In a framework of 1D membrane MEMS theory, we consider the MEMS boundary semi-linear elliptic problem with fringing field

where \(\lambda ^2\) and \(\delta \) are positive parameters, \(\varOmega =[-L,L] \subset {\mathbb {R}}\), and u is the deflection of the membrane. In this model, since the electric field \( {\mathbf {E}} \) on the membrane is locally orthogonal to the straight line tangent to the membrane at the same point, \( | {\mathbf {E}} | \), proportional to \(\lambda ^2/(1-u)^2\), is considered locally proportional to the curvature of the membrane. Thus, we achieve interesting results of existence writing it into its equivalent integral formulation by means of a suitable Green function and applying on it the Schauder–Tychonoff fixed point theory. Therefore, the uniqueness of the solution is proved exploiting both Poincaré inequality and Gronwall Lemma. Then once the instability of the only obtained equilibrium position is verified, an interesting limitation for the potential energy dependent on the fringing field capacitance is obtained and studied.

在1的框架d膜MEMS理论，我们认为所述MEMS边界半线性椭圆问题弥散场

其中\（\ lambda ^ 2 \）和\（\ delta \）是正参数，\（\ varOmega = [-L，L] \ subset {\ mathbb {R}} \），而u是膜。在此模型中，由于膜片上的电场\（{\ mathbf {E}} \\）局部正交于与膜在同一点处相切的直线，因此\（| {\ mathbf {E}} | \ ），与\（\ lambda ^ 2 /（1-u）^ 2 \）成正比被认为局部地与膜的曲率成比例。因此，我们获得了一个有趣的结果，即通过适当的格林函数将其写入等价的积分公式并在其上应用Schauder-Tychonoff不动点理论。因此，利用庞加莱不等式和Gronwall引理证明了该解决方案的独特性。然后，一旦验证了唯一获得的平衡位置的不稳定性，便获得并研究了取决于边缘场电容的势能的有趣限制。