Subjects: Optics >> Quantum optics submitted time 2023-02-19
Abstract： Optical modulators are required to have high modulation bandwidths and a compact footprint. In this paper we experimentally demonstrate a novel Si/In$_2$O$_3$ hybrid plasmonic waveguide modulator, which is realized by an asymmetric directional coupler (ADC) consisting of a silicon photonic waveguide and a Si/In$_2$O$_3$ hybrid plasmonic waveguide. The optical signal is modulated by radio-frequency (RF) signal applied on the Au electrodes at the top of MOS capacitor and contacting the In$_2$O$_3$ thin film. The record-high modulation bandwidth of >40 GHz is realized by a silicon-doping-free metal-oxide-In$_2$O$_3$ capacitor integrated in a 3.5-$\mu$m-long asymmetric directional coupler (ADC) for the first time.
Peer Review Status:
Awaiting Review
Subjects: Mathematics >> Mathematics （General） submitted time 2018-09-23
Abstract： " The aim of this paper is to study the heterogeneous optimization problem \begin{align*} \mathcal {J}(u)=\int_{\Omega}(G(|\nabla u|)+qF(u^+)+hu+\lambda_{+}\chi_{\{u>0\}} )\text{d}x\rightarrow\text{min}, \end{align*} in the class of functions $ W^{1,G}(\Omega)$ with $ u-\varphi\in W^{1,G}_{0}(\Omega)$, for a given function $\varphi$, where $W^{1,G}(\Omega)$ is the class of weakly differentiable functions with $\int_{\Omega}G(|\nabla u|)\text{d}x<\infty$. The functions $G$ and $F$ satisfy structural conditions of Lieberman's type that allow for a different behavior at $0$ and at $\infty$. Given functions $q,h$ and constant $\lambda_+\geq 0$, we address several regularities for minimizers of $\mathcal {J}(u)$, including local $C^{1,\alpha}-$, and local Log-Lipschitz continuities for minimizers of $\mathcal {J}(u)$ with $\lambda_+=0$, and $\lambda_+>0$ respectively. We also establish growth rate near the free boundary for each non-negative minimizer of $\mathcal {J}(u)$ with $\lambda_+=0$, and $\lambda_+>0$ respectively. Furthermore, under additional assumption that $F\in C^1([0,+\infty); [0,+\infty))$, local Lipschitz regularity is carried out for non-negative minimizers of $\mathcal {J}(u)$ with $\lambda_{+}>0$. " "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics （General） submitted time 2018-09-22
Abstract： "
In this paper, we prove several regularity results for the heterogeneous, two-phase free boundary problems $\mathcal {J}_{\gamma}(u)=\int_{\Omega}\big(f(x,\nabla
u)+(\lambda_{+}(u^{+})^{\gamma}+\lambda_{-}(u^{-})^{\gamma})+gu\big)\text{d}x\rightarrow \text{min}$
under non-standard growth conditions. Included in such problems are
heterogeneous jets and cavities of Prandtl-Batchelor type with
$\gamma=0$, chemical reaction problems with $0<\gamma<1$, and obstacle
type problems with $\gamma=1$. Our results hold not only in
the degenerate case of $p> 2$ for $p-$Laplace equations, but
also in the singular
case of $1
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics （General） submitted time 2018-09-22
Abstract： We establish regularity of solutions to the $G$-Laplace equation $-\text{div}\ \bigg(\frac{g(|\nabla u|)}{|\nabla u|}\nabla u\bigg)=\mu$, where $\mu$ is a nonnegative Radon measure satisfying $\mu (B_{r}(x_{0}))\leq Cr^{m}$ for any ball $B_{r}(x_{0})\subset\subset \Omega$ with $r\leq 1$ and $m>n-1-\delta\geq 0$. The function $g(t)$ is supposed to be nonnegative and $C^{1}$-continuous in $[0,+\infty)$, satisfying $g(0)=0$, and for some positive constants $\delta$ and $g_{0}$, $\delta\leq \frac{tg'(t)}{g(t)}\leq g_{0}, \forall t>0$, that generalizes the structural conditions of Ladyzhenskaya-Ural'tseva for an elliptic operator.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics （General） submitted time 2018-09-13
Abstract： The aim of this paper is to study the obstacle problem associated with an elliptic operator having degenerate coercivity, and with a low order term and $L^1-$data. We prove the existence of an entropy solution to the obstacle problem and show its continuous dependence on the $L^{1}-$data in $W^{1,q}(\Omega)$ with some $q>1$. " " " "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics （General） submitted time 2018-06-04
Abstract： " In this work, we establish several Lyapunov-type inequalities for a class of nonlinear higher order differential equations having a form \begin{align*} (\psi(u^{(m)}(x)))'+\sum_{i=0}^nr_i(x)f_i(u^{(i)}(x))=0, %\ \ \ \ \text{or}\ \ \ \ (\psi(u^{(m)}))^{(m)}+r_i(x)f(u)=0, \end{align*} with anti-periodic boundary conditions, where $m> n\geq 0$ are integers, $\psi$ and $f_i (i=0,1,2,...,n)$ satisfy certain structural conditions such that the considered equations have general nonlinearities. The obtained inequalities are extensions and complements of the existing results in the literature. "
Peer Review Status:Awaiting Review
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