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Path following around corank-2 bifurcation points of a semi-linear elliptic problem with symmetry. (English) Zbl 0758.65067

The partial differential equation \(\Delta u+\lambda f(u)=0\) is considered on the unit square with homogeneous Dirichlet conditions and a function \(f\) which is assumed to be smooth and odd. Bifurcations from the trivial solution curve with vanishing \(u\) for all \(\lambda\) are classified with respect to their symmetry properties. The notions of equivariant mappings, isotrop groups of solutions and restricted problems for symmetric solution curves are introduced. Basing on the eigenvalues and eigenfunctions of \(-\Delta\) also hidden symmetries are discussed.
For each of the 4 different nontrivial solution branches passing through a corank-2 bifurcation point on the trivial solution curve a representation in terms of eigenfunctions of \(-\Delta\) and the corresponding reduced problem is given.
For numerical treatment the reduced problems are shown to be equivalent to classical mixed boundary value problems on appropriate subdomains of the unit square. Near bifurcation points the problem is transformed into a nonsingular one. As an example a simplified buckling problem is solved using finite differences for the discretization of the Laplace operator.

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B32 Bifurcations in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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