In mathematics, the Lyapunov–Schmidt reduction or Lyapunov–Schmidt construction is used to study solutions to nonlinear equations in the case when the implicit function theorem does not work. It permits the reduction of infinite-dimensional equations in Banach spaces to finite-dimensional equations. It is named after Aleksandr Lyapunov and Erhard Schmidt.
Problem setup Let
be the given nonlinear equation, and are Banach spaces ( is the parameter space). is the -map from a neighborhood of some point to and the equation is satisfied at this point
For the case when the linear operator is invertible, the implicit function theorem assures that there exists a solution satisfying the equation at least locally close to .
In the opposite case, when the linear operator is non-invertible, the Lyapunov–Schmidt reduction can be applied in the following way.
Assumptions
One assumes that the operator is a Fredholm operator, and has finite dimension.
The range of this operator has finite co-dimension and is a closed subspace in .
Without loss of generality, one can assume that
Lyapunov–Schmidt construction
Let us split and into the direct sums and , respectively, where . Also, let be the projection operator onto .
Applying the operators and to the original equation and writing , one obtains the equivalent system
Viewing the first equation as a map
where
one finds that is an isomorphism. The implicit function theorem is thus applicable, in particular guaranteeing the existence of a such that
Now plugging into the second equation, one obtains the bifurcation equation
which is a finite-dimensional equation by the earlier assumption of being finite. This equation is now to be solved with respect to , which is finite-dimensional, and parameters :
Applications Lyapunov–Schmidt reduction has been used in economics, natural sciences, and engineering often in combination with bifurcation theory, perturbation theory, and regularization. LS reduction is often used to rigorously regularize partial differential equation models in chemical engineering resulting in models that are easier to simulate numerically but still retain all the parameters of the original model.
References
Bibliography Louis Nirenberg, Topics in nonlinear functional analysis, New York Univ. Lecture Notes, 1974. Aleksandr Lyapunov, Sur les figures d’équilibre peu différents des ellipsoides d’une masse liquide homogène douée d’un mouvement de rotation, Zap. Akad. Nauk St. Petersburg (1906), 1–225. Aleksandr Lyapunov, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse 2 (1907), 203–474. Erhard Schmidt, Zur Theory der linearen und nichtlinearen Integralgleichungen, 3 Teil, Math. Annalen 65 (1908), 370–399.