Global Bifurcation Curve for Fourth-Order MEMS/NEMS Models with Clamped Boundary Conditions

Summary

This study establishes a global bifurcation curve for fourth-order MEMS/NEMS models with clamped boundary conditions, extending a theorem by P. Korman (2004) to allow for singularities in the nonlinearity.

Highlights

• A global bifurcation curve is established for fourth-order MEMS/NEMS models with clamped boundary conditions.
• The result extends a theorem by P. Korman (2004) to allow for singularities in the nonlinearity.
• The study provides a global a priori bound for the C3-norm of positive solutions, which is optimal in terms of regularity.
• Examples arising in MEMS/NEMS models are presented to illustrate applications of the main results.
• The study uses a bifurcation approach to fourth-order Dirichlet problems, originally formulated by Korman (2004).
• The singularity of the nonlinearity poses new challenges, which are addressed by establishing a priori bounds.
• The study provides an explicit singular solution of the equation with more general nonlinearity.

Key Insights

• The study's main contribution lies in the derivation of a priori estimates and their subsequent applications to novel models arising from the recent monograph.
• The established global bifurcation curve reveals the exact multiplicity of positive solutions for the fourth-order equation with doubly clamped boundary conditions.
• The result extends the known findings for fourth-order MEMS/NEMS models, which previously did not characterize the complete structure of the solution curve when the nonlinearity exhibits a singularity.
• The study's approach, based on the bifurcation method, allows for the analysis of more general nonlinearities, including those with singularities.
• The obtained a priori bound for the C3-norm of positive solutions is optimal in terms of regularity, indicating the sharpness of the estimate.
• The study's findings have significant implications for the design and analysis of MEMS/NEMS devices, as they provide a more comprehensive understanding of the behavior of solutions under various conditions.
• The explicit singular solution obtained in the study can be used as a benchmark for numerical simulations and further analytical studies.

Mindmap

Citation

Lin, M., & Pan, H. (2024). Global Bifurcation Curve for Fourth-Order MEMS/NEMS Models with Clamped Boundary Conditions (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2412.18427