Communications in Mathematical Physics

, Volume 77, Issue 2, pp 149–172 | Cite as

Nonlinear desingularization in certain free-boundary problems

  • M. S. Berger
  • L. E. Fraenkel


We consider a nonlinear, elliptic, free-boundary problem involving an initially unknown setA that represents, for example, the cross-section of a steady vortex ring or of a confined plasma in equilibrium. The solutions are characterized by a variational principle which allows us to describe their behaviour under a limiting process such that the diameter ofA tends to zero, while the solutions degenerate to the solution of a related linear problem. This limiting solution is the sum of the Green function of the linear operator and of a smooth function satisfying the boundary conditions. Mathematically speaking, this limiting process, that we call “nonlinear desingularization”, is a novel kind of bifurcation phenomenon since the nonlinear effect here involves smoothing the singularity of the associated linear problem.


Boundary Condition Vortex Neural Network Statistical Physic Complex System 


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Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • M. S. Berger
    • 1
  • L. E. Fraenkel
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA
  2. 2.Mathematics DivisionUniversity of SussexBrightonEngland

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