Abstract
We analyze the modular geometry of the Lebesgue space with variable exponent, L p ( · ) . Our central result is that L p ( · ) possesses a modular uniform convexity property. Part of the novelty is that the property holds even in the case sup x ∈ Ω p ( x ) = ∞ . We present specific applications to fixed point theory.
Highlights
We prove a hitherto unknown modular convexity property of the Lebesgue spaces with variable exponent, L p(·), which has far reaching applications in fixed point theory, remarkably even in the case in which the exponent p(·) is unbounded
In the late 19th century these spaces were brought into the center stage of mathematical research as they were realized to be the natural solution space for partial differential equations exhibiting non-standard growth
Though it follows from Theorem 2 that there is no hope for norm-uniform convexity of L p(·) (Ω)
Summary
We prove a hitherto unknown modular convexity property of the Lebesgue spaces with variable exponent, L p(·) , which has far reaching applications in fixed point theory, remarkably even in the case in which the exponent p(·) is unbounded. Similar considerations arise in the study of the hydrodynamic equations governing non-Newtonian fluids [4,5]. These equations have non-standard growth and model, in particular, electrorheological fluids, i.e., fluids whose viscosity can be changed dramatically and in a few mili-seconds when exposed to a magnetic or an electric field. Electrorheological fluids are currently the object of intense research activity in both theoretical and applied fields Their applications include medicine, civil engineering and military science [6,7,8,9].
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