##  [Biharmonic Operator](/biharmonic-operator) 

  ##  [Biharmonic Operator](https://natural.quantumdictionary.io/biharmonic-operator-0) 

  

 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)



**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The biharmonic operator is the fourth‑order elliptic differential operator Δ² obtained by applying the Laplacian twice (Δ(Δ·)); it acts on scalar (or componentwise vector) fields and governs phenomena with bending or plate energies and fourth‑order boundary conditions.