During the past year a group of 12 graduate students, a visiting researcher and a postdoctoral fellow, supported in part by this contract, have studied various nonlinear partial differential equations. Seminars were held twice weekly on the topic of soliton theory. The graduate students involved have been studying the problem of singularity that arises in nonlinear equations.

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We begin by giving some results of continuity with respect to the domain for the Dirichlet problem (without any assumption of regularity on the domains). Then, following an idea of A. Beurling, a technique of subsolutions and supersolutions for the so-called quadrature surface free boundary problem is presented. This technique would apply to many free boundary problems in$R$$N$,$N$≥2, which have overdetermined Cauchy data on the free boundary. Some applications to concrete examples are also given.

A unified approach is presented for establishing a broad class of Cram\'er-Rao inequalities for the location parameter, including, as special cases, the original inequality of Cram\'er and Rao, as well as an $L^p$ version recently established by the authors. The new approach allows for generalized moments and Fisher information measures to be defined by convex functions that are not necessarily homogeneous. In particular, it is shown that associated with any log-concave random variable whose density satisfies certain boundary conditions is a Cram\'er-Rao inequality for which the given log-concave random variable is the extremal. Applications to specific instances are also provided.