course: Inversion of the divergence,I
speaker: Petru Mironescu (ENS Lyon)abstract: The equation (1) div \(X=f\) plays a central role in continuum mechanics. We will discuss the regularity of \(X\) in terms of \(f\) and of the domain \(\Omega\subset{\bf R}^n\) in which the equation is solved. One of the difficulties stems in the fact that the equation is underdetermined, and that in the quest of regularity one needs to select a good solution among all candidates.
To start with, we will briefly explain elliptic regularity theory, which roughly speaking implies that \(X\) gains a derivative with respect to \(f\). For example, if \(f\in L^p\), with \(1\)<\(p\)<\(\infty\), and if \(\Omega\) is not too rough, than we may pick \(X\) with derivatives in \(L^p\).
We next discuss what happens when \(\Omega\) is rough.
When \(f\in L^1\) (resp. \(f\in L^\infty\)), we will see that there may be no solution \(X\) of (1) with derivatives in \(L^1\) (resp. \(L^\infty\)).
Another interesting situation occurs when \(f\in L^n\) (with \(n\) the space dimension). In that case, one may pick \(X\) not only with derivatives in \(L^n\) (as explained above), but also continuous.
The first part of the mini-course is devoted to the proofs of the above results. The arguments make use of standard functional analysis, and of elementary harmonic analysis results that will be recalled during the course. The last result requires more involved analysis, and the course will be an opportunity to present a glimpse of the classical harmonic analysis.
Finally, we will explain how these results extend to more general setting, the one of Hodge systems, and how this is related to Hardy inequalities and some geometric inequalities.
timetable:
Mon 24 Mar, 11:30 - 12:25, Aula Dini
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