Pattern Formation in Nematic Liquid Crystals

We study the spherical droplet problem in 3D–Landau de Gennes theory with finite temperature. A solution with half–degree ring disclination and a solution with split–core disclination are constructed rigorously. The numerical results of Gartland–Mkaddem 2000 are theoretically confirmed. The main idea is to control the eigenvalues and eigenvectors of the \(Q\)-tensors by considering the thin obstacle problem for semilinear elliptic equations. This construction is novel and the main difficulty is to deal with the regularity on the obstacle.