司靈得 (Daniel Spector)

We study the asymptotic behavior, when k → ∞, of the minimizers of the energy \[\mathcal{G}_k(u) = \int_\Omega \left( \frac{k - 1}{2} |\nabla u|^2 + \frac{1}{2} \left(1 - |u|^2\right)^2 \right) dx,\] over the class of maps u ∈ H¹(Ω, ℝ²) satisfying the boundary condition u = g on ∂Ω, where Ω is a smooth, bounded and simply connected domain in ℝ² and g: ∂Ω → S¹. The motivation comes from a simplified version of Ericksen model for nematic liquid crystals. We will present similarities and differences with respect to the analog problem for the Ginzburg-Landau energy. Based on a joint work with Dmitry Golovaty.

Joint work with Almut Burchard (Toronto) and Ryan Gibara (Cincinnati). Let f be a function of bounded mean oscillation (BMO) on cubes in ℝⁿ, n > 1. If f is rearrangeable, we show that its symmetric decreasing rearrangement Sf belongs to BMO(ℝⁿ). We also improve the bounds for the decreasing rearrangement f^* by Bennett, DeVore and Sharpley, ||f^*||_BMO(ℝ⁺) ≤ C_n||f||_BMO(ℝⁿ), by eliminating the exponential dependence of C_n on the dimension n. The key is to switch from cubes to a comparable family of shapes. Using a family of rectangles that is preserved under bisections, one can prove a dimension-free Calderón-Zygmund decomposition, and the boundedness of the decreasing rearrangement with the same constant. Restricting to the subspace of functions of vanishing mean oscillation (VMO), we show that these rearrangements take VMO functions to VMO functions. Furthermore, while the map from f to f^* is not continuous in the BMO seminorm, we prove continuity when the limit is in VMO.

It is important to describe the motion of phase boundaries by macroscopic energy in the process of phase transitions. Typical energy describing the phenomena is the van der Waals energy, which is also called a Modica-Mortola functional with a double-well potential or the Allen-Cahn functional. It turns out that it is also important to consider the Modica-Mortola functional with a single-well potential since it is often used in various settings including the Kobayashi-Warren-Carter energy, which is popular in materials science. It is very fundamental to understand the singular limit of such a type of energies as the thickness parameter of a diffuse interface tends to zero. In the case of double-well potentials, such a problem is well-studied and it is formulated, for example, as the Gamma limit under L¹ convergence.

However, if one considers the Modica-Mortola functional, it turns out that L¹ convergence is too rough even in the one-dimensional problem.

We characterize the Gamma limit of a single-well Modica-Mortola functional under the topology which is finer than L¹ topology. In a one-dimensional case, we take the graph convergence. In higher-dimensional cases, it is more involved. As an application, we give an explicit representation of a singular limit of the Kobayashi-Warren-Carter energy. Since the higher-dimensional cases can be reduced to the one-dimensional case by a slicing argument, studying the one-dimensional case is very fundamental. A key idea to study the one-dimensional case is to introduce "an unfolding of a function" by changing an independent variable by the arc-length parameter of its graph. This is based on a joint work with Jun Okamoto (The University of Tokyo), Masaaki Uesaka (The University of Tokyo, Arithmer Inc.), and Koya Sakakibara (Okayama University of Science, RIKEN).

It has been an open question if maximal operators M satisfy the endpoint regularity bound \( \mathrm{var}(Mf) \leq C \mathrm{var}(f) \). So far the majority of the known results has been in one dimension. I give an overview of the progress on this question with a focus on the techniques. Next I present the techniques used in the recent proofs of \( \mathrm{var}(Mf) \leq C \mathrm{var}(f) \) for several maximal operators in higher dimensions. They are mostly geometric measure theoretic in the spirit of the relative isoperimetric inequality and involve a stopping time and various covering arguments.

I will report some of the recent progress regarding the boundedness and continuity of the map f → M_αf from the endpoint space W^1,1(ℝ^d) to L^{d/(d - α)}(ℝ^d), where M_α denotes the fractional version of either the centered or uncentered Hardy--Littlewood maximal function. After contributions by several authors, the problem is now totally solved in an affirmative way. I will focus on my contributions, which correspond to the radial case (in joint work with J. Madrid), and also to the general case for the continuity of the map (in joint work with C. González-Riquelme, J. Madrid and J. Weigt).

It has been an open question if maximal operators M satisfy the endpoint regularity bound \( \mathrm{var}(Mf) \leq C \mathrm{var}(f) \). So far the majority of the known results has been in one dimension. I give an overview of the progress on this question with a focus on the techniques. Next I present the techniques used in the recent proofs of \( \mathrm{var}(Mf) \leq C \mathrm{var}(f) \) for several maximal operators in higher dimensions. They are mostly geometric measure theoretic in the spirit of the relative isoperimetric inequality and involve a stopping time and various covering arguments.