Symmetry and automated branch following for a Semilinear elliptic PDE on a fractal region

SIAM Journal on Applied Dynamical Systems

Volumn 5, Issue 3, 2006, Pages 476-507

  Neuberger, John M ^a   Sieben, Nándor ^a   Swift, James W ^a  

^a NORTHERN ARIZONA UNIVERSITY   (United States)

Author keywords

Bifurcation; GNGA; Semilinear elliptic pde; Snowflake; Symmetry

Indexed keywords

BIFURCATION; GNGA; SEMILINEAR ELLIPTIC PDE; SNOWFLAKE; SYMMETRY;

AIRCRAFT ENGINES; BOUNDARY VALUE PROBLEMS; CODES (SYMBOLS); EIGENVALUES AND EIGENFUNCTIONS; NEWTON-RAPHSON METHOD; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; SNOW;

BIFURCATION (MATHEMATICS);

EID: 57249111985     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/050640485     Document Type: Article

References (36)

1. R. A. ADAMS, Sobolev Spaces, Pure Appl. Math. 65, Academic Press, New York, London, 1975.
2. A. AMBROSETTI AND P. H. RABINOWITZ, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), pp. 349-381.
3. M. S. BERGER, Nonlinearity and Functional Analysis. Lectures on Nonlinear Problems in Mathematical Analysis, Pure Appl. Math., Academic Press, New York, London, 1977.
4. A. CASTRO, J. COSSIO, AND J. M. NEUBERGER, Sign-changing solutions for a superlinear Dirichlet problem, Rocky Mountain J. Math., 27 (1997), pp. 1041-1053.
5. A. CASTRO, P. DRABEK, AND J. M. NEUBERGER, A sign-changing solution for a superlinear Dirichlet problem, II, in Proceedings of the Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electron. J. Differ. Equ. Conf. 10, Southwest Texas State University, San Marcos, TX, 2003, pp. 101-107.
6. P. CHOSSAT AND R. LAUTERBACH, Methods in Equivariant Bifurcations and Dynamical Systems, Adv. Ser. Nonlinear Dynam. 15, World Scientific, River Edge, NJ, 2000.
7. D. COSTA, Z. DLNG, AND J. M. NEUBERGER, A numerical investigation of sign-changing solutions to superlinear elliptic equations on symmetric domains, J. Comput. Appl. Math., 131 (2001), pp. 299-319.
8. GAP GROUP, GAP-Groups, Algorithms, and Programming, 2002, http://www.gap-system.org.
9. D. GILBARG AND N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer-Verlag, Berlin, 2001.
10. M. GOLUBITSKY, I. STEWART, AND D. G. SCHAEFER, Singularities and Groups in Bifurcation Theory, Vol. 2, Appl. Math. Sci. 69, Springer-Verlag, New York, 1988.
11. M. GOLUBITSKY AND I. STEWART, The Symmetry Perspective: From, Equilibrium to Chaos in Phase Space and Physical Space, Progr. Math. 200, Birkhäuser-Verlag, Basel, 2002.
12. J. HINEMAN AND J. M. NEUBERGER, GNGA for general regions: Semilinear elliptic PDE and, crossing eigenvalues, Commun. Nonlinear Sci. Numer. Simul., (2005), to appear.
13. C. T. KELLEY, Solving Nonlinear Equations with Newton's Method, Fundam. Algorithms 1, SIAM, Philadelphia, 2003.
14. C. T. KELLEY, Iterative Methods for Optimization, Frontiers Appl. Math. 18, SIAM, Philadelphia, 1999.
15. M. LAPIDUS, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Trans. Amer. Math. Soc, 325 (1991), pp. 465-529.
16. M. L. LAPIDUS, J. W. NEUBERGER, R. L. RENKA, AND C. A. GRIFFITH, Snowflake harmonics and computer graphics: Numerical computation of spectra on fractal drums, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), pp. 1185-1210.
17. M. LAPIDUS AND M. PANG, Eigenfunctions of the Koch snowflake domain, Comm. Math. Phys., 172 (1995), pp. 359-376.
18. R. LAUTERBACH AND S. MAIER, Symmetry-breaking at non-positive solutions of semilinear elliptic equations, Arch. Ration. Mech. Anal, 126 (1994), pp. 299-331.
19. R. B. LEHOUCQ, D. C. SORENSEN, AND C. YANG, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, Software Environ. Tools 6, SIAM, Philadelphia, 1998.
20. Y. LI AND J. ZHOU, Convergence results of a local minimax method for finding multiple critical points, SIAM J. Sci. Comput., 24 (2002), pp. 865-885.
21. P. C. MATTHEWS, Automated, symmetry-breaking calculations, LMS J. Comput. Math., 7 (2004) 101-119.
22. J. W. NEUBERGER, Sobolev Gradients and Differential Equations, Lecture Notes in Math. 1670, Springer-Verlag, Berlin, 1997.
23. J. M. NEUBERGER, A numerical method for finding sign-changing solutions of superlinear Dirichlet problems, Nonlinear World, 4 (1997), pp. 73-83.
24. J. M. NEUBERGER, GNGA: Recent progress and open problems for semilinear elliptic PDE, in Variational Methods: Open Problems, Recent Progress, and Numerical Algorithms, Contemp. Math. 357, AMS, Providence, RI, 2004, pp. 201-237.
25. J. M. NEUBERGER, Nonlinear elliptic partial difference equations on graphs, J. Exper. Math., 15 (2005), pp. 91-107.
26. J. M. NEUBERGER AND J. W. SWIFT, Newton's method and Morse index for semilinear elliptic PDEs, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), pp. 801-820.
27. J. M. NEUBERGER, N. SIEBEN, AND J. W. SWIFT, Computing eigenfunctions on the Koch snowflake: A new grid and symmetry, J. Comput. Appl. Math., 191 (2005), pp. 126-142.
28. P. RABINOWITZ, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, AMS, Providence, RI, 1986.
29. W. R. SCOTT, Group Theory, Prentice-Hall, Englewood Cliffs, NJ, 1964.
30. J. SMOLLER AND A. G. WASSERMAN, Symmetry-breaking for positive solutions of semilinear elliptic equations, Arch, Ration. Mech. Anal., 95 (1986), pp. 217-225.
31. S. STERNBERG, Group Theory and Physics, Cambridge University Press, Cambridge, UK, 1994.
32. A. D. THOMAS AND G. V. WOOD, Group Tables, Shiva Math. Ser. 2, Shiva Publishing, Nantwich, UK, 1980; distributed by Birkhäuser Boston, Cambridge, MA.
33. T. J. THOMPSON, Estimating Solutions for the Ginzburg-Landau Superconductivity Model in Thin Disks, M.S. thesis, Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ, 2005.
34. M. TINKHAM, Group Theory and Quantum Mechanics, McGraw-Hill, New York, Toronto, London, 1964.
35. Z.-Q. WANG AND J. ZHOU, A local minimax-Newton method for finding multiple saddle points with symmetries, SIAM J. Numer. Anal., 42 (2004), pp. 1745-1759.
36. Z.-Q. WANG AND J. ZHOU, An efficient and stable method for computing saddle points with symmetries, SIAM J. Numer. Anal., 43 (2005), pp. 891-907.