Elliptic regularity in $L^1$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T08:45:09Z http://mathoverflow.net/feeds/question/106407 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106407/elliptic-regularity-in-l1 Elliptic regularity in $L^1$ Richard Gustier 2012-09-05T07:20:27Z 2012-10-18T10:22:00Z <p>Dear all,</p> <p>I am looking for a good reference for elliptic regularity in $L^1$. To be more precise</p> <p>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</p> <p>$$Au=f, \mbox{ in }\Omega,\qquad Bu=0, \mbox{ on }\partial\Omega.$$</p> <p>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 $u$?</p> <p>I would be very grateful for any useful hints on this problem.</p> <p>Richard</p> http://mathoverflow.net/questions/106407/elliptic-regularity-in-l1/106485#106485 Answer by Richard Gustier for Elliptic regularity in $L^1$ Richard Gustier 2012-09-06T07:25:23Z 2012-09-06T07:25:23Z <p>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.</p>