| The technology of superconductivity and liquid crystals has been used in modern physics widely. As the superconductor an example, it plays an important role in the developing of many fields, such as, energy, medical, information and national defense. On the other hand, the applications of liquid crystal are very common in the nature world. LCD, as an example, has been successfully used in consumer devices, such as watches, computers, telephones and video players. It builds a bridge for communications between human beings and machines. In the future years, the research of superconductivity and liquid crystals will still be concerned by physicists and mathematicians.First, my research focuses on the mathematical theory of superconductiv-ity. Especially, we study the nodal sets of the solutions to magnetic Shrodinger equations involving the magnetic field. whereψis a complex-valued solution.The nodal set ofψis usually called vortex set. At the zero point ofψ, the superconductor loses its superconductivity. The research on the nodal set may give us a clear profile of the vortex set.It is known that the complexity of a nodal set mainly comes from the critical set. So we first establish a global 1-dimensional Hausdorff measure estimate for the critical sets under the boundary condition:▽Aψ·v=0.We will borrow the geometric measure methods from [29] to estimate the Hausdorff measure of S(ψ), but their results do not apply to our case directly where lies in three aspects. One is the equation involving a magnetic Schrodinger operator, the second one is that the solutionψis complex-valued function, the last one is we need a global estimates to S(ψ), not a local one. Next we turn to study the set N(ψ). Sinceψis a complex-valued function, the structure of N(ψ) might be complicated. Locally, it may be either a single point, a curve, or even a surface. To overcome the problem, we add a restriction onψ, which is called quasi-conformal condition. Under this assumption, we proved that N(ψ)) is 1-rectifiable, and its 1-dimensional Hausdorff measure is bounded.To explore more properties of the nodal set, we work on the complexity of the solution to PDEs in the point of topological view, that is, Betti numbers.In 1963, J. Milnor gave a beautiful proof on the sum of Betti numbers for polynomials. However, when the problem comes to the solutions of PDEs. we can not only work on polynomials. So it is necessary to obtain the corresponding estimates on the total Betti numbers for solutions of second-order elliptic PDEs.We consider a solution to a general second-order elliptic equation and prove that the total Betti numbers of N(u)∩B1/2(0) is bounded.This theorem tells us that the topology of the nodal set can be controlled by a constant C, which only depends on the coefficients of PDEs and the uniformly upper bound of the vanishing order.I am also interested in the mathematical theory in liquid crystals. Landau-de Gennes model is the most important model in the research of liquid crystals, which can describe phase transitions of smectic state liquid crystals.Here we consider a simplified functional of Landau-de Gennes model, We establish the interior partial regularity results:After removing some relatively closed set, whose 1-dimension Hausdorff measure equals to zero, we show that the minimizers are analytic.We should point out what differ from the research of Ossen-Frank model lies in two respects. One is that Landau-de Gennes functional involves the order parameterΨ, which is regarded as a dependent variable in our case. On the other hand, we will discuss the problem with the extra term They will bring some complexity in the proof. The partial regularity theorem we obtained gives us a further understanding about the physical properties of the liquid crystals.
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