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

Abstract

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 $Lu=div (A(x) \nabla u)$, 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 $||u||_{ W^{1,p}(\mathcal{\Omega})}\le C||f||_{L^2(\mathcal{\Omega})}$, for some $p>2$. The number $p$ depends on the geometry of the singularities of $\delta \mathcal{\Omega}$, but not on $u$.