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BIRS Workshop Lecture Videos
On a type of second order variational problem in L-infinity. Moser, Roger
Let K be an elliptic (not necessarily linear) second order differential operator. Suppose that we want to minimise the L-infinity norm of K(u) for functions u satisfying suitable boundary conditions. Here K may represent, e.g., the curvature of a curve in the plane or the scalar curvature of a Riemannian manifold in a fixed conformal class, but the problem is not restricted to questions with a geometric background. If the operator and the boundary conditions are such that the equation K(u) = 0 has a solution, then the problem is of course trivial. But since this is a second order variational problem, it may be appropriate to prescribe u as well as its first derivative on the boundary of its domain, which in general rules out this situation. In the cases studied so far, the solution, while not trivial, still has a nice structure, and one feature is that |K(u)| is always constant. The sign of K(u) may jump, but we have a characterisation of the jump set in terms of a linear PDE. Furthermore, in some cases we have a unique solution, even though the underlying functional is not strictly convex. This talk is based on joint papers with H. Schwetlick and with N. Katzourakis and the PhD thesis of Z. Sakellaris.
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