Complex varieties and higher integrability of Dir-minimizing Q-valued functions

We provide new elementary proofs of the following two results: every complex variety is locally the graphs of a Dir-minimizing function, first proved by Almgren; the gradients of Dir-minimizing functions, in principle square-summable, are p-integrable for some p > 2, proved by De Lellis and the author. In the planar case, we prove that our integrability exponents are optimal.Comment: 13 page

On the measure and the structure of the free boundary of the lower dimensional obstacle problem

We provide a thorough description of the free boundary for the lower dimensional obstacle problem in $\mathbb{R}^{n+1}$ up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, we prove (i) local finiteness of the $(n-1)$-dimensional Hausdorff measure of the free boundary, (ii) $\mathcal{H}^{n-1}$-rectifiability of the free boundary, (iii) classification of the frequencies up to a set of dimension at most (n-2) and classification of the blow-ups at $\mathcal{H}^{n-1}$ almost every free boundary point

An Epiperimetric Inequality for the Thin Obstacle problem

We prove an epiperimetric inequality for the thin obstacle problem, extending the pioneering results by Weiss on the classical obstacle problem (Invent. Math. 138 (1999), no. 1, 23-50). This inequality provides the means to study the rate of converge of the rescaled solutions to their limits, as well as the regularity properties of the free boundary

Regularity of area minimizing currents III: blow-up

This is the last of a series of three papers in which we give a new, shorter proof of a slightly improved version of Almgren's partial regularity of area minimizing currents in Riemannian manifolds. Here we perform a blow-up analysis deducing the regularity of area minimizing currents from that of Dir-minimizing multiple valued functions

Regularity of area minimizing currents II: center manifold

This is the second paper of a series of three on the regularity of higher codimension area minimizing integral currents. Here we perform the second main step in the analysis of the singularities, namely the construction of a center manifold, i.e. an approximate average of the sheets of an almost flat area minimizing current. Such center manifold is complemented with a Lipschitz multi-valued map on its normal bundle, which approximates the current with a highe degree of accuracy. In the third and final paper these objects are used to conclude a new proof of Almgren's celebrated dimension bound on the singular set.Comment: In the new version the proofs and the structure are improved and some minor errors have been correcte

Non-Uniqueness of Minimizers for Strictly Polyconvex Functionals

In this note we solve a problem posed by Ball (in Philos Trans R Soc Lond Ser A 306(1496):557-611, 1982) about the uniqueness of smooth equilibrium solutions to boundary value problems for strictly polyconvex functionals, $$\mathcal {F}(u)=\int_\Omega f(\nabla u(x)) {\rm d}x\quad{\rm and}\quad u\vert_{\partial\Omega}=u_0,$$ where Ω is homeomorphic to a ball. We give several examples of non-uniqueness. The main example is a boundary value problem with at least two different global minimizers, both analytic up to the boundary. All these examples are suggested by the theory of minimal surface