Variable exponent Bochner-Lebesgue spaces with symmetric gradient structure

TL;DR

This paper develops new variable exponent function spaces focusing on the symmetric gradient, introduces a boundary Poincaré inequality, constructs a smoothing operator, and proves an existence result for non-linear parabolic equations.

Contribution

It introduces novel variable exponent Bochner-Lebesgue spaces with symmetric gradient structure and establishes key inequalities and operators for analyzing non-linear PDEs.

Findings

Established a boundary Poincaré inequality involving only the symmetric gradient.

Constructed a smoothing operator with desirable properties for these spaces.

Proved an abstract existence theorem for non-linear parabolic equations.

Abstract

We introduce function spaces for the treatment of non-linear parabolic equations with variable $\log$-H\"older continuous exponents, which only incorporate information of the symmetric part of a gradient. As an analogue of Korn's inequality for these functions spaces is not available, the construction of an appropriate smoothing method proves itself to be difficult. To this end, we prove a point-wise Poincar\'e inequality near the boundary of a bounded Lipschitz domain involving only the symmetric gradient. Using this inequality, we construct a smoothing operator with convenient properties. In particular, this smoothing operator leads to several density results, and therefore to a generalized formula of integration by parts with respect to time. Using this formula and the theory of maximal monotone operators, we prove an abstract existence result.