Skip to main content
SpringerLink
Log in

Finding Multiple Solutions to Elliptic PDE with Nonlinear Boundary Conditions

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, in order to solve an elliptic partial differential equation with a nonlinear boundary condition for multiple solutions, the authors combine a minimax approach with a boundary integral-boundary element method, and identify a subspace and its special expression so that all numerical computation and analysis can be carried out more efficiently based on information of functions only on the boundary. Some mathematical justification of the new approach is established. An efficient and reliable local minimax-boundary element method is developed to numerically search for solutions. Details on implementation of the algorithm are also addressed. The existence and multiplicity of solutions to the problem are established under certain regular assumptions. Some conditions related to convergence of the algorithm and instability of solutions found by the algorithm are verified. To illustrated the method, numerical multiple solutions to some examples on domains with different geometry are displayed with their profile and contour plots.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Amann, H., Fila, M.: A Fujita-type theorem for the Laplace equation with a dynamic boundary condition. Acta Math. Univ. Comenianae 2, 321–328 (1997)

    MathSciNet  Google Scholar 

  2. Ando, T., Fowleer, A.B., Stern, F.: Electronic properties of two dimensional system. Rev. Mod. Phys. 54, 437–621 (1982)

    Article  Google Scholar 

  3. Armontano, M.G.: The effect of reduced integration in the Steklov eigenvalue problem. Math. Model. Numer. Anal. 38, 27–36 (2004)

    Article  Google Scholar 

  4. Auchmuty, G.: Steklov eigenproblems and the representation of solutions of elliptic boundary value problems. Numer. Func. Anal. Optim. 25, 321–348 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Auchmuty, G.: Spectral characterization of the trace spaces \(H^s(\partial \Omega )\). SIAM J. Math. Anal. 38, 894–905 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Atkinson, K.T.: The numerical solution of a nonlinear boundary integral equation on smooth surface. IMA J. Numer. Anal. 14, 461–483 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Aziz, A.K., Dorr, M.R., Kellogg, R.B.: A new approximation mehtod for the Helmholtz equation in an exterior domain. SIAM J. Num. Anal. 19, 899–908 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bartsch, T., Chang, K.-C., Wang, Z.-Q.: On the Morse indices of sign changing solutions of nonlinear elliptic problems. Math. Z. 233, 655–677 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bartsch, T., Liu, Z.L., Weth, T.: Sign changing solutions of superlinear Schrödinger equations. Comm. Partial Differ. Equ. 29, 25–42 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bartsch, T., Wang, Z.-Q.: On the existence of sign changing solutions for semilinear Dirichlet problems. Topol. Meth. Nonl. Anal. 7, 115–131 (1996)

    MathSciNet  MATH  Google Scholar 

  11. Bialecki, R., Nowak, A.J.: Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions. Appl. Math. Model. 5, 417–421 (1981)

    Article  MATH  Google Scholar 

  12. Cabré, X., Tan, J.: Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224, 2052–2093 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Partial Diff. Equ. 32, 1245–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cushing, J.M.: Nonlinear Steklov problems on the unit circle II–a hydro-dynamical application. JMAA 39, 267–278 (1972)

    MathSciNet  MATH  Google Scholar 

  15. Chen, G., Zhou, J.: Boundary Element Methods. Academic Press, London-San Diego (1992)

    MATH  Google Scholar 

  16. Fasino, D., Inglese, G.: Recovering unknown terms in a nonlinear boundary condition for Laplace’s equation. IMA J. Appl. Math. 71, 832–852 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fila, M., Quittner, P.: Global solutions of the Laplace equation with a nonlinear dynamical boundary condition. Math. Methods Appl. Sci. 20, 1325–1333 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ganesh, M.: A BIE method for a nonlinear BVP. J. Comput. Appl. Math. 45, 299–308 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kavian, O., Vogelius, M.: On the existence and ‘blow-up‘ of solutions to a two-dimensional nonlinear boundary-value problem arising in corrosion modelling. Proceedings of the Royal Society of Edinburgh 133A, 119–149 (2003)

