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Capacitary estimates for solutions of the Dirichlet problem for second order elliptic equations in divergence form
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.ORCID iD: 0000-0002-1238-6751
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
2000 (English)In: Potential Analysis, ISSN 0926-2601, Vol. 12, no 1, 81-113 p.Article in journal (Refereed) Published
Abstract [en]

We consider the Dirichlet problem for A-harmonic functions, i.e. the solutions of the uniformly elliptic equation div(A(x)del u(x)) = 0 in an n-dimensional domain Omega, n greater than or equal to 3. The matrix A is assumed to have bounded measurable entries. We obtain pointwise estimates for the A-harmonic functions near a boundary point. The estimates are in terms of the Wiener capacity and the so called capacitary interior diameter. They imply pointwise estimates for the A-harmonic measure of the domain Omega, which in turn lead to a sufficient condition for the Holder continuity of A-harmonic functions at a boundary point. The behaviour of A-harmonic functions at infinity and near a singular point is also studied and theorems of Phragmen-Lindelof type, in which the geometry of the boundary is taken into account, are proved. We also obtain pointwise estimates for the Green function for the operator -div(A(.)del u(.)) in a domain Omega and for the solutions of the nonhomogeneous equation -div(A(x)del u(x)) = mu with measure on the right-hand side.

Place, publisher, year, edition, pages
2000. Vol. 12, no 1, 81-113 p.
Keyword [en]
second order elliptic equations in divergence form, Dirichlet problem, Holder continuity, capacitary interior diameter, Phragmen-Lindelof theorem
National Category
Engineering and Technology
Identifiers
URN: urn:nbn:se:liu:diva-49846OAI: oai:DiVA.org:liu-49846DiVA: diva2:270742
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2016-05-04

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Björn, JanaMaz´ya, Vladimir G.

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CiteExportLink to record
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  • de-DE
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  • nn-NB
  • sv-SE
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