Fucik Spectrum of the Laplace Operator

A simplified model of a suspension bridge [4] can serve as an example of an application related to the Fucik spectrum. The model involves the term , where stands for the (vertical) deflection of the roadbed at a certain point of the bridge, , and the two coefficients (spring constants) describe different forces applied by the cables when they are tight or slack . (Here is more on another model of a suspension bridge.)

The Fucik spectrum of the Laplace operator is the set of all pairs for which the following problem has a nontrivial solution in the Sobolev space :

where is a bounded domain.

In the late 70's it was discovered by S. Fucik [2] and E. N. Dancer [1] that the existence of solutions of the nonlinear problem

with , depends on the position of the pair of numbers in the real plane with respect to the "curves" of the Fucik spectrum.


Figure 1Figure 2
Fig. 1 Fig. 2 

Together with Wolfgang Reichel we took a look at this problem in [3]. We studied it using both a variational formulation and an implicit function argument. In addition to analytical results we also performed numerical computations on several domains in the plane. The two figures show some of these computations. Figure 1 shows a part of the Fucik spectrum computed for a triangular domain (one should note the bifurcation of the curves and their crossing), Figure 2 shows one Fucik eigenfunction on this domain.


References

[1]  E. N. Dancer, On the Dirichlet problem for weakly nonlinear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 76 (1997) 283-300.
[2]  S. Fucik, Solvability of nonlinear equations and boundary value problems, Mathematics and its applications, Vol. 4, Reidel, Dordrecht/Boston, 1980.
[3]  J. Horak, W. Reichel, Analytical and numerical results for the Fucik spectrum of the Laplacian, J. Comput. Appl. Math. 161 (2003) 313-338.
[4]  A.C. Lazer, P. J. McKenna, Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities, Trans. Amer. Math. Soc. 315 (1989) 721-739.


Updated on December 2, 2004