    Article  MathSciNet  Google Scholar 

  20. Li, S.J., Wang, Z.-Q.: Ljusternik-Schnirelman theory in partially ordered Hilbert spaces. Trans. Amer. Math. Soc. 354, 3207–3227 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, Y., Zhou, J.: A minimax method for finding multiple critical points and its applications to semilinear elliptic PDEs. SIAM J. Sci. Comp. 23, 840–865 (2001)

    Article  MATH  Google Scholar 

  22. Li, Y., Zhou, J.: Convergence results of a local minimax method for finding multiple critical points. SIAM J. Sci. Comp. 24, 865–885 (2002)

    Article  MATH  Google Scholar 

  23. Liu, Z., Wang, Z.-Q.: Sign-changing solutions of nonlinear elliptic equations. Frontiers Math. China 3, 1–18 (2008)

    Article  Google Scholar 

  24. Medville, K., Vogelius, M.: Existence and blow up of solutions to certain classes of two-dimensional nonlinear Neumann problems. Ann. I. H. Poincare 23, 499–538 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rabinowitz, P.H.: Minimax Method in Critical Point Theory with Applications to Differential Equations, CBMB Reginal Conference Series in Mathematics, vol. 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1986)

  26. Ruotsalainen, K., Saramen, J.: On the collocation method for a nonlinear boundary integral equation. J. Comput. Appl. Math. 28, 339–348 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ruotsalainen, K., Wendland, W.L.: On the boundary element method for some nonlinear boundary value problems. Numer. Math. 53, 299–314 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  28. Vitillaro, E.: On the Laplace equation with non-linear dynamic boundary conditions. Proc. London Math. Soc. 93, 418–446 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Vogelius, M., Xu, J.-M.: A nonlinear elliptic boundary value problem related to corrosion modeling. Q. Appl. Math. 56, 479–505 (1998)

    MathSciNet  MATH  Google Scholar 

  30. Wang, Z.-Q.: On a superlinear elliptic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 8, 43–57 (1991)

    MATH  Google Scholar 

  31. Wang, Z.-Q.: Minimax methods, invariant sets, and applications to nodal solutions of nonlinear elliptic problems, Proceedings of EquaDiff 03, International Conference on Differential Equations, Hasselt: World Scientific. Singapore 2005, 561–566 (2003)

  32. Wang, Z.Q., Zhou, J.: An efficient and stable method for computing multiple saddle points with symmetries. SIAM J. Num. Anal. 43, 891–907 (2005)

    Article  MATH  Google Scholar 

  33. Wang, Z.Q., Zhou, J.: A local minimax-Newton method for finding critical points with symmetries. SIAM J. Num. Anal. 42, 1745–1759 (2004)

    Article  MATH  Google Scholar 

  34. Yao, X., Zhou, J.: A minimax method for finding multiple critical points in Banach spaces and its application to quasilinear elliptic PDE. SIAM J. Sci. Comp. 26, 1796–1809 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhou, J.: A local min-orthogonal method for finding multiple saddle points. JMAA 291, 66–81 (2004)

    MATH  Google Scholar 

  36. Zhou, J.: Instability analysis of saddle points by a local minimax method. Math. Comp. 74, 1391–1411 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhou, J.: Global sequence convergence of a local minimax method for finding multiple solutions in Banach Spaces. Num. Funct. Anal. Optim. 32, 1365–1380 (2011)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

We would like to thank the reviewer for making helpful comments.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Utah State University, Logan, UT, 84322, USA

    An Le & Zhi-Qiang Wang

  2. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Jianxin Zhou

Authors
  1. An Le
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhi-Qiang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianxin Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianxin Zhou.

Additional information

Z.-Q. Wang’s research was supported in part by NSF DMS-0820327 and Jianxin Zhou’s research was supported in part by NSF DMS-0713872/0820327/1115384.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Le, A., Wang, ZQ. & Zhou, J. Finding Multiple Solutions to Elliptic PDE with Nonlinear Boundary Conditions. J Sci Comput 56, 591–615 (2013). https://doi.org/10.1007/s10915-013-9689-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-013-9689-9

Keywords

AMS Mathematical Subject Classification

Search

Navigation