# Definition of elliptic pde

http://en.wikipedia.org/wiki/Elliptic_partial_differential_equation

Very confusing. So are there non-linear elliptic pdes? It seems to be the case. And what about $$A u_{xx} + C u_{yy} + F u = 0$$ That seems to be elliptic, too...

I know there isn't a precise mathematical definition, but can somebody give me some hints to a bit more complete (and more accessible than this one) definition?

Thank you!

• It depends on $A$ and $C$, namely on the sign of $AC$. – Kaster Jul 26 '14 at 18:21
• There is a very precise mathematical definition of ellipticity: you can find it in pretty much any sensible textbook on the subject! – Mariano Suárez-Álvarez Jul 26 '14 at 19:39
• @MarianoSuárez-Alvarez It would be more correct to say that there are very precise definitions, each of which applies to a specific type of PDE, and none of which define ellipticity for general PDE. – user147263 Jul 26 '14 at 19:44

are there non-linear elliptic pdes?

Yes. The $p$-Laplace equation $\operatorname{div}(|\nabla u|^{p-2}\nabla u)=0$ is recognized as elliptic (degenerate elliptic at the singular points of $u$). It is nonlinear when $p\ne 2$.

The above is a special case of an Euler-Lagrange equation for a convex functional, namely $\int |\nabla u|^p$. All such equations are recognized as elliptic.

To give a nonvariational example: the Monge-Ampère equation $\det D^2u=f$ is considered elliptic when $u$ is being sought in the class of convex functions. The introduction of Caffarelli's ICM 2002 address explains why.

what about $A u_{xx} + C u_{yy} + F u = 0$

Ellipticity is determined by the highest-order derivatives the equation contains. So, this PDE qualifies as elliptic if $A u_{xx} + C u_{yy}=0$ does.

### So what is ellipticity anyway?

It's what makes boundary value problems work. If you have a PDE for which boundary value problems are sensibly behaved, your PDE is elliptic. Sensible behavior includes: existence and uniqueness of solutions in some function space, and a bit of extra regularity of solutions compared to general elements of that function space.