Optimal pointwise stimates for derivatives of solutions to Laplace, Lamé, and Stokes equations (Q467573)

Let \(\Omega \) be a domain in \(\mathbb{R}^{n}\), and \(\mathcal{U}\) be the collection of harmonic functions \(u\) on \(\Omega \) such that \(|u|\leq 1\). It is well known that \(\left| \nabla u(x)\right| \leq n/\mathrm{dist}(x,\partial \Omega )\). A sample result of the present paper says that, if \(B\subset \Omega \,\)\ is a ball and \(\xi \in \partial B\cap \partial \Omega \), then \[ \lim \sup_{x\rightarrow \xi ,x\in E}\sup_{u\in \mathcal{U}}\left| x-\xi \right| \left| \nabla u(x)\right| \leq C_{n}, \] where \(E\) is the radius of \(B\) ending at \(\xi \) and \(C_{n}\) is an explicit sharp constant. Analogous results are established for directional derivatives and the divergence of a vector field with harmonic components, and for derivatives of analytic functions with bounded real part. The authors also obtain estimates for divergence of an elastic displacement field and pressure in a fluid.