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The Double Obstacle Problem on Metric Spaces
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we investigate the double obstacle problem for p-harmonic functions on metric spaces. We minimize the p-energy integral among all functions which have prescribed boundary values and lie between two given obstacles. This is a generalization of the Dirichlet problem for p-harmonic functions, in which case the obstacles are — and .

We show the existence and uniqueness of solutions, and their continuity when the obstacles are continuous. Moreover we show that the continuous solution is p-harmonic in the open set where it does not touch the continuous obstacles. If the obstacles are not continuous, but satisfy a Wiener type regularity condition, we prove that the solution is still continuous. The Hölder continuity for solutions is shown, when the obstacles are Hölder continuous. Boundary regularity of the solutions is also studied.

Furthermore we study two kinds of convergence problems for the solutions. First we let the obstacles and the boundary values vary and show the convergence of the solutions. We also consider generalized solutions for insoluble obstacle problems, using the convergence results. Moreover we show that for soluble obstacle problems the generalized solution coincides, locally, with the standard solution.

Second we consider an increasing sequence of open sets, with union Ω, and fix the obstacles and the boundary values. We show that the solutions of the obstacle problems in these sets converge to the solution of the corresponding problem in Ω.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press , 2009. , 7 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1283
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-51588ISBN: 978-91-85831-00-5 (print)OAI: oai:DiVA.org:liu-51588DiVA: diva2:275934
Public defence
2009-12-15, Planck, Fysikhuset, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2009-11-09 Created: 2009-11-09 Last updated: 2016-05-04Bibliographically approved
List of papers
1. The double obstacle problem on metric spaces
Open this publication in new window or tab >>The double obstacle problem on metric spaces
2009 (English)In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 34, no 1, 261-277 p.Article in journal (Refereed) Published
Abstract [en]

We study the double obstacle problem on a metric measure space equipped with a doubling measure. and supporting a p-Poincare inequality. We prove existence and uniqueness. We also prove the continuity of the solution of the double obstacle. problem with continuous obstacles and show that. the continuous solution is a minimizer in the open set where it. does not touch the two obstacles. Moreover we consider the regular boundary points and show that the solution of the double obstacle problem on a regular open set with continuous obstacles is continuous up to the boundary. Regularity of boundary points is further characterized in some other ways using the solution of the double obstacle problem.

Keyword
Double obstacle problem, doubling measure, metric space, nonlinear, p-harmonic, Poincare inequality, potential theory, regularity
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-17169 (URN)
Available from: 2009-03-07 Created: 2009-03-07 Last updated: 2017-12-13Bibliographically approved
2. