Emanuele Spadaro

valued functions revisited

In this note we revisit Almgrens theory of Q-valued functions, that are functions taking values in the space of unordered Q-tuples of points in R^n. In particular: 1) we give shorter versions of Almgrens proofs of the existence of Dir-minimizing Q-valued functions, of their Hoelder regularity and of the dimension estimate of their singular set; 2) we propose an alternative intrinsic approach to these results, not relying on Almgrens biLipschitz embedding; 3) we improve upon the estimate of the singular set of planar Dir-minimizing functions by showing that it consists of isolated points.

Uniform energy and density distribution: diblock copolymers' functional

We study a nonlocal variational problem arising in diblock copolymers models, whose energy is given by the Cahn‐Hilliard functional plus a long-range interaction term. We prove that minimizers develop uniform energy and density distributions, thus justifying partially the highly regular microphase separation observed in diblock copolymers’ melts. We also give a new proof of the scaling law for the minimum energy. This work extends the techniques introduced in [1] where analogous results are proved for the sharp interface limit of the functional considered.

On the Measure and the Structure of the Free Boundary of the Lower Dimensional Obstacle Problem

AbstractWe provide a thorough description of the free boundary for the lower dimensional obstacle problem in

Regularity of higher codimension area minimizing integral currents

{\mathbb{R}^{n+1}}

Quasi-Monotonicity Formulas for Classical Obstacle Problems with Sobolev Coefficients and Applications

Rn+1 up to sets of null

Regularity of area minimizing currents III: blow-up

{\mathcal{H}^{n-1}}