liu.seSearch for publications in DiVA
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I
Université Paris-Sud.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
2011 (English)In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 184, no 1, p. 47-115Article in journal (Refereed) Published
Abstract [en]

We develop new solvability methods for divergence form second order, real and complex, elliptic systems  above Lipschitz graphs, with $L_2$ boundary data.    The coefficients $A$ may depend on all variables, but are assumed to be close to coefficients $A_0$ that are independent of the coordinate transversal to the boundary, in the Carleson sense $\|A-A_0\|_C$ defined by Dahlberg.  We obtain a number of {\em a priori} estimates and boundary behaviour results under finiteness of $\|A-A_0\|_C$.  Our methods yield full characterization of weak solutions, whose gradients have $L_2$ estimates of a non-tangential maximal function or of the square function, via an integral representation acting on the conormal gradient, with a singular operator-valued kernel.   Also, the non-tangential maximal function of a weak solution is controlled in $L_2$ by the square function of its   gradient. This estimate is new for systems in such generality, and even for real non-symmetric equations in dimension $3$  or higher. The existence of a proof {\em a priori} to well-posedness, is also a new fact.  As corollaries, we obtain well-posedness of the Dirichlet, Neumann and Dirichlet regularity problems under   smallness of $\|A-A_0\|_C$ and well-posedness for $A_0$, improving earlier results for real symmetric equations.  Our methods build on an algebraic reduction to a first order system first made for coefficients $A_0$ by the two authors   and A. McIntosh in order to use functional calculus related to the Kato conjecture solution,   and the main analytic tool for coefficients $A$ is an operational calculus to prove weighted maximal regularity estimates.

Place, publisher, year, edition, pages
Springer , 2011. Vol. 184, no 1, p. 47-115
Keywords [en]
elliptic systems, maximal regularity, Dirichlet and Neumann problems, square function, non-tangential maximal function, Carleson measure, functional and operational calculus
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-63350DOI: 10.1007/s00222-010-0285-4ISI: 000288674100002OAI: oai:DiVA.org:liu-63350DiVA, id: diva2:378797
Available from: 2010-12-16 Created: 2010-12-16 Last updated: 2017-12-11Bibliographically approved

Open Access in DiVA

No full text in DiVA

Other links

Publisher's full text

Authority records

Axelsson Rosén, Andreas

Search in DiVA

By author/editor
Axelsson Rosén, Andreas
By organisation
Applied MathematicsThe Institute of Technology
In the same journal
Inventiones Mathematicae
Mathematics

Search outside of DiVA

GoogleGoogle Scholar

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 102 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf