Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form

In this paper we consider minimizers of the functional$\min \{{\lambda }_1(\mathrm{\Omega })+\cdots +{\lambda }_k(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,:\mathrm{\Omega }\subset D\text{ open}\}$ where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_k(\mathrm{\Omega })$ are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets ${\mathrm{\Omega }}^{⁎}$ have finite perimeter and that their free boundary $\partial {\mathrm{\Omega }}^{⁎}\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,\alpha }$-regular function, and a singular part, which is empty if $d<d^{⁎}$, discrete if $d=d^{⁎}$ and of Hausdorff dimension at most $d-d^{⁎}$ if $d>d^{⁎}$, for some $d^{⁎}\in \{5,6,7\}$.     

Regularity and Rigidity for Nonlocal Curvatures in Conformal Geometry

In this paper, we will explore the geometric effects of conformally covariant operators and the induced nonlinear curvature equations in certain nonlocal nature. Mainly, we will prove some regularity and rigidity results for the distributional solutions to those equations.     

On the Generalized Porous Medium Equation in Fourier-Besov Spaces

We study a kind of generalized porous medium equation with fractional Laplacian and abstract pressure term. For a large class of equations corresponding to the form: $u_t+\nu \Lambda^{\beta}u=\nabla\cdot(u\nabla Pu)$, we get their local well-posedness in Fourier-Besov spaces for large initial data. If the initial data is small, then the solution becomes global. Furthermore, we prove a blowup criterion for the solutions.     

基于图像处理的光学谐振腔自准直研究方法

针对我国氧碘化学激光器输出能量不断提高的情况下,主要限制光束质量的因素之一是出光过程中的光腔失调,提出了一种基于图像处理的光学谐振腔自准直研究方法。该研究方法利用氦氖穿腔光通过凹腔镜中心孔,并在凸腔镜后方放置工业相机,在屏幕上方形成一个干涉同心圆环。通过借助OpenCV库中二值化、霍夫圆检测等图像处理的方法,获得干涉同心圆环的圆心坐标,再通过以圆心为中心将图像分成上下左右四个部分,通过判断这四部分亮点像素的数量差值来预估干涉同心圆环的偏移状态,进而获得空腔条件下光学谐振腔发生变化的理论判据。实验结果表明,该方法可以获得相当精准的光学谐振腔失调判据,其失调判据选取左右亮点像素差值30000,上下亮点像素差值45000,为今后失调判据与步进电机自动调腔的结合做好充足的准备。     

Regularity results of the thin obstacle problem for the p(x)-Laplacian

We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and Hölder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent $p(x)$ is Hölder continuous.     

Explicit estimates of sums related to the Nyman–Beurling criterion for the Riemann Hypothesis

We give an estimate for sums appearing in the Nyman–Beurling criterion for the Riemann Hypothesis. These sums contain the Möbius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably sharp in comparison to other sums containing the Möbius function. The bound is smaller than the trivial bound – essentially the number of terms – by a fixed power of that number. The exponent is made explicit. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.     

Orientation dependent size effects in thermal buckling and post-buckling of nanoplates with cubic anisotropy

In this work, a continuum model is presented for size and orientation dependent thermal buckling and post-buckling of anisotropic nanoplates considering surface and bulk residual stresses. The model with von-Karman nonlinear strains and material cubic anisotropy of single crystals contains two parameters that reflect the orientation effects. Using Ritz method, closed form solutions are given for buckling temperature and post-buckling deflections. Regarding self-instability states of nanoplates and their recovering at higher temperatures, an experiment is discussed based on low pressurized membranes to verify the predictions. For simply supported nanoplates, the size effects are lowest when they are aligned in [100] direction. When the edges get clamped, the orientation dependence is ignorable and the behavior becomes symmetric about [510] axis. The surface residual stress makes drastic increase in buckling temperature of thinner nanoplates for which a minimum thickness is pointed to stay far from material softening at higher temperatures. Deflection of [100]-oriented buckled nanoplates is higher than [110] ones but this reverses at higher temperatures. The results for long nanoplates show that the buckling mode numbers are changed by orientation which is verified by FEM.