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Register a new userConfinement of dislocations inside a crystal with a prescribed external strain
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A system of $n$ screw dislocations in an isotropic crystal undergoing antiplane shear is studied in the framework of linear elasticity. Imposing a suitable boundary condition for the strain, namely requesting the non-vanishing of its boundary integral, results in a confinement effect. More precisely, in the presence of an external strain with circulation equal to n times the lattice spacing, it is energetically convenient to have n distinct dislocations lying inside the crystal. The result is obtained by formulating the problem via the core radius approach and by studying the asymptotics as the core size vanishes. An iterative scheme is devised to prove the main result. This work sets the basis for studying the upscaling problem, i.e., the limit as $ntoinfty$, which is treated in [17].
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Experimental results on the dislocation dynamics in a two-dimensional plasma crystal are presented. Edge dislocations were created in pairs in lattice locations where the internal shear stress exceeded a threshold and then moved apart in the glide plane at a speed higher than the sound speed of shear waves, $C_T$. The experimental system, a plasma crystal, allowed observation of this process at an atomistic (kinetic) level. The early stage of this process is identified as a stacking fault. At a later stage, supersonically moving dislocations generated shear-wave Mach cones.
We analyze a mathematical model of elastic dislocations with applications to geophysics, where by an elastic dislocation we mean an open, oriented Lipschitz surface in the interior of an elastic solid, across which there is a discontinuity of the displacement. We model the Earth as an infinite, isotropic, inhomogeneous, elastic medium occupying a half space, and assume only Lipschitz continuity of the Lame parameters. We study the well posedness of very weak solutions to the forward problem of determining the displacement by imposing traction-free boundary conditions at the surface, continuity of the traction and a given jump on the displacement across the fault. We employ suitable weighted Sobolev spaces for the analysis. We utilize the well posedness of the forward problem and unique-continuation arguments to establish uniqueness in the inverse problem of determining the dislocation surface and the displacement jump from measuring the displacement at the surface of the Earth. Uniqueness holds for tangential or normal jumps and under some geometric conditions on the surface.
We study the relaxation to equilibrium for a class linear one-dimensional Fokker-Planck equations characterized by a particular subcritical confinement potential. An interesting feature of this class of Fokker-Planck equations is that, for any given probability density $e(x)$, the diffusion coefficient can be built to have $e(x)$ as steady state. This representation of the equilibrium density can be fruitfully used to obtain one-dimensional Wirtinger-type inequalities and to recover, for a sufficiently regular density $e(x) $, a polynomial rate of convergence to equilibrium.Numerical results then confirm the theoretical analysis, and allow to conjecture that convergence to equilibrium with positive rate still holds for steady states characterized by a very slow polynomial decay at infinity.
We consider the semilinear heat equation, to which we add a nonlinear gradient term, with a critical power. We construct a solution which blows up in finite time. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. Thanks to the interpretation of the parameters of the finite-dimensional problem in terms of the blow-up time and point, we also show the stability of the constructed solution with respect to initial data. This note presents the results and the main arguments. For the details, we refer to our paper cite{TZ15}.
In this paper, we are concern with the multiplicity of solutions for a p-Laplacian problem. A weaker super-quadratic assumptions is required on the nonlinearity. Under the weaker condition we give a new proof for the infinite solutions having a prescribed number of nodes to the problem. It turns out that the weaker condition on nonlinearity suffices to guarantee the infinitely many solutions. At the same time, a global characterization of the critical values of the nodal radial solutions are given.
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