In this talk I will discuss the problem of finding Einstein metrics in the homogeneous and cohomogeneity one setting.
In particular, I will describe a recent result concerning existence of solutions to the Dirichlet problem for cohomogeneity one Einstein metrics.
1) The Hardt-Lin's problem and a new approximation of a relaxed energy for harmonic maps.
We introduce a new approximation for the relaxed energy $F$ of the Dirichlet energy and prove that the minimizers of the approximating functional converge to a minimizer $u$ of the relaxed energy for harmonic maps, and that $u$ is partially regular without using the concept of Cartesian currents.
2) Partial regularity in liquid crystals for the Oseen-Frank model: a new proof of the result of Hardt, Kinderlehrer and Lin.
Hardt, Kinderlehrer and Lin (\cite {HKL1}, \cite {HKL2}) proved that a minimizer $u$ is smooth on some open subset
$\Omega_0\subset\Omega$ and moreover $\mathcal H^{\b} (\Omega\backslash \Omega_0)=0$ for some positive $\b <1$, where
$\mathcal H^{\b}$ is the Hausdorff measure. We will present a new proof of Hardt, Kinderlehrer and Lin.
3) Global existence of solutions of the Ericksen-Leslie system for the Oseen-Frank model.
The dynamic flow of liquid crystals is described by the Ericksen-Leslie system. The Ericksen-Leslie system is a system of the Navier-Stokes equations coupled with the gradient flow for the Oseen-Frank model, which generalizes the heat flow for harmonic maps into the $2$-sphere. In this talk, we will outline a proof of global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank model in 2D.
1) The Hardt-Lin's problem and a new approximation of a relaxed energy for harmonic maps.
We introduce a new approximation for the relaxed energy $F$ of the Dirichlet energy and prove that the minimizers of the approximating functional converge to a minimizer $u$ of the relaxed energy for harmonic maps, and that $u$ is partially regular without using the concept of Cartesian currents.
2) Partial regularity in liquid crystals for the Oseen-Frank model: a new proof of the result of Hardt, Kinderlehrer and Lin.
Hardt, Kinderlehrer and Lin (\cite {HKL1}, \cite {HKL2}) proved that a minimizer $u$ is smooth on some open subset
$\Omega_0\subset\Omega$ and moreover $\mathcal H^{\b} (\Omega\backslash \Omega_0)=0$ for some positive $\b <1$, where
$\mathcal H^{\b}$ is the Hausdorff measure. We will present a new proof of Hardt, Kinderlehrer and Lin.
3) Global existence of solutions of the Ericksen-Leslie system for the Oseen-Frank model.
The dynamic flow of liquid crystals is described by the Ericksen-Leslie system. The Ericksen-Leslie system is a system of the Navier-Stokes equations coupled with the gradient flow for the Oseen-Frank model, which generalizes the heat flow for harmonic maps into the $2$-sphere. In this talk, we will outline a proof of global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank model in 2D.
The dynamic flow of liquid crystals is described by the Ericksen-Leslie system. The Ericksen-Leslie system is a system of the Navier-Stokes equations coupled with the gradient flow for the Oseen-Frank model, which generalizes the heat flow for harmonic maps into the $2$-sphere. In this talk, we will outline a proof of global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank model in 2D.