论文标题
使用硬化的弹性塑性理论中两个标准模型的规律性结果
Regularity results for two standard models in elasto-perfect-plasticity theory with hardening
论文作者
论文摘要
我们考虑了在弹性塑料理论中研究的两个最受研究的标准模型，并在任意维度上硬化$ d \ ge 2 $，即运动硬化和各向同性硬化问题。尽管解决方案的存在和独特性是众所周知的，但到边界的最佳规律性仍然是一个开放的问题。在这里，我们表明，在内部，我们具有应力和硬化的sobolev规律性，而对于它们的时间衍生物，我们具有带有空间和时间变量的“一半”衍生物。这是有限问题而闻名的，但我们表明这些估计是统一的，并且独立于近似顺序。主要的新颖性由边界附近的估计组成。我们表明，对于应力和硬化参数，我们控制Lebesgue空间中的切向导数〜$ l^2 $，对于应力的时间导数和固化，我们控制了“半个时间衍生物”，并且还控制了空间切向衍生物。最后，对于正常导数，我们表明，应力和硬化具有相对于正常的$ 3/5 $衍生物，并且在压力的时间和钢化度上，我们表明它们具有相对于正常方向的$ 1/5 $衍生产品，只要我们考虑我们考虑运动型硬化或附近的迪克尔特边界。这些估计与维度无关。如果我们考虑在Neumann边界附近的各向同性硬化，我们将获得$ w^{α，2} $正常性的压力和耐加工，并用一些$α> 1/2 $ $α> 1/2 $，具体取决于尺寸和$ w^{β，2} $，并带有一些$β> 1/6 $，以使压力和硬化的时间衍生性和硬化。最后，在运动硬化的情况下，相同的规律性估计也适用于速度梯度。
We consider two most studied standard models in the theory of elasto-plasticity with hardening in arbitrary dimension $d\ge 2$, namely, the kinematic hardening and the isotropic hardening problem. While the existence and uniqueness of the solution is very well known, the optimal regularity up to the boundary remains an open problem. Here, we show that in the interior we have Sobolev regularity for the stress and hardening while for their time derivatives we have the "half" derivative with the spatial and time variable. This was well known for the limiting problem but we show that these estimates are uniform and independent of the order of approximation. The main novelty consist of estimates near the boundary. We show that for the stress and the hardening parameter, we control tangential derivative in the Lebesgue space~$L^2$, and for time derivative of the stress and the hardening we control the "half" time derivative and also spatial tangential derivative. Last, for the normal derivative, we show that the stress and the hardening have the $3/5$ derivative with respect to the normal and for the time derivative of the stress and the hardening we show they have the $1/5$ derivative with respect to the normal direction, provided we consider the kinematic hardening or near the Dirichlet boundary. These estimates are independent of dimension. In case, we consider the isotropic hardening near the Neumann boundary we shall obtain $W^{α,2}$ regularity for the stress and the hardening with some $α>1/2$ depending on the dimension and $W^{β,2}$ with some $β> 1/6$ for the time derivative of the stress and the hardening. Finally, in case of kinematic hardening the same regularity estimate holds true also for the velocity gradient.
