$W^{1,p}$ 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 $f\in L^2(\mathcalΩ)$ and $Lu=div (A(x) \nabla u)$, and, given a bounded subanalytic domain $\mathcalΩ$, 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 $||u||_{ W^{1,p}(\mathcalΩ)}\le C||f||_{L^2(\mathcalΩ)}$, for some $p>2$. The number $p$ depends on the geometry of the singularities of $δ\mathcalΩ$, but not on $u$.