论文标题
双线性进化方程的粘度解决方案
Viscosity Solutions for Doubly-Nonlinear Evolution Equations
论文作者
论文摘要
我们扩展了粘度解决方案的理论，以处理标量值双线性进化方程。这种方程在包括干摩擦在内的许多机械模型中自然出现。在此环境中为不连续的粘度解决方案提供了合适的定义后，我们表明Perron的构造仍然可用，即我们证明了存在结果。此外，我们将证明这些问题的比较原则和稳定性结果。理论上的考虑伴随着几个示例，例如，我们证明了存在速率无关的水平集平均曲率流的解决方案。最后，我们详细讨论了与非凸能问题有关的与速率无关的普通微分方程。我们表明，通过最大化运动获得的解决方案和通过消失的粘度方法与上和下perron溶液相一致，并显示了与速率无关的滞后环的出现。
We extend the theory of viscosity solutions to treat scalar-valued doubly-nonlinear evolution equations. Such equations arise naturally in many mechanical models including a dry friction. After providing a suitable definition for discontinuous viscosity solutions in this setting, we show that Perron's construction is still available, i.e., we prove an existence result. Moreover, we will prove comparison principles and stability results for these problems. The theoretical considerations are accompanied by several examples, e.g., we prove the existence of a solution to a rate-independent level-set mean curvature flow. Finally, we discuss in detail a rate-independent ordinary differential equation stemming from a problem with non-convex energy. We show that the solution obtained by maximal minimizing movements and the solution obtained by the vanishing viscosity method coincide with the upper and lower Perron solutions and show the emergence of a rate-independent hysteresis loop.
