zbMATH — the first resource for mathematics

Positivity preserving results for a biharmonic equation under Dirichlet boundary conditions. (English) Zbl 1331.35121
Summary: We prove a dichotomy result giving the positivity preserving property for a biharmonic equation with Dirichlet boundary conditions arising in MEMS models. We adapt some ideas in [H.-C. Grunau and G. Sweers, Math. Ann. 307, No. 4, 589–626 (1997; Zbl 0892.35031)].

35J40 Boundary value problems for higher-order elliptic equations
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
35B50 Maximum principles in context of PDEs
Full Text: DOI
[1] T. Boggio, Sulle funzioni di Green dordine m, Rend. Circ. Mat. Palermo 20 (1905), 97–135. · JFM 36.0827.01
[2] F. Gazzola, H.-Ch. Grunau, G. Sweers, Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains, Lecture Notes in Math. 1991, Springer-Verlag, Berlin (2010). · Zbl 1239.35002
[3] H.-Ch. Grunau, Positivity, change of sign and buckling eigenvalues in a one-dimensional fourth order model problem, Adv. Differential Equations 7 (2002), 177–196. · Zbl 1031.34085
[4] H.-Ch. Grunau, G. Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann. 307 (1997), 589–626. · Zbl 0892.35031
[5] H.-Ch. Grunau, G. Sweers, Sign change for the Green function and for the first eigenfunction of equations of clamped-plate type, Arch. Rational Mech. Anal. 150 (1999), 179–190. · Zbl 0973.74044
[6] P. Laurençot, C. Walker, Sign-preserving property for some fourth oreder elliptic operators in one dimension and radial symmetry, J. Anal. Math., to appear.
[7] F.H. Lin, Y.S. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. A 463 (2007), 1323–1337. · Zbl 1143.78001
[8] J.A. Pelesko, D.H. Bernstein, Modelling MEMS and NEMS, New York, NY: Chapman & Hall/CRC, 2003. · Zbl 1031.74003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.