Divergence Operator and Related Inequalities by Gabriel Acosta, Ricardo G. Durán

By Gabriel Acosta, Ricardo G. Durán

This short is especially dedicated to classical and comparable effects: the lifestyles of a correct inverse of the divergence operator and the so-called Korn Inequalities. it really is renowned that either effects are primary instruments within the research of a few vintage differential equations, fairly in these coming up in fluid dynamics and elasticity. a number of connections among those issues and better Poincaré inequalities are largely handled. From basic key principles the e-book is transforming into easily in complexity. starting with the learn of those difficulties on star-shaped domain names the arguments are prolonged first to John domain names after which to Hölder α domain names the place the necessity of weighted areas arises certainly. during this style, the authors reach offering in an unified and concise manner a number of vintage and up to date advancements within the box. those gains definitely makes this short beneficial for college kids, post-graduate scholars, and researchers as well.

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Buckley and P. Koskela [16] for the study of several inequalities. On the other hand, John domains are closely related with the extension domains of P. Jones [62]. Indeed the (ε , ∞) domains, also called uniform domains, are John domains (but the converse is not true: a John domain can have an internal cusp while a uniform domain cannot). As we will show in this section, the approach used to construct solutions of the divergence on star-shaped domains can be generalized to John domains. This generalization has been done in [3].

Since the class of John domains is very general and excludes external cusps it seems a natural question whether Ω satisfies div p ⇐⇒ Ω is a John domain. 1) As far as we know the answer is not known. However, a partial answer can be given. 1) is true if the bounded domain Ω satisfies the separation property. We omit the technical definition of this property and refer the reader to [16] where it was introduced. In that paper it is also proved that, in the two dimensional case, any simply connected domain satisfies the separation property.

7). We can also use the following argument: in [35] it is proved that, if the improved Poincar´e is valid for some q ≥ 1 then it is valid for all r such that q ≤ r < ∞, actually, in that paper the proof is written 44 2 Divergence Operator for q = 1 but the argument can be easily extended to any q. Now, assuming div p for some p > n we have that Ω satisfies the improved Poincar´e for p , and therefore, for p. 10, we obtain that Ω satisfies div p . But p < n, and so, Ω is a John domain. Finally, the case p = n was proved in [59].

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