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