Regularity of the free boundary for the two-phase Bernoulli problem

We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we also show regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian.

Average-distance problem with curvature penalization for data parameterization: regularity of minimizers

We propose a model for finding one-dimensional structure in a given measure. Our approach is based on minimizing an objective functional which combines the the average-distance functional to measure the quality of the approximation and penalizes the curvature, similarly to the elastica functional. Introducing the curvature penalization overcomes some of the shortcomings of the average-distance functional, in particular the lack of regularity of minimizers. We establish existence, uniqueness and regularity of minimizers of the proposed functional. In particular we establish $C^{1,1}$ estimates on the minimizers.

Variational formulae and estimates of O’Hara’s knot energies

O’Hara’s energies, introduced by Jun O’Hara, were proposed to answer the question what the canonical shape in a given knot type is, and were configured so that the less the energy value of a knot is, the “better” its shape is. The existence and regularity of minimizers has been well studied. In this paper, we calculate the first and second variational formulae of the [Formula: see text]-O’Hara energies and show absolute integrability, uniform boundedness, and continuity properties. Although several authors have already considered the variational formulae of the [Formula: see text]-O’Hara energies, their techniques do not seem to be applicable to the case [Formula: see text]. We obtain the variational formulae in a novel manner by extracting a certain function from the energy density. All of the [Formula: see text]-energies are made from this function, and by analyzing it, we obtain not only the variational formulae but also the estimates in several function spaces.