Variable exponent Bochner–Lebesgue spaces with symmetric gradient structure

Abstract

We introduce function spaces for the treatment of non-linear parabolic partial differential equations with variable log–Hölder continuous exponents that 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 to be difficult. Using a point-wise Poincaré inequality near the boundary of a bounded Lipschitz domain involving only the symmetric gradient, 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.