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Global bifurcation curves for fourth-order MEMS/NEMS models II

Manting Lin, Hongjing Pan

TL;DR

This work extends global bifurcation theory to a one-dimensional, clamped fourth-order beam model with a singular nonlinear source $f(u)$, examining the problem $u''''(x)=\lambda f(u(x))$ under $u(0)=u(1)=u'(0)=u'(1)=0$ for $\lambda>0$. Building on Korman's framework, the authors establish a priori bounds in $C^3$ and show Hölder-optimal $C^{2+\alpha}$ regularity, even when $f$ has a finite singularity at $r$. They prove the existence of a single, globally continued solution curve that turns at a critical point and terminates, as $\lambda\downarrow0$, at an explicit axisymmetric endpoint $w$ with $\max w=r$; the endpoint is not $C^3$ but lies in $C^{2+\alpha}$, and the positive solutions converge to $w$ in $C^{2+\alpha}$. The endpoint profile $w$ is given explicitly by a piecewise polynomial with symmetry about $x=\tfrac{1}{2}$, enabling precise characterization of MEMS/NEMS models with clamped boundaries. The paper also outlines two open directions: (i) extending the global bifurcation results to a more general MEMS model with a linear term $-T u''$ (i.e., $u''''-T u''$) and (ii) analyzing a non-monotone, regularized nonlinearity arising in other MEMS formulations.

Abstract

Global solution curve and exact multiplicity of positive solutions for a class of fourth-order beam equations with clamped boundary conditions are derived. The results extend atheorem of P. Korman (2004) by allowing the presence of a singularity in the nonlinearity. The paper also establishes an a priori estimate for C^3-norm of positive solutions, which is optimal in Holder regularity. Applications to MEMS/NEMS models are presented.

Global bifurcation curves for fourth-order MEMS/NEMS models II

TL;DR

This work extends global bifurcation theory to a one-dimensional, clamped fourth-order beam model with a singular nonlinear source , examining the problem under for . Building on Korman's framework, the authors establish a priori bounds in and show Hölder-optimal regularity, even when has a finite singularity at . They prove the existence of a single, globally continued solution curve that turns at a critical point and terminates, as , at an explicit axisymmetric endpoint with ; the endpoint is not but lies in , and the positive solutions converge to in . The endpoint profile is given explicitly by a piecewise polynomial with symmetry about , enabling precise characterization of MEMS/NEMS models with clamped boundaries. The paper also outlines two open directions: (i) extending the global bifurcation results to a more general MEMS model with a linear term (i.e., ) and (ii) analyzing a non-monotone, regularized nonlinearity arising in other MEMS formulations.

Abstract

Global solution curve and exact multiplicity of positive solutions for a class of fourth-order beam equations with clamped boundary conditions are derived. The results extend atheorem of P. Korman (2004) by allowing the presence of a singularity in the nonlinearity. The paper also establishes an a priori estimate for C^3-norm of positive solutions, which is optimal in Holder regularity. Applications to MEMS/NEMS models are presented.

Paper Structure

This paper contains 4 sections, 6 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lemmas
  3. Proof of Theorem \ref{['th2:u bounded']}
  4. Concluding Remarks

Key Result

Theorem 1.1

Assume that $f(u) \in C^{2}(0,\infty) \cap C^1[0,\infty)$ satisfies $f(u)>0$ for $u \geq 0$, $f^{\prime}(0) \geqslant 0$, $f^{\prime \prime}(u) >0$ for $u>0$, and Then all positive solutions of eq:4order lie on a unique smooth curve of solutions. This curve starts at $(\lambda, u)=(0,0)$, it continues for $\lambda> 0$ until a critical $\lambda_{0}$, where it bends back, and continues for decreas

Figures (1)

  • Figure 1: Global bifurcation diagrams provided by Theorems \ref{['thm:korman']} and \ref{['th2:u bounded']}. (i) $r=+\infty$ and $\lim _{u \rightarrow +\infty} \frac{f(u)}{u}=+\infty$. (ii) $r<+\infty$ and $\liminf_{u \rightarrow r^{-}} \,(r-u)f(u) >0$.

Theorems & Definitions (11)

  • Theorem 1.1: Korman2004
  • Theorem 1.2
  • Lemma 2.1
  • Remark 2.2
  • proof : Proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 3.1: Crandall1973
  • ...and 1 more