Third derivative estimates for Dirichlet's problem in convex domains (Q1325163)

Let \(\Omega \subset \mathbb{R}^ n\) be a bounded, convex domain. Consider the inhomogeneous Dirichlet problem \(\Delta u = f \text{ in }\Omega\), \(u = 0\text{ on } \partial\Omega\). Let \(\delta(x) = \text{dist}(x,\partial\Omega)\). In this paper we prove that if \(f \in C^ \infty(\overline{\Omega})\), then for any \(\varepsilon > 0\) \[ \int_ \Omega | \delta(x)^ \varepsilon \nabla^ 3 u(x)| dx < \infty.\tag{1} \] This estimate is sharp in the sense that \(\varepsilon > 0\) cannot be replaced by \(\varepsilon = 0\). An important feature of (1) is that it is valid for all convex regions independent of smoothness. Our bounds depend only on the eccentricity of \(\Omega\), which we define as \(\text{ecc }\Omega = \text{diam }\Omega/\text{inradius }\Omega.\)