I'm stucked in the following (maybe classical) issue concerning the $W^{2,p}$ regularity of solutions of a second order elliptic equations like $Lu=f$ in a bounded domain (say a ball) $\Omega$. I have that $u$ satisfies an inhomogeneous Dirichlet condition $u=g$ on a PART $\Gamma$ of the boundary $\partial\Omega$. Is it true that taking $f\in L^p(\Omega)$ and $g\in W^{1-1/p,p}(\partial\Omega)$ the solution $u$ is in $W^{2,p}(\Omega)$? Where can I get a good reference for such question?

Best regards, Bruno

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    $\begingroup$ What is the boundary condition on the other part of the boundary? And how, if at all, are the parts joined together? $\endgroup$ Sep 15, 2013 at 22:31
  • $\begingroup$ Hi Michael. I was thinking on a cone $B\times (-R,R)$ as a domain $\Omega$, on which I have nonhomogeneous Dirichlet condition on $u=g$ on the two faces $y=\pm R$ (which is the part $\Gamma$ of the boundary in my question), while I have zero Neumann condition on the lateral boundary. $\endgroup$ Sep 16, 2013 at 7:48
  • $\begingroup$ There is a rather extensive literature on elliptic boundary value problems for domains with corners. The monograph by Grisvard is a good place to start looking. $\endgroup$ Sep 16, 2013 at 16:07


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