Excerpt from The Small Dispersion Limit of the Korteweg-Devries Equations
In Section 3 we show that Q is continuous in a weak sequential topology, and that the space of admissible functions is compact in that topology. We further show that Q is a strictly convex function; since the admissible functions form a convex set, this implies not only that the minimum of Q is taken on at a unique function, but that this function is the only one which satisfies variational conditions.
The variational conditions are then converted to a riemann-hilbert problem, i.e. To the problem of determining an analytic function of class Hp in the upper half plane whose real and imaginary parts are prescribed on alternate intervals of the real axis.
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