Regularity of Minimizers of a Tensor-valued Variational Obstacle Problem in Three Dimensions

TL;DR

This paper investigates the regularity properties of energy-minimizing tensor fields in a 3D obstacle problem inspired by liquid crystal models, establishing interior and boundary regularity results and characterizing contact sets.

Contribution

It provides new regularity results for tensor-valued minimizers in a 3D obstacle problem, including interior and boundary partial regularity and contact set characterization.

Findings

Higher interior regularity of the minimizer $Q$

Contact set is either empty or has small Hausdorff dimension

Boundary partial regularity of the energy-minimizer

Abstract

Motivated by Ball and Majumdar's modification of Landau-de Gennes model for nematic liquid crystals, we study energy-minimizer $Q$ of a tensor-valued variational obstacle problem in a bounded 3-D domain with prescribed boundary data. The energy functional is designed to blow up as $Q$ approaches the obstacle. Under certain assumptions, especially on blow-up profile of the singular bulk potential, we prove higher interior regularity of $Q$, and show that the contact set of $Q$ is either empty, or small with characterization of its Hausdorff dimension. We also prove boundary partial regularity of the energy-minimizer.