We prove here that the Poincaré-Sobolev pointwise inequalities for the relative rearrangement can be considered as the root of a great number of inequalities in various sets not necessarily vector spaces. In particular, new interpolation inequalities can be derived.
We study a general mathematical model linked with various physical models. Especially, we focus on those models established by King or Spencer-Davis-Voorhees related to thin films extending the lubrication model studied by Bernis-Friedman. According to the initial data, we prove that, either, blow up or global existence can be obtained.
We give two examples of periodic Gaussian processes, having entropy numbers of exactly the same order but radically different small deviations. Our construction is based on Knopp's classical result yielding existence of continuous nowhere differentiable functions, and more precisely on Loud's functions. We also obtain a general lower bound for small deviations using the majorizing measure method. We show by examples that our bound is sharp. We also apply it to Gaussian independent sequences and...
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