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.
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.
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