In this note, we establish a boundary maximum principle for a class of stationary pairs of varifolds satisfying a fixed contact angle condition in any compact Riemannian manifold with smooth boundary.
In this note, we establish a boundary maximum principle for a class of stationary pairs of varifolds satisfying a fixed contact angle condition in any compact Riemannian manifold with smooth boundary.
For a function f which foliates a one-sided neighborhood of a closed hypersurface M, we give an estimate of the distance of M to a Wulff shape in terms of the Lp-norm of the traceless F-Hessian of f, where F is the su...
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For a function f which foliates a one-sided neighborhood of a closed hypersurface M, we give an estimate of the distance of M to a Wulff shape in terms of the Lp-norm of the traceless F-Hessian of f, where F is the support function of the Wulff shape. This theorem is applied to prove quantitative stability results for the anisotropic Heintze-Karcher inequality, the anisotropic Alexandrov problem, as well as for the anisotropic overdetermined boundary value problem of Serrin-type. (c) 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).
Characterizing engineering properties of rock especially associated with tension is crucial for stability assessment of rock structures. This study integrates physical and numerical experiments to investigate the elec...
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Characterizing engineering properties of rock especially associated with tension is crucial for stability assessment of rock structures. This study integrates physical and numerical experiments to investigate the electromagnetic radiation (EMR) and acoustic emission (AE) responses of sandstone under Brazilian disc testing. During the Brazilian splitting process, the EMR and AE responses reflect well the cracking evolution of the disc sandstone specimen. The cracking evolution process and failure mechanism are vividly illustrated. When approaching the peak stress, massive EMR and AE activities occur abruptly. Importantly, the fractal dimension and b-value of EMR and AE switch from increase to decrease once the failure initiates. Such significant decrease in the fractal dimension and b-value of EMR and AE upon failure initiation could be applied to identify the rock failure initiation.
In this paper, we study a Serrin-type partially overdetermined problem proposed by Guo-Xia [Calc. Var. Partial Differential Equations (2019), Paper No. 160, 15], and prove a rigidity result that characterizes capillar...
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In this paper, we study a Serrin-type partially overdetermined problem proposed by Guo-Xia [Calc. Var. Partial Differential Equations (2019), Paper No. 160, 15], and prove a rigidity result that characterizes capillary spherical caps in a half ball.
In this paper, we prove the following Willmore-type inequality: on an unbounded closed convex set K subset of Rn+1$K\subset \mathbb {R}{n+1}$ (n >= 2$(n\geqslant 2$), for any embedded hypersurface Sigma subset of K...
In this paper, we prove the following Willmore-type inequality: on an unbounded closed convex set K subset of Rn+1$K\subset \mathbb {R}<^>{n+1}$ (n >= 2$(n\geqslant 2$), for any embedded hypersurface Sigma subset of K${\Sigma }\subset K$ with boundary partial derivative Sigma subset of partial derivative K$\partial {\Sigma }\subset \partial K$ satisfying a certain contact angle condition, there holds 1n+1 integral Sigma HndA >= AVR(K)|Bn+1|.$$\begin{equation*} {\frac{1}{n+1}\int _{{\Sigma }}{\left|{H}\right|}<^>n d A\geqslant \rm AVR}(K)\vert \mathbb {B}<^>{n+1}\vert . \end{equation*}$$Moreover, equality holds if and only if Sigma${\Sigma }$ a part of a sphere and K\Omega$K\setminus \Omega$ is a part of the solid cone determined by Sigma${\Sigma }$. Here, Omega$\Omega$ is the bounded domain enclosed by Sigma${\Sigma }$ and partial derivative K$\partial K$, H$H$ is the normalized mean curvature of Sigma${\Sigma }$, and AVR(K)${\rm AVR}(K)$ is the asymptotic volume ratio of K$K$. We also prove an anisotropic version of this Willmore-type inequality. As a special case, we obtain a Willmore-type inequality for anisotropic capillary hypersurfaces in a half-space.
