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Variational problems on classes of rearrangements and multiple configurations for steady vortices. (English) Zbl 0677.49005

Summary: A Mountain Pass Lemma is proved for a convex functional restricted to the class \({\mathcal F}\) of rearrangements of a fixed \(L^ p\) function. Together with results on maximization and minimization relative to \({\mathcal F}\), this proves the existence of at least four solutions for a problem on the steady configurations of a vortex in an ideal fluid.

MSC:

49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
76B47 Vortex flows for incompressible inviscid fluids
35J20 Variational methods for second-order elliptic equations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:

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