Geometric Variational Problems and Rearrangement Inequalities

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This proposal presents a research agenda on non-local geometric variational problems. Non-local functionals arise in many places in Mathematical Physics and Geometry. For example, the Coulomb energy appears in Electrostatics, Celestial Mechanics and Quantum Mechanics, path integrals take the form of multiple convolutions, and interactions of particles are described by more complicated collision kernels in Statistical Mechanics. ***Many of these functionals satisfy geometric inequalities. An important consequence is that their extremals, known as "ground states", are often rotationally symmetric. These inequalities can be viewed as functional versions of the Brunn-Minkowski inequality of convex geometry, and, by extension, the isoperimetric inequality. ***A key problem is the geometric stability of nonlocal functionals --- a question with fundamental implications for continuum limits in Statistical Mechanics. Geometric stability results where a "deficit" (the deviation of a functional from its optimal value) controls some measure of "asymmetry" (the distance from the manifold of optimizers) have been established for many geometric functionals. Little is known for non-local functionals that involve convolutions, but there has been notable progress in the last two years. ***Another fundamental problem concerns the stability of Riesz' rearrangement inequality in higher dimensions, which would imply new stability results for the Brunn-Minkowski inequality in the non-convex case. The ultimate generalization of Riesz' inequality to multiple integrals is the Brascamp-Lieb-Luttinger inequality. Stability for the BLL inequality will require to first classify the equality cases, a long-standing problem that has also seen very recent progress. Exensions of Riesz' rearrangement inequality to spheres will also be considered.***Other topics topics in this proposal are the approximation of the symmetric decreasing rearrangement by sequences of simpler symmetrization, and the symmetry and variational characterization of dispersion-managed solitons.***The proposed work seeks to address the questions described above, and to develop analytical tools that can be applied more broadly.**