## View abstract

#### Conference abstracts

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. document.getElementById('cloak7c99e02381b6ecced69470b42b92864a').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy7c99e02381b6ecced69470b42b92864a = 'rd&#117;r&#97;n' + '&#64;'; addy7c99e02381b6ecced69470b42b92864a = addy7c99e02381b6ecced69470b42b92864a + 'dm' + '&#46;' + '&#117;b&#97;' + '&#46;' + '&#97;r'; var addy_text7c99e02381b6ecced69470b42b92864a = 'rd&#117;r&#97;n' + '&#64;' + 'dm' + '&#46;' + '&#117;b&#97;' + '&#46;' + '&#97;r';document.getElementById('cloak7c99e02381b6ecced69470b42b92864a').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy7c99e02381b6ecced69470b42b92864a + '\'>'+addy_text7c99e02381b6ecced69470b42b92864a+'<\/a>';

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.