Direct arylation polycondensation(DArP)has emerged as an eco-friendly and atom-efficient methodology for the syntheses ofπ-conjugated polymers(CPs).This approach features the direct C—H arylation of an aromatic hydr...
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Direct arylation polycondensation(DArP)has emerged as an eco-friendly and atom-efficient methodology for the syntheses ofπ-conjugated polymers(CPs).This approach features the direct C—H arylation of an aromatic hydrocarbon with an aryl *** the prevalence of thiophene-containing CPs,achieving efficient and defect-free DArP of thiophene-based C−H monomers is of great *** review presents a mechanistic insight into DArP and describes the development of DArP catalytic systems for varied thiophene-based C−H ***,the control of the primary defects(i.e.,branching and homo-coupling)in thiophene-based DArP is also *** emphasizing the principles behind monomer selection and catalytic system optimization,this review intends to provide a framework for future advancements in the DArP of thiophene-containing CPs.
In this paper, we first prove a rigidity result for a Serrin-type partially overdetermined problem in the half-space, which gives a characterization of capillary spherical caps by the overdetermined problem. In the se...
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In this paper, we first prove a rigidity result for a Serrin-type partially overdetermined problem in the half-space, which gives a characterization of capillary spherical caps by the overdetermined problem. In the second part, we prove quantitative stability results for the Serrin-type partially overdetermined problem, as well as capillary almost constant mean curvature hypersurfaces in the half-space.
It is of great theoretical significance to effectively reveal the energy driving-damage degradation-structural failure mechanism of rocks for the probability evaluation and prevention of rock burst in deep coal mines....
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It is of great theoretical significance to effectively reveal the energy driving-damage degradation-structural failure mechanism of rocks for the probability evaluation and prevention of rock burst in deep coal mines. Therefore, uniaxial loading and unloading tests of layered sandstones were conducted. Subsequently, the progressive failure process and stress-strain curves were obtained. Then, the evolution characteristics of strength, strain, energy, damage and macroscopic failure was characterized. Meanwhile, the damage proportion at each stage was quantified, and the linear damage deterioration law was obtained. Finally, the energy driving-damage degradation-structural failure mechanism was revealed. The results showed that: (1) increasing the unloading stress level did not necessarily reduce the bearing capacity, but the unloading effect could significantly affect the energy storage capacity;(2) there was the significant linear evolution relationship between the damage proportion and the unloading stress level under identical inclination angle in stages II and IV;(3) When the unloading stress level was over 0.7, the pre-peak structural adjustment behavior could strengthen the stability of post-peak structures and reduce the impact degree of rock burst. The conclusions could provide certain of theoretical basis for the prevention of rock burst in deep coal mines.
In this paper, we show that any embedded capillary hypersurface in the half-space with anisotropic constant mean curvature is a truncated Wulff shape. This extends Wente's result (Pac J Math 88:387-397, 1980. ) to...
In this paper, we show that any embedded capillary hypersurface in the half-space with anisotropic constant mean curvature is a truncated Wulff shape. This extends Wente's result (Pac J Math 88:387-397, 1980. ) to the anisotropic case and He-Li-Ma-Ge's result (Indiana Univ Math J 58(2):853-868, 2009. ) to the capillary boundary case. The main ingredients in the proof are a new Heintze-Karcher inequality and a new Minkowski formula, which have their own interest.
In this paper, by isometrically embedding (Sn+1, gS(n+1)) into Rn+2, and using nonlinear analysis on the codimension-2 graphs, we will show that the level-sets of the distance function from the boundary of any open se...
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In this paper, by isometrically embedding (Sn+1, gS(n+1)) into Rn+2, and using nonlinear analysis on the codimension-2 graphs, we will show that the level-sets of the distance function from the boundary of any open set in sphere, are C-1,C-1-rectifiable. As a by-product, we establish a Heintze-Karcher-type inequality on sphere.
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