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On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. (English) Zbl 1143.35037
In this paper the authors recall, survey and strenghten properties of \( W_0^{1,\infty }\)-truncations of \(W_0^{1,p}\)-functions that are useful from the point of view of existence theory concerning nonlinear PDE’s. One illustrates the potential of this tool by establishing the weak stability for the system of \(p\)-Laplace equations with very general right-hand sides. Then one studies the properties of Lipschitz truncations of Sobolev functions with constant and variable exponent. As non-trivial applications the Lipschitz truncations are utilized to provide a simplified proof of an existence result for incompressible power-law like fluids. New existence results are established to a class of incompressible electro-rheological fluids.

MSC:
35J60 Nonlinear elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35J70 Degenerate elliptic equations
35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76W05 Magnetohydrodynamics and electrohydrodynamics
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