Optimal pointwise stimates for derivatives of solutions to Laplace, Lamé, and Stokes equations
type: Item
instance of: scholarly article
title: Optimal pointwise stimates for derivatives of solutions to Laplace, Lamé, and Stokes equations
zbMATH Open document ID: 1302.31008
published in: Journal of Mathematical Sciences (New York)
publication date: 2014-11-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1310.6711
review text: 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.
reviewed by: Stephen J. Gardiner
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 35A23
Mathematics Subject Classification ID: 30A10
zbMATH DE Number: 6363709
zbMATH Keywords: harmonic function
zbMATH Keywords: gradient estimate
zbMATH Keywords: analytic function
author: Gershon Kresin
author: V. Maz'ya
MaRDI profile type: Publication
OpenAlex ID: W2006101154
arXiv ID: 1310.6711
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DOI: 10.1007/S10958-014-1660-2
description: scientific article; zbMATH DE number 6363709