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LAYER POTENTIALS BEYOND SINGULAR INTEGRAL OPERATORS
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
2013 (English)In: Publicacions matemàtiques, ISSN 0214-1493, E-ISSN 2014-4350, Vol. 57, no 2, 429-454 p.Article in journal (Refereed) Published
Abstract [en]

We prove that the double layer potential operator and the gradient of the single layer potential operator are L-2 bounded for general second order divergence form systems. As compared to earlier results, our proof shows that the bounds for the layer potentials are independent of well posedness for the Dirichlet problem and of De Giorgi-Nash local estimates. The layer potential operators are shown to depend holomorphically on the coefficient matrix A is an element of L-infinity, showing uniqueness of the extension of the operators beyond singular integrals. More precisely, we use functional calculus of differential operators with non-smooth coefficients to represent the layer potential operators as bounded Hilbert space operators. In the presence of Moser local bounds, in particular for real scalar equations and systems that are small perturbations of real scalar equations, these operators are shown to be the usual singular integrals. Our proof gives a new construction of fundamental solutions to divergence form systems, valid also in dimension 2.

Place, publisher, year, edition, pages
UNIV AUTONOMA BARCELONA, DEPT MATHEMATICS, 08193 BELLATERRA, SPAIN , 2013. Vol. 57, no 2, 429-454 p.
Keyword [en]
Double layer potential, fundamental solution, divergence form system, functional calculus
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:liu:diva-98670DOI: 10.5565/PUBLMAT_57213_08ISI: 000324282400008OAI: oai:DiVA.org:liu-98670DiVA: diva2:655263
Note

Funding Agencies|Swedish research council, VR|621-2011-3744|

Available from: 2013-10-10 Created: 2013-10-10 Last updated: 2017-12-06

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Rosen, Andreas

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  • de-DE
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