Existence and nonexistence of solutions for fractional elliptic equations arising from closed MEMS model

TL;DR

This work addresses the fractional elliptic MEMS equation $(-\Delta)^s u=\frac{\lambda}{(a-u)^2}$ in bounded domains with Dirichlet-type exterior data, incorporating a boundary-vanishing membrane function $a$. It develops a Green-operator framework and boundary-decay analysis to establish the existence of a finite pull-in voltage $\lambda^*$ and a minimal solution branch $u_\lambda$ for $\lambda\in(0,\lambda^*)$, along with nonexistence results for larger $\lambda$ and a detailed description of boundary behavior. The authors derive sharp near-boundary estimates, prove weak regularity of $u_{\lambda^*}$, and establish stability properties of the minimal solutions; in the unit ball they obtain explicit bounds on $\lambda^*$ and $\lambda_*$ and show monotonicity with respect to the boundary-decay exponent $\gamma$ and parameter $\kappa$. Under radial symmetry and a specific decay profile, they further obtain classical regularity for certain dimensional regimes, highlighting how boundary geometry and fractional diffusion influence pull-in voltage and solution structure. Overall, the paper advances understanding of MEMS-type models with fractional diffusion, clarifying how boundary decay controls existence, regularity, and stability of solutions and providing quantitative bounds useful for design considerations.

Abstract

The objective of our paper is to investigate fractional elliptic equations of the form $(-Δ)^s u=\frac{λ}{(a-u)^2}$ within a bounded domain $Ω$, subject to zero Dirichlet boundary conditions. Here, $s\in(0,1)$, $λ>0$, and the function $a$ vanishes at the boundary while satisfying additional conditions. This problem originates from Micro-Electromechanical Systems (MEMS) devices, particularly when the elastic membrane makes contact with the ground plate at the boundary. We establish both existence and nonexistence results, illustrating how the boundary decay of the membrane influences the solutions and pull-in voltage.

Theorems & Definitions (23)

• Theorem 1.1
• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.1
• Proposition 2.1
• Lemma 2.1
• Lemma 2.2
• Corollary 2.1
• Proposition 3.1