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    Title: No Touchdown at Zero Points of the Permittivity Profile for the MEMS Problem
    Authors: Guo, Jong-Shenq;Philippe Souplet
    Contributors: 淡江大學數學學系
    Date: 2015-02-01
    Issue Date: 2015-05-20 07:01:32 (UTC+8)
    Publisher: Philadelphia: Society for Industrial and Applied Mathematics
    Abstract: We study the quenching behavior for a semilinear heat equation arising in models of micro-electro-mechanical systems (MEMS). The problem involves a source term with a spatially dependent potential, given by the dielectric permittivity profile, and quenching corresponds to a touchdown phenomenon. It is well known that quenching does occur. We prove that touchdown cannot occur at zero points of the permittivity profile. In particular, we remove the assumption of compactness of the touchdown set, made in all previous work on the subject and whose validity is unknown in most typical cases. This answers affirmatively a conjecture made in [W. Guo, Z. Pan, and M. J. Ward, SIAM J. Appl. Math., 66 (2005), pp. 309--338] on the basis of numerical evidence. The result crucially depends on a new type I estimate of the quenching rate, that we establish. In addition we obtain some sufficient conditions for compactness of the touchdown set, without a convexity assumption on the domain. These results may be of some qualitative importance in applications to MEMS optimal design, especially for devices such as microvalves.
    Relation: SIAM Journal on Mathematical Analysis 47(1), pp.614-625
    DOI: 10.1137/140970070
    Appears in Collections:[數學學系暨研究所] 期刊論文

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