Abstract
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.
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Communicated by J. Glimm
Research partially supported by A FOSR and NSF grants
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Berger, M.S., Fraenkel, L.E. Nonlinear desingularization in certain free-boundary problems. Commun.Math. Phys. 77, 149–172 (1980). https://doi.org/10.1007/BF01982715
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DOI: https://doi.org/10.1007/BF01982715