Invited talk

Monday, July 19, 13:30 ~ 14:30 UTC-3

Solutions of the divergence, decomposition of $L^p$-functions, and applications.

Ricardo Durán

University of Buenos Aires and CONICET, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The variational analysis of the classical equations of mechanics is strongly based on some inequalities involving a function and its derivatives (for example, Poincare and Korn type inequalities). Many of these results can be obtained from the so-called Lions lemma .

The Lions lemma is equivalent to the existence of appropriate solutions of the equation $div\,u=f$. In this talk we recall how solutions of the divergence equation can be constructed by elementary arguments in a very general class of bounded domains. Then, we show how these solutions can be used to obtain a decomposition of a function of vanishing integral in a domain into a sum of locally supported functions with the same property.

Finally, we show how such a decomposition can be used to prove a local version of the classic result of Fefferman and Stein on the boundedness of the sharp maximal function. We apply these results to obtain weighted a priori estimates for elliptic problems and give some applications to the analysis of finite element approximations of elliptic problems with singular data.

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