This paper studies a shape optimization problem which reduces to a nonlocal free boundary problem involving perimeter. It is motivated by a study of liquid crystal droplets with a tangential ...anchoring boundary condition and a volume constraint. We establish in 2D the existence of an optimal shape that has two cusps on the boundary. We also prove that the boundary of the droplet is a chord–arc curve with its normal vector field in the VMO space, and its arc-length parameterization belongs to the Sobolev space
H
3
/
2
. In fact, the boundary curves of such droplets closely resemble the so-called Weil–Petersson class of planar curves. In addition, the asymptotic behavior of the optimal shape when the volume becomes extremely large or small is studied.
Liquid Crystal Flows in Two Dimensions Lin, Fanghua; Lin, Junyu; Wang, Changyou
Archive for rational mechanics and analysis,
07/2010, Letnik:
197, Številka:
1
Journal Article
Recenzirano
This paper is concerned with a simplified hydrodynamic equation, proposed by Ericksen and Leslie, modeling the flow of nematic liquid crystals. In dimension two, we establish both interior and ...boundary regularity theorems for such a flow under smallness conditions. As a consequence, we establish the existence of global (in time) weak solutions on a bounded smooth domain in
which are smooth everywhere with possible exceptions of finitely many singular times.
This book provides a broad yet comprehensive introduction to the analysis of harmonic maps and their heat flows. The first part of the book contains many important theorems on the regularity of ...minimizing harmonic maps by Schoen–Uhlenbeck, stationary harmonic maps between Riemannian manifolds in higher dimensions by Evans and Bethuel, and weakly harmonic maps from Riemannian surfaces by Helein, as well as on the structure of a singular set of minimizing harmonic maps and stationary harmonic maps by Simon and Lin.
In this survey on extremum problems of Laplacian-Dirichlet eigenvalues of Euclidian domains, the author briefly presents some relevant classical results and recent progress. The main goal is to ...describe the well-known conjecture due to Polya, its connections to Weyl’s asymptotic formula for eigenvalues and shape optimizations. Many related open problems and some preliminary results are also discussed.
We study some geometric and potential theoretic properties of nodal domains of solutions to certain uniformly elliptic equations. In particular, we establish corkscrew conditions, Carleson type ...estimates and boundary Harnack inequalities on a class of nodal domains.
For a family of second-order elliptic systems with rapidly oscillating periodic coefficients in a C^{1,\alpha } estimates, Lipschitz estimates, and nontangential maximal function estimates on ...solutions with Neumann boundary conditions.
The aim of this article is to provide a simple and unified way to obtain the sharp upper bounds of nodal sets of eigenfunctions for different types of eigenvalue problems on real analytic domains. ...The examples include biharmonic Steklov eigenvalue problems, buckling eigenvalue problems and champed-plate eigenvalue problems. The geometric measure of nodal sets are derived from doubling inequalities and growth estimates for eigenfunctions. It is done through analytic estimates of Morrey–Nirenberg and Carleman estimates.
For any n-dimensional compact Riemannian manifold （M,g） without boundary and another compact Riemannian manifold （N,h）, the authors establish the uniqueness of the heat flow of harmonic maps from M ...to N in the class C（0,T）,W1,n）. For the hydrodynamic flow （u,d） of nematic liquid crystals in dimensions n = 2 or 3, it is shown that the uniqueness holds for the class of weak solutions provided either （i） for n = 2, u ∈ Lt∞ L2x∩L2tHx1, ▽P ∈ Lt4/3 Lx4/3 , and ▽d ∈ L∞t Lx2∩Lt2Hx2; or （ii） for n = 3, u ∈ Lt∞ Lx2∩L2tHx1∩ C（0,T）,Ln）, P ∈ Ltn/2 Lxn/2 , and ▽d ∈ L2tLx2 ∩ C（0,T）,Ln）. This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.
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