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

TL;DR

This paper establishes $W^{1,p}$ a priori estimates for solutions of linear elliptic PDEs on subanalytic domains with boundary singularities, linking the estimates to the domain's geometric singularity structure.

Contribution

It provides explicit geometric conditions on boundary singularities that guarantee $W^{1,p}$ estimates for elliptic PDE solutions, extending regularity results to non-smooth subanalytic domains.

Findings

$W^{1,p}$ estimates depend on boundary singularity geometry

Explicit conditions on tangent cones ensure regularity

Estimates hold uniformly for solutions with $L^2$ data

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 $u$.