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.
Similar content being viewed by others
References
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)
Ando, T., Fowleer, A.B., Stern, F.: Electronic properties of two dimensional system. Rev. Mod. Phys. 54, 437–621 (1982)
Armontano, M.G.: The effect of reduced integration in the Steklov eigenvalue problem. Math. Model. Numer. Anal. 38, 27–36 (2004)
Auchmuty, G.: Steklov eigenproblems and the representation of solutions of elliptic boundary value problems. Numer. Func. Anal. Optim. 25, 321–348 (2004)
Auchmuty, G.: Spectral characterization of the trace spaces \(H^s(\partial \Omega )\). SIAM J. Math. Anal. 38, 894–905 (2006)
Atkinson, K.T.: The numerical solution of a nonlinear boundary integral equation on smooth surface. IMA J. Numer. Anal. 14, 461–483 (1994)
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)
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)
Bartsch, T., Liu, Z.L., Weth, T.: Sign changing solutions of superlinear Schrödinger equations. Comm. Partial Differ. Equ. 29, 25–42 (2004)
Bartsch, T., Wang, Z.-Q.: On the existence of sign changing solutions for semilinear Dirichlet problems. Topol. Meth. Nonl. Anal. 7, 115–131 (1996)
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)
Cabré, X., Tan, J.: Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224, 2052–2093 (2010)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Partial Diff. Equ. 32, 1245–1260 (2007)
Cushing, J.M.: Nonlinear Steklov problems on the unit circle II–a hydro-dynamical application. JMAA 39, 267–278 (1972)
Chen, G., Zhou, J.: Boundary Element Methods. Academic Press, London-San Diego (1992)
Fasino, D., Inglese, G.: Recovering unknown terms in a nonlinear boundary condition for Laplace’s equation. IMA J. Appl. Math. 71, 832–852 (2006)
Fila, M., Quittner, P.: Global solutions of the Laplace equation with a nonlinear dynamical boundary condition. Math. Methods Appl. Sci. 20, 1325–1333 (1997)
Ganesh, M.: A BIE method for a nonlinear BVP. J. Comput. Appl. Math. 45, 299–308 (1993)
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)
Li, S.J., Wang, Z.-Q.: Ljusternik-Schnirelman theory in partially ordered Hilbert spaces. Trans. Amer. Math. Soc. 354, 3207–3227 (2002)
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)
Li, Y., Zhou, J.: Convergence results of a local minimax method for finding multiple critical points. SIAM J. Sci. Comp. 24, 865–885 (2002)
Liu, Z., Wang, Z.-Q.: Sign-changing solutions of nonlinear elliptic equations. Frontiers Math. China 3, 1–18 (2008)
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)
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)
Ruotsalainen, K., Saramen, J.: On the collocation method for a nonlinear boundary integral equation. J. Comput. Appl. Math. 28, 339–348 (1989)
Ruotsalainen, K., Wendland, W.L.: On the boundary element method for some nonlinear boundary value problems. Numer. Math. 53, 299–314 (1988)
Vitillaro, E.: On the Laplace equation with non-linear dynamic boundary conditions. Proc. London Math. Soc. 93, 418–446 (2006)
Vogelius, M., Xu, J.-M.: A nonlinear elliptic boundary value problem related to corrosion modeling. Q. Appl. Math. 56, 479–505 (1998)
Wang, Z.-Q.: On a superlinear elliptic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 8, 43–57 (1991)
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)
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)
Wang, Z.Q., Zhou, J.: A local minimax-Newton method for finding critical points with symmetries. SIAM J. Num. Anal. 42, 1745–1759 (2004)
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)
Zhou, J.: A local min-orthogonal method for finding multiple saddle points. JMAA 291, 66–81 (2004)
Zhou, J.: Instability analysis of saddle points by a local minimax method. Math. Comp. 74, 1391–1411 (2005)
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)
Acknowledgments
We would like to thank the reviewer for making helpful comments.
Author information
Authors and Affiliations
Corresponding author
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
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-013-9689-9