This proposal is focused on three problems in three somewhat overlapping areas, namely geometric measure theory, harmonic analysis, and the calculus of variations. The first problem is Besicovitch's 1/2- conjecture regarding the sharpest bound for the (a.e.) lower spherical density for purely unrectifiable 1-sets in Euclidean space (or more general spaces as well). The second is the Harmonic Lip_1-Capacity problem (Mattila's conjecture and the related David-Semmes problem), regarding the characterization (in terms of rectifiability) of sets of co-dimension one which allow a Lipschitz harmonic function defined off the set to be extended to all of Euclidean space. This is precisely the higher dimensional analogue of the analytic capacity problem. The third problem is regarding the question of regularity of minimizers of the relaxed energy functional (introduced by Bethuel, Brezis, and Coron) which is basically the usual Dirichlet energy plus a cost function on singularities. On all three problems the PI has done a good deal of work (the third problem is joint with R. Hardt) advancing the state of knowledge. Our program on the first problem has met significant progress in the recent months exceeding the PI's expectations and appears close to completion. Our work on the other two appears to also be heading in direction of further progress as well.
The first problem addresses one of the most fundamental questions regarding the geometrical properties of purely unrectifiable sets going back to Besicovitch's two classic papers which established the foundation of the subject (1928, 1938). These sets are fractal-like, appearing in several contexts in the physical world and hence the sciences (e.g. dynamical systems, number theory etc.). Our work on the problem has required finding new algorithms dealing with very general closed sets, and we expect it to have an impact on the understanding of other related geometrical questions of densities and rectifiability not to mention our understanding of the properties of these natural sets.. Our second problem combines geometry and harmonic analysis in a natural way, having impact on several areas of analysis and also to Physics, where harmonic functions have a special role (e.g. in electrostatics). Our third problem overlaps geometry, topology, partial differential equations, and geometric measure theory. It addresses deep questions in its area. It is of special importance in the theory of liquid crystals since the quantities being minimized are similar to the energies of such physical systems
