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The main objective of this paper is to extend the pioneering work of Sims in [9] on secondorder linear differential equations with a complex coefficient, in which he obtains an analogue of the Titchmarsh-Weyl theory and classification. The generalisation considered exposes interesting features not visible in the special case in [9]. An m-function is constructed (which is either unique or a point on a "limit-circle") and the relationship between its properties and the spectrum of underlying maccretive differential operators analysed. The paper is a contribution to the study of non-self-adjoint operators; in general the spectral theory of such operators is rather fragmentary, and further study is being driven by important physical applications, to hydrodynamics, electro-magnetic theory and nuclear physics, for instance.

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We consider the nonlinear curl-curl problem ∇×∇×U +V (x)U = Γ(x)|U | p−1 U in R 3 related to the nonlinear Maxwell equations for monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of H(curl; R 3 ). Under a cylindrical symmetry assumption on the functions V and Γ the variational problem can be posed in a symmetric subspace of H(curl; R 3 ). For a strongly defocusing case ess sup Γ < 0 with large negative values of Γ at infinity we obtain ground states by the direct minimization method. For the focusing case ess inf Γ > 0 the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear operator ∇ × ∇ × +V (x). Examples of cylindrically symmetric functions V are provided for which this holds.

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Abstract:We consider a continuous sealed-bid auction model for an indivisible object with two bidders and incomplete information on both sides where the bidders' evaluations are assumed to be independently distributed on some real intervals. The price the winner (the highest bidder) has to pay is a given convex combination of the highest and the second highest (lowest) bid. It is shown that, for all but the second highest bid-price auction, all equilibrium-strategies are continuously differentiable and strictly monotonically increasing, and moreover, that the set of Nash-equilibria is completely characterized by a boundary value problem for a system of singular differential equations. In the case of symmetric data (independently and identically distributed true values) and for a particular class of asymmetric distributions (including uniform distributions), the boundary value problem is solved explicitly and uniquely.

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