Penalization method

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The penalization method (also called the regularization method or viscosity method) constructs solutions to an equation by the following general procedure:

  1. Regularize the equation somehow (for instance, by adding penalty terms to the Hamiltonian or Lagrangian to try to prevent the solution from becoming too rough or too high in amplitude, or by inserting dissipation terms into the nonlinearity).
  2. Construct solutions of the regularized equations (which is often much easier than with the original equation).
  3. Establish uniform bounds on all of these regularized solutions.
  4. Exploiting some sort of compactness to then extract a convergent subsequence of regularized solutions. The convergence might only be in a weak topology.
  5. Show that the limit of this subsequence is an exact solution to the original equation.

This method is thus an example of a compactness method. Like all compactness methods, penalization methods tend to excel at establishing existence of solutions, but have difficulty with establishing uniqueness, continuity, regularity, or stability, in large part due to the weakness of the topology in which compactness is attained.