We consider the following eigenvalue optimization problem: Given a bounded domain ⊂ R and numbers α > 0, A ∈ [0, | |], find a subset D ⊂ of area A for which the first Dirichlet eigenvalue of the operator − + αχ D is as small as possible.We prove existence of solutions and investigate their qualitative properties. For example, we show that for some symmetric domains (thin annuli and dumbbells with narrow handle) optimal solutions must possess fewer symmetries than ; on the other hand, for convex reflection symmetries are preserved.Also, we present numerical results and formulate some conjectures suggested by them.

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.

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.

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We give a condition which ensures that the Paneitz operator of an embedded three-dimensional CR manifold is nonnegative and has kernel consisting only of the CR pluriharmonic functions. Our condition requires uniform positivity of the Webster scalar curvature and the stability of the CR pluriharmonic functions for a real analytic deformation. As an application, we show that the real ellipsoids in C 2 are such that the CR Paneitz operator is nonnegative with kernel consisting only of the CR pluriharmonic functions.