4
$\begingroup$

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

$\endgroup$
1
  • 2
    $\begingroup$ Given $u\in W^{1,1}$, how do you define $B u|_{\partial \Omega}$? $\endgroup$ – Liviu Nicolaescu Sep 5 '12 at 8:11
1
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ Can you be more specific with this reference (journal, year of publication). I cannot seem to find the reference on MathSciNet $\endgroup$ – Liviu Nicolaescu Sep 6 '12 at 8:20
  • $\begingroup$ amazon.com/… $\endgroup$ – Michael Renardy Sep 6 '12 at 9:29
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.