Multiconstrained variational problems of nonlinear eigenvalue type: new formulations and algorithms

Authors:
Alexander Eydeland, Joel Spruck and Bruce Turkington

Journal:
Math. Comp. **55** (1990), 509-535

MSC:
Primary 49R05; Secondary 35J85, 76W05

DOI:
https://doi.org/10.1090/S0025-5718-1990-1035931-7

MathSciNet review:
1035931

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Abstract: A new variational approach is proposed for a class of semilinear elliptic eigenvalue problems involving many eigenvalue parameters. These problems arise, for instance, in the modelling of magnetohydrodynamic equilibria with one spatial symmetry. In this case, the physical variational principle imposes a continuously infinite family of constraints, which prescribes the mass and helicity within every flux tube. The equilibrium equations therefore contain unspecified profile functions that are determined along with the solution as multipliers for those constraints. A prototype problem for this general class is formulated, and a natural discretization of its constraint family is introduced. The resulting multiconstrained minimization problem is solved by an iterative algorithm, which is based on relaxation of the given nonlinear equality constraints to linearized inequalities at each iteration. By appealing to convexity properties, the monotonicity and global convergence of this algorithm is proved. The explicit construction of the iterative sequence is obtained by a dual variational characterization.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1990-1035931-7

Keywords:
Semilinear elliptic eigenvalue problems,
constrained minimization,
iterative procedure,
magnetohydrodynamics

Article copyright:
© Copyright 1990
American Mathematical Society