Perturbation methods

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The term perturbation methods refers to a variety of methods to prove existence, uniqueness or regularity for equations that are in some sense close to an equation that is well understood. If an equation can be written as the sum of a well understood equation plus an extra term (perturbation) that is small with respect to some relevant quantity, then sometimes a property of the well understood term can be transfered to the original equation.

The simplest example of a perturbation method is bootstrapping. For the method of bootstrapping to work out successfully, the equation must be the sum of a well understood equation plus another term whose loss of regularity is smaller than the gain of the first term.

There are other methods besides bootstrapping that can also be characterized as perturbations techniques. For example in the proof of the Schauder estimates or the Cordes-Nirenberg estimates, the solution to an elliptic equation with variable coefficients is compared locally to the solution to an elliptic equation with constant coefficients. In this case the method is perturbative because the assumption is that the oscillation of the coefficients is small enough in small scales, and thus the equation can be written as the sum of a constant coefficient equation plus a small perturbation.

Typically all the following types of results are proved using perturbation methods

  1. Well posedness results which require a smallness assumption on the data.
  2. Well posedness results for evolution equations locally in time.
  3. Most proofs involving the contraction mapping principle.
  4. Most proofs involving the implicit function theorem for Banach spaces.
  5. Any regularity for an elliptic or parabolic equation with variable coefficients that if we scale it in (zoom in), would converge to an equation with constant coefficients that is well understood.
  6. Any result which requires a Kato class assumption.

On the other hand, perturbation methods are rarely useful for the following type of problems

  1. Fully nonlinear or quasilinear equations with constant coefficients and only one term.
  2. Scale invariant equations with arbitrarily large data. More precisely, if there is some scaling that preserves the equations and its most relevant a priori estimate.
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