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Variational problems and PDEs in affine differential geometry

H. Z. Li (2005)

Banach Center Publications

This paper is part of the autumn school on "Variational problems and higher order PDEs for affine hypersurfaces". We discuss variational problems in equiaffine differential geometry, centroaffine differential geometry and relative differential geometry, which have been studied by Blaschke [Bla], Chern [Ch], C. P. Wang [W], Li-Li-Simon [LLS], and Calabi [Ca-II]. We first derive the Euler-Lagrange equations in these settings; these equations are complicated, strongly nonlinear fourth order PDEs. We...

Vortex rings for the Gross-Pitaevskii equation

Fabrice Bethuel, G. Orlandi, Didier Smets (2004)

Journal of the European Mathematical Society

We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension N ≥ 3 . We also extend the asymptotic analysis of the free field Ginzburg–Landau equation to a larger class of equations, including the Ginzburg–Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if N = 3 ).

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