Fanghua Lin

A new proof of the Caffarelli‐Kohn‐Nirenberg theorem

Here we give a self-contained new proof of the partial regularity theorems for solutions of incompressible Navier-Stokes equations in three spatial dimensions. These results were originally due to Scheffer and Caffarelli, Kohn, and Nirenberg. Our proof is much more direct and simpler.

An Optimal Partition Problem for Eigenvalues

In this paper, we studied an optimal partition problem for the Dirichlet eigenvalue. We established the existence of classical solutions to such problems as well as the regularity of free interfaces.

Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries

Here we study the asymptotic limits of solutions of some singularly perturbed elliptic systems. The limiting problems involve multiple valued harmonic functions or, in general, harmonic maps to singular spaces and free interfaces between supports of various components of the maps. The main results of the paper are the uniform Lipschitz regularity of solutions as well as the regularity of free interfaces.

Homogenization of elliptic systems with Neumann boundary conditions

The main purpose of this work is to study uniform regularity estimates for a family of elliptic operators

On the Dirichlet problem for minimal graphs in hyperbolic space

\{\mathcal{L}_\varepsilon, \varepsilon>0\}

Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds

, arising in the theory of homogenization, with rapidly oscillating periodic coefficients. We establish sharp

A quantization property for static Ginzburg-Landau vortices

W^{1,p}

Homogenization of elliptic problems withL p boundary data

estimates, Lipschitz estimates, and nontangential maximal function estimates, which are uniform in the parameter

Global Small Solutions to a Complex Fluid Model in Three Dimensional

\varepsilon

A partial answer to a conjecture of B. Simon concerning unique continuation

, on solutions with Neumann boundary conditions in