We develop a theory to connect the following three areas: (a) the mean field equations on flat tori, (b) the classical Lame equations and (c) modular forms. A major theme in part I is a classification of developing maps f attached to solutions of the mean field equation according to the types of transformation laws (or monodromy) with respect to the period lattice satisfied by f.

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We prove$L$^{$q$}-inequalities for the gradient of the Green potential ($Gf$) in bounded, connected NTA-domains in$R$^{$n$},$n$≥2. These domains may have a highly non-rectifiable boundary and in the plane the set of all bounded simply connected NTA-domains coincides with the set of all quasidiscs. We get a restriction on the exponent$q$for which our inequalities are valid in terms of the validity of a reverse Hölder inequality for the Green function close to the boundary.

The Lp dual curvature measure was introduced by Lutwak, Yang & Zhang in an attempt to unify the Lp Brunn– Minkowski theory and the dual Brunn–Minkowski theory. The characterization problem for Lp dual curvature measure, called the Lp dual Minkowski problem, is a fundamental problem in this unifying theory. The Lp dual Minkowski problem contains the Lp Minkowski problem and the dual Minkowski problem, two major problems in modern convex geometry that remain open in general. In this paper, existence results on the Lp dual Minkowski problem in the weak sense will be provided. Moreover, existence and uniqueness of the solution in the smooth category will also be demonstrated.