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A gradient bound for the Grad‐Kruskal‐Kulsrud functional
Author(s) -
Laurence Peter,
Stredulinsky Edward W.
Publication year - 1996
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199603)49:3<237::aid-cpa2>3.0.co;2-e
Subject(s) - mathematics , bounded function , boundary (topology) , limiting , regular polygon , calculus of variations , variational principle , mathematical analysis , function (biology) , nonlinear system , geometry , physics , mechanical engineering , evolutionary biology , engineering , biology , quantum mechanics
We establish the existence of minimizers with a bounded gradient for a variational problem arising in the Grad‐Kruskal‐Kulsrud model for the equilibrium of a confined plasma. The variational problem involves derivatives of the nondecreasing rearrangement of minimizers. Results are derived for bounded convex domains in R 2 . Our results answer in the negative (in the case of convex domains) a question raised by Grad concerning the possibility of singular behavior of the magnetic field at the point of maximum flux. The main approach is to use an approximating free boundary problem to handle the nonlinear nonlocal nature of the variational functional. Limiting minimizers of the variational problem are shown to have bounded gradient and to satisfy a weak equation that for one model problem takes the form $$\Delta\psi=-\psi^{*\,\prime \prime }(\mu_\psi(\psi)),$$ where ψ*, μ ψ are, respectively, the nondecreasing rearrangement and the distribution function of ψ. © 1996 John Wiley & Sons, Inc.