Vortex rings in a cylinder and rearrangements. (English) Zbl 0648.35029

Burton, G. R.

J. Differ. Equations 70, 333-348 (1987).

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

Keywords:

vortex rings; Steiner symmetrization; cylindrical coordinates; distribution function; variational problems; rearrangement

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