# Elliptic regularity in $L^1$

Dear all,

I am looking for a good reference for elliptic regularity in $L^1$. To be more precise

Let $\Omega\subset\mathrm{R}^n$ be a bounded smooth domain, let $A$ be a properly elliptic differential operator in $\Omega$ and let $B$ be a first order differential operator on $\partial\Omega$ satisfying the complementing condition. For $f\in L^1(\Omega)$ consider the problem

$$Au=f, \mbox{ in }\Omega,\qquad Bu=0, \mbox{ on }\partial\Omega.$$

It seems to be well-known that there is at least a weak solution $u\in W^{1,1}(\Omega)$. Is it possible to give a reference for this? Is it possible to state more regularity for $u$?

I would be very grateful for any useful hints on this problem.

Richard

• Given $u\in W^{1,1}$, how do you define $B u|_{\partial \Omega}$? – Liviu Nicolaescu Sep 5 '12 at 8:11

## 2 Answers

I found a reference where elliptic equations in L1 are dealt with: Tanabe, "Functional analytical methods for partialdifferential equations" There it is also explained in what way the boundary values are to be understood.

• Can you be more specific with this reference (journal, year of publication). I cannot seem to find the reference on MathSciNet – Liviu Nicolaescu Sep 6 '12 at 8:20
• amazon.com/… – Michael Renardy Sep 6 '12 at 9:29

Give a look vere http://www.sciencedirect.com/science/article/pii/0022247X9290285L where Poisson type equation with f in some particular subclass of L1 is studied.