Geometric Measure Theory with Applications to Shape Optimization Problems

This thesis mainly focuses on geometric measure theory with applications to some shape optimization problems being considered over rough sets. We extend previous theory of traces for rough vector fields over rough domains and proved the compactness of uniform domains without uniformly bounded perimeter assumption. As application of these results, together with some other tools from geometric analysis, we can give partial results on the existence and uniqueness of minimizers of the nematic liquid droplets problem and the thermal insulation problem. Two other geometric minimization problems with averaged property, which include the generalized Cheeger set problem, are also studied

Local and Global Results for Shape optimization problems with weighted source

We consider shape optimization problems of maximizing the averaged heat under various boundary conditions. Assuming that the heat source is radial, we obtain several local stability and global optimality results on ball shape. As a byproduct of stability analysis, we show that Talenti type pointwise comparison result is no longer true under Robin conditions even if the domain is a smooth small perturbation of a ball

Traces and Extensions of Bounded Divergence-Measure Fields on Rough Open Sets

We prove that an open set $\Omega \subset \mathbb{R}^n$ can be approximated by smooth sets of uniformly bounded perimeter from the interior if and only if the open set $\Omega$ satisfies \begin{align*} &\qquad \qquad\qquad\qquad\qquad\qquad\qquad \mathscr{H}^{n-1}(\partial \Omega \setminus \Omega^0)<\infty, \qquad &&\quad\qquad\qquad \qquad\qquad (*) \end{align*} where $\Omega^0$ is the measure-theoretic exterior of $\Omega$. Furthermore, we show that condition (*) implies that the open set $\Omega$ is an extension domain for bounded divergence-measure fields, which improves the previous results that require a strong condition that $\mathscr{H}^{n-1}(\partial \Omega)<\infty$. As an application, we establish a Gauss-Green formula up to the boundary on any open set $\Omega$ satisfying condition (*) for bounded divergence-measure fields, for which the corresponding normal trace is shown to be a bounded function concentrated on $\partial \Omega \setminus \Omega^0$. This new formula does not require the set of integration to be compactly contained in the domain where the vector field is defined. In addition, we also analyze the solvability of the divergence equation on a rough domain with prescribed trace on the boundary, as well as the extension domains for bounded $BV$ functions.Comment: 29 page

A stronger constant rank theorem

Motivated from one-dimensional rigidity results of entire solutions to Liouville equation, we consider the semilinear equation \begin{align} \label{liouvilleequationab} \Delta u=G(u) \quad \mbox{in $\mathbb{R}^n$}, \end{align}where $G>0, G'0$. Let $u$ be a smooth convex solution and $\sigma_k(D^2 u)$ be the $k$-th elementary symmetric polynomial with respect to $D^2u$. We prove stronger constant rank theorems in the following sense. (1) When $A\le 2$, if $\sigma_2(D^2u)$ takes a local minimum, then $D^2 u$ has constant rank $1$. (2) When $A\le \frac{n}{n-1}$, if $\sigma_n(D^2 u)$ takes a local minimum, then $\sigma_n(D^2 u)$ is always zero in the domain

On location of maximum of gradient of torsion function

It has been a widely belief that for a planar convex domain with two coordinate axes of symmetry, the location of maximal norm of gradient of torsion function is either linked to contact points of largest inscribed circle or connected to points on boundary of minimal curvature. However, we show that this is not quite true in general. Actually, we derive the precise formula for the location of maximal norm of gradient of torsion function on nearly ball domains in $\mathbb{R}^n$, which displays nonlocal nature and thus does not inherently establish a connection to the aforementioned two types of points. Consequently, explicit counterexamples can be straightforwardly constructed to illustrate this deviation from conventional understanding. We also prove that for a rectangular domain, the maximum of the norm of gradient of torsion function exactly occurs at the centers of the faces of largest $(n-1)$-volume

Worst-Case Linear Discriminant Analysis as Scalable Semidefinite Feasibility Problems

In this paper, we propose an efficient semidefinite programming (SDP) approach to worst-case linear discriminant analysis (WLDA). Compared with the traditional LDA, WLDA considers the dimensionality reduction problem from the worst-case viewpoint, which is in general more robust for classification. However, the original problem of WLDA is non-convex and difficult to optimize. In this paper, we reformulate the optimization problem of WLDA into a sequence of semidefinite feasibility problems. To efficiently solve the semidefinite feasibility problems, we design a new scalable optimization method with quasi-Newton methods and eigen-decomposition being the core components. The proposed method is orders of magnitude faster than standard interior-point based SDP solvers. Experiments on a variety of classification problems demonstrate that our approach achieves better performance than standard LDA. Our method is also much faster and more scalable than standard interior-point SDP solvers based WLDA. The computational complexity for an SDP with $m$ constraints and matrices of size $d$ by $d$ is roughly reduced from $\mathcal{O}(m^3+md^3+m^2d^2)$ to $\mathcal{O}(d^3)$ ($m>d$ in our case).Comment: 14 page

On a variational problem of nematic liquid crystal droplets

Let $\mu>0$ be a fixed constant, and we prove that minimizers to the following energy functional \begin{align*} E_f(u,\Omega):=\int_{\Omega}|\nabla u|^2+\mu P(\Omega) \end{align*}exist among pairs $(\Omega,u)$ such that $\Omega$ is an $M$-uniform domain with finite perimeter and fixed volume, and $u \in H^1(\Omega,\mathbb{S}^2)$ with $u =\nu_{\Omega}$, the measure-theoretical outer unit normal, almost everywhere on the reduced boundary of $\Omega$. The uniqueness of optimal configurations in various settings is also obtained. In addition, we consider a general energy functional given by \begin{align*} E_f(u,\Omega):=\int_{\Omega} |\nabla u(x)|^2 \,dx + \int_{\partial^* \Omega} f\big(u(x)\cdot \nu_{\Omega}(x)\big) \,d\mathcal{H}^2(x), \end{align*}where $\partial^* \Omega$ is the reduced boundary of $\Omega$ and $f$ is a convex positive function on $\mathbb R$. We prove that minimizers of $E_f$ also exist among $M$-uniform outer-minimizing domains $\Omega$ with fixed volume and $u \in H^1(\Omega,\mathbb{S}^2)$.Comment: 19 page

The Effects of Flagging Propaganda Sources on News Sharing:Quasi-Experimental Evidence from Twitter

While research on flagging misinformation and disinformation has received much attention, we know very little about how the flagging of propaganda sources could affect news sharing on social media. Using a quasi-experimental design, we test the effect of source flagging on people’s actual sharing behaviors. By analyzing tweets (N = 49,126) posted by 30 China's media accounts before and after Twitter's practice of labeling state-affiliated media, we reveal the corrective role that flagging plays in preventing people's sharing of information from propaganda sources. The findings suggest that the corrective effect occurs immediately after these accounts are labeled as state-affiliated media and it leads to a long-term reduction in news sharing, particularly for political content. The results contribute to the understanding of how flagging efforts affect user engagement in real-world conversations and highlight that the effect of corrective measures takes place in a dynamic process