SpringerLink
Log in

Weak Formulation for Singular Diffusion Equation with Dynamic Boundary Condition

  • Chapter
  • First Online:
Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs

Part of the book series: Springer INdAM Series ((SINDAMS,volume 22))

  • 641 Accesses

Abstract

In this paper, we propose a weak formulation of the singular diffusion equation subject to the dynamic boundary condition. The weak formulation is based on a reformulation method by an evolution equation including the subdifferential of a governing convex energy. Under suitable assumptions, the principal results of this study are stated in forms of Main Theorems A and B, which are respectively to verify: the adequacy of the weak formulation; the common property between the weak solutions and those in regular problems of standard PDEs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York (2006)

    MATH  Google Scholar 

  2. Andreu, F., Ballester, C., Caselles, V., Mazón, J.M.: The Dirichlet problem for the total variation flow. J. Funct. Anal. 180(2), 347–403 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Anzellotti, G.: The Euler equation for functionals with linear growth. Trans. Am. Math. Soc. 290, 483–501 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  4. Attouch, H.: Variational Convergence for Functions and Operators. In: Applicable Mathematics Series. Pitman, Boston (1984)

    MATH  Google Scholar 

  5. Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. In: MPS-SIAM Series on Optimization, vol. 6. SIAM/MPS, Philadelphia (2006)

    Google Scholar 

  6. Barbu, V.: Nonlinear Differential Equations of Monotone Type in Banach Spaces. In: Springer Monographs in Mathematics. Springer, New York (2010)

    Book  MATH  Google Scholar 

  7. Brézis, H.: Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. In: North-Holland Mathematics Studies, vol. 5, Notas de Matemática (50). North-Holland/American Elsevier Publishing, Amsterdam (1973)

    Google Scholar 

  8. Calatroni, L., Colli, P.: Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions. Nonlinear Anal. 79, 12–27 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Colli, P., Fukao, T.: The Allen–Cahn equation with dynamic boundary conditions and mass constraints. Math. Methods Appl. Sci. 38, 3950–3967 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  10. Colli, P., Sprekels, J.: Optimal control of an Allen–Cahn equation with singular potentials and dynamic boundary condition. SIAM J. Control Optim. 53, 213–234 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  11. Colli, P., Gilardi, G., Nakayashiki, R., Shirakawa, K.: A class of quasi-linear Allen–Cahn type equations with dynamic boundary conditions. Nonlinear Anal. 158, 32–59 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  12. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, Revised edn. In: Textbooks in Mathematics. CRC Press, Boca Raton (2015)

    Google Scholar 

  13. Gagliardo, E.: Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. Mat. Univ. Padova (Italian) 27, 284–305 (1957)

    MATH  Google Scholar 

  14. Gal, C.G., Grasselli, M., Miranville, A.: Nonisothermal Allen-Cahn equations with coupled dynamic boundary conditions. In: Nonlinear phenomena with energy dissipation. GAKUTO International Series. Mathematical Sciences and Applications, vol. 29, pp. 117–139. Gakktōsho, Tokyo (2008)

    Google Scholar 

  15. Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. In: Monographs in Mathematics, vol. 80. Birkhäuser, Cambridge, MA (1984)

    Google Scholar 

  16. Ito, A., Kenmochi, N., Yamazaki, N.: A phase-field model of grain boundary motion. Appl. Math. 53(5), 433–454 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ito, A., Kenmochi, N., Yamazaki, N.: Weak solutions of grain boundary motion model with singularity. Rend. Mat. Appl. (7) 29(1), 51–63 (2009)

    Google Scholar 

  18. Kenmochi, N.: Solvability of nonlinear evolution equations with time-dependent constraints and applications. Bull. Fac. Educ. Chiba Univ. 30, 1–87 (1981). http://ci.nii.ac.jp/naid/110004715232

    MATH  Google Scholar 

  19. Kenmochi, N., Mizuta, Y., Nagai, T.: Projections onto convex sets, convex functions and their subdifferentials. Bull. Fac. Educ. Chiba Univ. 29, 11–22 (1980). http://ci.nii.ac.jp/naid/110004715212

    Google Scholar 

  20. Kobayashi, R., Warren, J.A., Carter, W.C.: A continuum model of grain boundaries. Phys. D 140(1–2), 141–150 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kobayashi, R., Warren, J.A., Carter, W.C.: Grain boundary model and singular diffusivity. In: Free Boundary Problems: Theory and Applications. GAKUTO International Series. Mathematical Sciences and Applications, vol. 14, pp. 283–294. Gakkōtosho, Tokyo (2000)

    Google Scholar 

  22. Moll, J.S.: The anisotropic total variation flow. Math. Ann. 332(1), 177–218 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Moll, S., Shirakawa, K.: Existence of solutions to the Kobayashi-Warren-Carter system. Calc. Var. Partial Differ. Equ. 51, 621–656 (2014). doi:10.1007/ s00526-013-0689-2

    Article  MATH  MathSciNet  Google Scholar 

  24. Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3, 510–585 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  25. Savaré, G., Visintin, A.: Variational convergence of nonlinear diffusion equations: applications to concentrated capacity problems with change of phase. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 8(1), 49–89 (1997)

    Google Scholar 

  26. Shirakawa, K., Watanabe, H., Yamazaki, N.: Solvability of one-dimensional phase field systems associated with grain boundary motion. Math. Ann. 356, 301–330 (2013). doi:10.1007/s00208-012-0849-2

    Article  MATH  MathSciNet  Google Scholar 

  27. Shirakawa, K., Watanabe, H., Yamazaki, N.: Phase-field systems for grain boundary motions under isothermal solidifications. Adv. Math. Sci. Appl. 24, 353–400 (2014)

    MATH  MathSciNet  Google Scholar 

  28. Temam, R.: On the continuity of the trace of vector functions with bounded deformation. Appl. Anal. 11, 291–302 (1981)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This paper is dedicated to Professor Gianni Gilardi on the occasion of his 70th birthday. On a final note, we appreciate very much to the anonymous referee for taking great efforts to review our manuscript, and for giving us a lot of valuable comments and remarks. Ken Shirakawa is supported by Grant-in-Aid No. 16K05224, JSPS.

Author information

Authors and Affiliations

  1. Department of Mathematics and Informatics, Graduate School of Science, Chiba University, 1-33, Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan

    Ryota Nakayashiki

  2. Department of Mathematics, Faculty of Education, Chiba University, 1-33, Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan

    Ken Shirakawa

Authors
  1. Ryota Nakayashiki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ken Shirakawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ken Shirakawa .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Pierluigi Colli

  2. Dipartimento di Matematica, Università di Bologna , Bologna, Italy

    Angelo Favini

  3. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Elisabetta Rocca

  4. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Giulio Schimperna

  5. für Angewandte Analysis und Stochastik (WIAS), Weierstraß-Institut , Berlin, Germany

    Jürgen Sprekels

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Nakayashiki, R., Shirakawa, K. (2017). Weak Formulation for Singular Diffusion Equation with Dynamic Boundary Condition. In: Colli, P., Favini, A., Rocca, E., Schimperna, G., Sprekels, J. (eds) Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs. Springer INdAM Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-64489-9_16

Download citation

Publish with us

Policies and ethics

Search

Navigation