In this note we give an overview of recent results in the theory of electrorheological fluids and the theory of function spaces with variable exponents. Moreover, we present a detailed and self-contained exposition of shifted -functions that are used in the studies of generalized Newtonian fluids and problems with -structure.
Mathematics Subject Classification: 26D10, 46E30, 47B38 We prove the Hardy inequality and a similar inequality for the dual Hardy operator for variable exponent Lebesgue spaces.
The trace space of consists of those functions on ℝⁿ that can be extended to functions of (as in the fixed-exponent case). Under the assumption that p is globally log-Hölder continuous, we show that the trace space depends only on the values of p on the boundary. In our main result we show how to define an intrinsic norm for the trace space in terms of a sharp-type operator.
We establish a local Lipschitz regularity result for local minimizers of asymptotically convex variational integrals.
We establish a local Lipschitz regularity result for local minimizers of asymptotically convex variational integrals.
We study properties of Lipschitz truncations of Sobolev functions with constant and variable exponent. As non-trivial applications we use the Lipschitz truncations to provide a simplified proof of an existence result for incompressible power-law like fluids presented in [Frehse (2003) 1064–1083]. We also establish new existence results to a class of incompressible electro-rheological fluids.
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