I know of several uses for the Laplacian operator Del^2 in physics and the like.

 

Is there a mathematical/physical meaning of Del^3. What would be this operators name?

 

Does this hold for Del^n. Would this have applications in hyperdimmensional geometry?

It's an operator so you've gotta be very careful about what the index actually means.

 

There is such a thing as Del^4 which is called the bilapacian operator:

 

http://icl.pku.edu.cn/yujs/MathWorld/math/b/b194.htm

The first question you should be asking yourself is: How do I contruct [math]\textdisplay \nabla ^3[/math]?

 

The biharmonic operator provided in the link by Aeschylus should be written as [math]\textdisplay \nabla ^2 \nabla ^2[/math], not as [math]\textdisplay \nabla ^4[/math] (since that doesn't make any sense, but it apparently is defined that way to shorten the notation). Just remember that these operators are short-hand for large expressions which mathematicians are too lazy to write.