The regularity of elliptic equation

Question

Define $$ Lu:=-\partial_j(a_{ij}\partial_iu)+b_i\partial_iu+cu $$ where $a_{ij}$, $b_i$, and $c$ are all of $C^\infty(\bar \Omega)$ and we assume $\Omega$ is open bounded in $R^N$, smooth boundary, $N\geq 2$. Also, here $A=(a_{ij})$ satisfies uniformly elliptic condition, i.e., $A\xi\xi\geq\theta \,|\xi|^2$

We consider the standard PDE \begin{cases} Lu=f&x\in\Omega\\ u=0&x\in\partial\Omega \end{cases} Then by regularity, we have if $f\in C^\infty(\bar \Omega)$, then $u\in C^\infty(\bar \Omega)$ as well. However, if in addition we assume that $f$ is analytic, could I have $u$ is analytic as well? If yes, where can I find a proof?

Secondly, unless we are in one dimension, we know that if $f\in C^2(\bar\Omega)$, we may not have $u\in C^4(\bar\Omega)$(this is why we need Sobolev and Holder space). But could anybody provide me a counterexample? Thank you!

For the first question, you need to assume that $a_{ij}$, $b_i$ and $c$ are also analytic. — Yiorgos S. Smyrlis, Jan 10 '15 at 23:16
@YiorgosS.Smyrlis Thank you sir. But could you point out where I can find a prove for this? — spatially, Jan 10 '15 at 23:18

score 3 · Accepted Answer · edited Apr 13 '17 at 12:21

3

For your first question, the answer can be found in paper [1] and references therein.

The answer for your second question can be found here.

[1]: Morrey, C. B., Jr.; Nirenberg, L. On the analyticity of the solutions of linear elliptic systems of partial differential equations. Comm. Pure Appl. Math. 10 (1957), 271–290.

edited Apr 13 '17 at 12:21

Community

1

answered Jan 11 '15 at 11:20

Tomás

22,985

You are welcome @wisher. Note that the paper cited above, shows analitycity even in points of the boundary, if the boundary itself is analytic. – Tomás Jan 11 '15 at 14:13
thank you! It looks to me if the data and coff are good enough, then the solution will be good as well, or even better, in some suitable space. – spatially Jan 11 '15 at 17:32

The regularity of elliptic equation

1 Answers1