Korn’s Inequality in L ^p

Abstract

In Chapter 7, we propose a path that culminates in a proof of Korn’s inequality in W ^1,p when p>2. This inequality, which generalizes the well-known one for p=2, furnishes an upper bound in the Sobolev space $W^{1, p}_{0}(\Omega, \mathbb{R}^{n})$ for the L ^p norm of the gradient of any function with values in $\mathbb{R}^{N}$ whose components belong to L ^p(Ω), namely a constant times the L ^p norm of the symmetric part of the gradient (linearized stress tensor in mechanics). To achieve this, we use Riesz inequalities to prove that if the gradient of a distribution belongs to the dual of the Sobolev space W ^1,p′(Ω), then this distribution belongs to L ^p′. We prove the Riesz inequalities by using the Fourier transform to study the convolution of functions in L ^p with Riesz kernels. To achieve this, we need to introduce the Hardy–Littlewood maximal function, the Hilbert transform, and the Hilbert maximal function. This is the object of the first sections of the chapter, inspired in part by the book of Stein and Weiss.

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-1-4471-2807-6_8