$W^{1,p}$ priori estimates for solutions of linear elliptic PDEs on subanalytic domains

We prove a priori estimates for solutions of order $2$ linear elliptic PDEs in divergence form on subanalytic domains. More precisely, we study the solutions of a strongly elliptic equation $Lu=f$, with $f\in L^2(\mathcal{\Omega })$ and $Lu=div(A(x)\mathrm{\nabla }u)$, and, given a bounded subanalytic domain $\mathcal{\Omega }$, possibly admitting non metrically conical singularities within its boundary, we provide explicit conditions on the tangent cone of the singularities of the boundary which ensure that $\mathrm{\mid }\mathrm{\mid }u\mathrm{\mid }{\mathrm{\mid }}_{W^{1,p}(\mathcal{\Omega })}\le C\mathrm{\mid }\mathrm{\mid }f\mathrm{\mid }{\mathrm{\mid }}_{L^2(\mathcal{\Omega })}$, for some $p>2$. The number $p$ depends on the geometry of the singularities of $\delta \mathcal{\Omega }$, but not on $u$.