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!

  • $\begingroup$ It depends on $A$ and $C$, namely on the sign of $AC$. $\endgroup$ – Kaster Jul 26 '14 at 18:21
  • $\begingroup$ There is a very precise mathematical definition of ellipticity: you can find it in pretty much any sensible textbook on the subject! $\endgroup$ – Mariano Suárez-Álvarez Jul 26 '14 at 19:39
  • $\begingroup$ @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. $\endgroup$ – 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.

This isn't helpful!

Probably. But if you are interested in elliptic PDE, your goal should not be "I must understand what ellipticity is in general". Rather, you should understand how the Laplace equation works, how the Poisson equation works, what methods to use for divergence-type vs non-divergence type equations, and so on. The often-cited book by Gilbarg and Trudinger is an excellent guide, provided you have firm foundation of real and functional analysis.

On the other hand, if you are just taking an introductory course in PDE, then elliptic means whatever your professor tells you, and you should memorize his/her definition.


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