Solutions of the dirichlet problem for elliptic systems in a disk
We study 2×2 second-order elliptic systems, which can be written as a single equation with complex coefficients. In an arbitrary bounded region with smooth boundary, we obtain necessary and sufficient conditions on the trace relation of a solution, which we apply in the case of a disk. We prove existence and uniqueness theorems for a solution in a Sobolevskii space of an equation which is not properly elliptic. In particular, we prove that the properties of the problem determine the angle between the bicharacteristics. If it is π-rational, then there is no uniqueness, but if it is π-irrational, then the smoothness of the solution of the Dirichlet problem depends on the order of its approximation by π-rational numbers; but if it is nonreal, then the problem has the usual properties for the elliptic case.
Abstract We discuss random matrix-model representations of D = 1 string theory, with particular emphasis on the case in which the target space is a circle of finite radius. The duality properties of discretized strings are analyzed and shown to depend on the dynamics of vortices. In the representation in terms of a continuous circle of matrices we find an exact expression for the free energy, neglecting non-singlet states, as a function of the string coupling and the radius which exhibits exact duality. In a second version, based on a discrete chain of matrices, we find that vortices induce, for a finite radius, a Kosterlitz-Thouless phase transition that takes us to a c = 0 theory.