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Rearrangements of functions, maximization of convex functionals, and vortex rings. (English) Zbl 0592.35049
This paper considers the maximization of a convex functional \(\psi\) over the set \({\mathcal J}\) of rearrangements of a fixed function \(f_ 0\in L^ p(\mu)\), where \(1\leq p\leq \infty\) and \(\mu\) is a finite, separable, nonatomic, positive measure on a set \(\Omega\). If \(\psi\) is weakly \((weak^*)\) sequentially continuous on \(L^ p(\mu)\) and strictly convex, it is shown that a maximizer \(f^*\) exists, and if \(g\in \partial \psi (f^*)\) \((\subset L^ q(\mu)\) where q is the conjugate exponent of p) then \(f^*=\phi \circ g\) almost everywhere in \(\Omega\) for some increasing function \(\phi\). Under suitable assumptions this ensures the existence of a solution to a semilinear elliptic equation \({\mathcal L}u=\phi (u-v)\), where v is prescribed, \({\mathcal L}u\) is a rearrangement of a prescribed function, and \(\phi\) is a priori unknown. A particular case studied is a partial differential equation for a vortex ring in fluid mechanics. Maximization of \(\psi\) on \({\mathcal J}\) subject to a linear constraint is also considered. The results rest on a detailed study of the maximization of linear functionals relative to \({\mathcal J}\). A dual problem for the maximization of \(\psi\) on \({\mathcal J}\) is formulated; in the case of vortex rings this gives rise to a variational problem for a ”queer differential equation”.

MSC:
35J60 Nonlinear elliptic equations
76B47 Vortex flows for incompressible inviscid fluids
49K20 Optimality conditions for problems involving partial differential equations
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