Lyapunov–Schmidt reduction

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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[edit]

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[edit]

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[edit]

Let us split into the direct product , where .

Let be the projection operator onto .

Let us consider also the direct product .

Applying the operators and to the original equation, one obtains the equivalent system

Let and , then the first equation

can be solved with respect to by applying the implicit function theorem to the operator

(now the conditions of the implicit function theorem are fulfilled).

Thus, there exists a unique solution satisfying

Now substituting into the second equation, one obtains the final finite-dimensional equation

Indeed, the last equation is now finite-dimensional, since the range of is finite-dimensional. This equation is now to be solved with respect to , which is finite-dimensional, and parameters :

References[edit]

  • 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.