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Vortex rings in a cylinder and rearrangements. (English) Zbl 0648.35029
Let $$\Omega \subset {\mathbb{R}}^ 3$$ be a cylindrical paper with circular cross section. In cylindrical coordinates, let $$Lu=-(1/r)((1/r)u_ r)_ r-(1/r^ 2)u_{zz}$$. Given the distribution function of Lu, there exists a $$\lambda_ 0>0$$ such that for $$0<\lambda <\lambda_ 0$$ there exists a monotone nonlinearity $$\phi$$ and a solution $$u\in W^{2,p}_{loc}(\Omega)$$ of the problem $$Lu=\phi (u-\lambda r^ 2/2)$$. Moreover u satisfies various natural properties and estimates. This result is based on a previous paper of the author on variational problems over classes of functions with prescribed rearrangement [Math. Ann. 276, 225-253 (1987; Zbl 0592.35049)]. Related results have recently been obtained by P. Laurence and E. Stredulinsky.
Reviewer: B.Kawohl

##### MSC:
 35J20 Variational methods for second-order elliptic equations 76B47 Vortex flows for incompressible inviscid fluids
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##### References:
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