5
$\begingroup$

I was wondering if there were any recent references dealing with the theory of systems of elliptic PDEs: in particular, someone was telling me about something which sounded like 'Schur complementarity' and reducing a system of elliptic PDE down to one PDE, does anyone know of a recent reference which explains this theory?

I did try the book by Ladyzhenskaya but there was not that much on systems of PDE, perhaps this is quite an old reference now though.

Improve this question
$\endgroup$
5
  • $\begingroup$ A not so recent reference are the papers of Agmon, Douglis, and Nirenberg. $\endgroup$
    – Deane Yang
    Commented Feb 12, 2020 at 13:41
  • $\begingroup$ Since, for the part on the Schur complement, @DenisSerre has adequately answere, I'd like to point out the book of Carlo Miranda, Partial differential equations of elliptic type, 2nd rev. ed. Translated from the Italian by Zane C. Motteler. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Berlin-Heidelberg-New York: Springer-Verlag pp. XII+370 (1970), MR0284700, ZBL0198.14101. $\endgroup$
    – Daniele Tampieri
    Commented Feb 12, 2020 at 16:17
  • 1
    $\begingroup$ While the book of Ladyzhenskaya et al is more oriented to the then recently introduced De Giorgi's regularity theory and therefore is oriented to the analysis of single, divergence type partial , this book tries to embrace the whole field of elliptic PDEs and thus is focused more on systems than on single equations. It is an "old" reference, thus there are problems and methods not even touched nor imagined at the time of its writing: nevertheless, it is worth reading. $\endgroup$
    – Daniele Tampieri
    Commented Feb 12, 2020 at 16:33
  • $\begingroup$ @Deane Yang: Were there any papers in particular which you would recommend? $\endgroup$
    – Hollis Williams
    Commented Feb 16, 2020 at 2:26
  • 1
    $\begingroup$ There are papers written jointly by Agmon, Douglis, Nirenberg. $\endgroup$
    – Deane Yang
    Commented Feb 16, 2020 at 3:22

1 Answer 1

Reset to default
5
$\begingroup$

In matrix analysis, the Schur complement is an object that you obtain after eliminating a part of the unknowns. It works that way: you have to solve $Mx=b$ where $M$ is a square, invertible matrix. You write the system in block form $$\begin{pmatrix} A & B \\ C & D \end{pmatrix}\binom{y}{z}=\binom{c}{d},$$ where you are lucky enough that $A$ is invertible too. Then elimination of $y$ yields a system $(D-CA^{-1}B)z=b'$. The Schur complement of $A$ is precisely $D-CA^{-1}B$. Remark that we have the formula $\det M=\det A\cdot\det (D-CA^{-1}B)$.

The situation is similar if you have an elliptic system of PDEs. It can be written in an abstract way as a matrix $M$ of differential operators (including the information of their domains). Then write $M$ blockwise as above, with $A$ an elliptic operator, invertible over its domain. Then the Schur complement is $D-CA^{-1}B$, with domain that of $D$.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .