MEMS问题的小介电常数点无触地和非平凡触地集
No touchdown at points of small permittivity and nontrivial touchdown sets for the MEMS problem
摘要：
我们考虑一个著名的模型，微机电系统（MEMS）的可变介电常数，涉及一个抛物型方程奇异非线性。我们研究触地或淬火现象。最近，对于内点的情况，是否可以在介电率曲线f的零点处发生触地的问题得到了否定的回答，而介电率曲线f的零点长期以来一直是开放的。本文的第一个目的是进一步考虑相同的问题，在点的积极，但小介电常数。我们表明，在任何有界域，接地不能发生在一个内点的介电常数配置文件是适当的小。我们也得到了类似的结果，在边界的情况下，在一个小的假设下，f在边界的邻域。这特别允许构造f，产生集中在任何给定球体附近的触地集。我们的下一个目标是获得更多的信息的结构和属性的降落集。特别是，我们表明，接地集一般不需要本地化的极大点附近的介电率曲线f。在球的径向情况下，我们表明存在的M-形的轮廓，其触地集位于远离极大点的f，我们甚至得到严格凸f的触地发生在唯一的极小点的f。这些结果表明，为了排除在一点触地，上述某种小条件是不可避免的。另一方面，我们构造的配置文件f产生更复杂的行为：在任何有界域的触地集可以集中在附近的两个任意给定的点，或两个任意给定的（n-1）维球在一个球。这些例子是作为接地时间和接地设置的介电常数分布的小扰动下的稳定性结果获得的。
英文摘要：
We consider a well-known model for micro-electromechanical systems (MEMS) with variable dielectric permittivity, involving a parabolic equation with singular nonlinearity. We study the touchdown, or quenching, phenomenon. Recently, the question whether or not touchdown can occur at zero points of the premittivity profile f, which had long remained open, was answered negatively for the case of interior points. The first aim of this article is to go further by considering the same question at points of positive but small permittivity. We show that, in any bounded domain, touchdown cannot occur at an interior point where the permittivity profile is suitably small. We also obtain a similar result in the boundary case, under a smallness assumption on f in a neighborhood of the boundary. This allows in particular to construct f producing touchdown sets concentrated near any given sphere. Our next aim is to obtain more information on the structure and properties of the touchdown set. In particular, we show that the touchdown set need not in general be localized near the maximum points of the premittivity profile f. In the radial case in a ball, we show the existence of M-shaped profiles for which the touchdown set is located far away from the maximum points of f and we even obtain strictly convex f for which touchdown occurs at the unique minimum point of f. These results show that some kind of smallness condition as above cannot be avoided in order to rule out touchdown at a point. On the other hand, we construct profiles f producing more complex behaviors: in any bounded domain the touchdown set may be concentrated near two arbitrarily given points, or two arbitrarily given (n-1)-dimensional spheres in a ball. These examples are obtained as a consequence of stability results for the touchdown time and touchdown set under small perturbations of the permittivity profile.
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