A Variational and Regularization Framework for
Stable Strong Solutions of Nonlinear
Boundary-Value Problems
By: Joseph W. Jerome
We study a variational approach introduced by S.D. Fisher and the author
in the 1970s in the context of norm minimization for differentiable
mappings occurring in nonlinear elliptic boundary value problems.
It may be viewed as an abstract version of the calculus of variations.
A strong hypothesis,
initially limiting the scope of this approach, is the assumption of a
bounded minimizing sequence in the least squares formulation. In this
article, we employ regularization and invariant regions to overcome this
obstacle. A consequence of the framework is the convergence of
approximations for regularized problems to a desired solution.
The variational method is closely associated with the implicit function
theorem, and it can be jointly studied, so that
continuous parameter stability is
naturally deduced. A significant aspect of the theory is that the reaction
term in a reaction-diffusion equation can be selected to act globally as
in the steady Schroedinger-Hartree equation. Local action, as in the
non-equilibrium Poisson-Boltzmann equation, is also included.
Both cases are studied at length prior to the development of a general
theory.
This paper has appeared
in the Journal of
Numerical Functional Analysis and Optimization: vol. 44 (2023), 394-419.
It has also been published online:
https://doi.org/10.1080/01630563.2023.2178010.
It is presently accessible in the publisher's online version at: