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Register a new userBoundary blow-up analysis of gradient estimates for Lam{e} systems in the presence of $M$-convex hard inclusions
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In high-contrast elastic composites, it is vitally important to investigate the stress concentration from an engineering point of view. The purpose of this paper is to show that the blowup rate of the stress depends not only on the shape of the inclusions, but also on the given boundary data, when hard inclusions are close to matrix boundary. First, when the boundary of inclusion is partially relatively parallel to that of matrix, we establish the gradient estimates for Lam{e} systems with partially infinite coefficients and find that they are bounded for some boundary data $varphi$ while some $varphi$ will increase the blow-up rate. In order to identify such novel blowup phenomenon, we further consider the general $m$-convex inclusion cases and uncover the dependence of blow-up rate on the inclusions convexity $m$ and the boundary datas order of growth $k$ in all dimensions. In particular, the sharpness of these blow-up rates is also presented for some prescribed boundary data.
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In this paper, we derive the pointwise upper bounds and lower bounds on the gradients of solutions to the Lam{e} systems with partially infinite coefficients as the surface of discontinuity of the coefficients of the system is located very close to the boundary. When the distance tends to zero, the optimal blow-up rates of the gradients are established for inclusions with arbitrary shapes and in all dimensions.
In this paper, we establish the asymptotic expressions for the gradient of a solution to the Lam{e} systems with partially infinity coefficients as two rigid $C^{1,gamma}$-inclusions are very close but not touching. The novelty of these asymptotics, which improve and make complete the previous results of Chen-Li (JFA 2021), lies in that they show the optimality of the gradient blow-up rate in dimensions greater than two.
In a viscous incompressible fluid, the hydrodynamic forces acting on two close-to-touch rigid particles in relative motion always become arbitrarily large, as the interparticle distance parameter $varepsilon$ goes to zero. In this paper we obtain asymptotic formulas of the hydrodynamic forces and torque in $2mathrm{D}$ model and establish the optimal upper and lower bound estimates in $3mathrm{D}$, which sharply characterizes the singular behavior of hydrodynamic forces. These results reveal the effect of the relative convexity between particles, denoted by index $m$, on the blow-up rates of hydrodynamic forces. Further, when $m$ degenerates to infinity, we consider the particles with partially flat boundary and capture that the largest blow-up rate of the hydrodynamic forces is $varepsilon^{-3}$ both in 2D and 3D. We also clarify the singularities arising from linear motion and rotational motion, and find that the largest blow-up rate induced by rotation appears in all directions of the forces.
In the perfect conductivity problem of composites, the electric field may become arbitrarily large as $varepsilon$, the distance between the inclusions and the matrix boundary, tends to zero. The main contribution of this paper lies in developing a clear and concise procedure to establish a boundary asymptotic formula of the concentration for perfect conductors with arbitrary shape in all dimensions, which explicitly exhibits the singularities of the blow-up factor $Q[varphi]$ introduced in [29] by picking the boundary data $varphi$ of $k$-order growth. In particular, the smoothness of inclusions required for at least $C^{3,1}$ in [27] is weakened to $C^{2,alpha}$, $0<alpha<1$ here.
We study boundary gradient estimates for second-order divergence type parabolic and elliptic systems in $C^{1,alpha}$ domains. The coefficients and data are assumed to be Holder in the time variable and all but one spatial variables. This type of systems arises from the problems of linearly elastic laminates and composite materials.
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