The inhomogeneous mean-field thermodynamic limit is constructed and evaluated for both the canonical thermodynamic functions and the states of systems of classical point particles with logarithmic interactions in two space dimensions. The results apply to various physical models of translation invariant plasmas, gravitating systems, as well as to planar fluid vortex motion. For attractive interactions a critical behavior occurs which can be classified as an extreme case of a second-order phase transition. To include in particular attractive interactions a new inequality for configurational integrals is derived from the arithmetic-geometric mean inequality. The method developed in this paper is easily seen to apply as well to systems with fairly general interactions in all space dimensions. In addition, it also provides us with a new proof of the Trudinger-Moser inequality known from differential geometry -in its sharp form.

Abstract:The method of moving planes is used to establish a weak set of conditions under which the nonlinear equation -Au(x) = V(\x\)e u(x \ xeR 2 admits only rotationally symmetric solutions. Additional structural invariance properties of the equation then yield another set of conditions which are not originally covered by the moving plane technique. For instance, nonmonotonic V can be accommodated. Results for -Au(y) = V(y)e u(y) -c, with yeS 2 , are obtained as well. A nontrivial example of broken symmetry is also constructed. These equations arise in the context of extremization problems, but no extremization arguments are employed. This is of some interest in cases where the extremizing problem is neither manifestly convex nor monotone under symmetric decreasing rearrangements. The results answer in part some conjectures raised in the literature. Applications to logarithmically interacting particle systems and geometry are emphasized.

No abstract

We establish a strict isoperimetric inequality and a Pohozaev-Rellich identity for the system. . , N}, under certain reasonable conditions on the γ i,j and u i . Thus we prove that under these conditions, all solutions u i are radial symmetric and decreasing about some point.

No abstract