A note on a fourth order PDE with critical nonlinearity

A Note on a Fourth Order Pde with Critical Nonlinearity

(1.3) 2(1 + α)∆w + 2β div ( αD(|Dw|2)− (α∆w + β|Dw|2)Dw) = div(T (Dw, ·))− (E′(w − w)− E′(w − w)) where E′(w − w) = 1 volM ∫ E′(w − w). Chang, Gursky and Yang [CGY] proved that any F -minimizing solution u ∈ W 2,2(M) to (1.3) is actually smooth. It was asked in [CGY] whether any weak solution u ∈ W 2,2(M) is smooth. Indeed, Uhlenbeck and Viaclovsky [UV] confirmed this recently and proved the sm...

Multiple Quenching Solutions of a Fourth Order Parabolic PDE with a Singular Nonlinearity Modeling a MEMS Capacitor

Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of microelectromechanical systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a ...

Regularity of a Fourth Order Non-linear Pde with Critical Exponent

On a Fourth Order Pde in Conformal Geometry

A Semilinear Fourth Order Elliptic Problem with Exponential Nonlinearity

We study a semilinear fourth order elliptic problem with exponential nonlinearity. Motivated by a question raised in [Li], we partially extend known results for the corresponding second order problem. Several new difficulties arise and many problems still remain to be solved. We list the ones we feel particularly interesting in the final section. Mathematics Subject Classification: 35J65; 35J40.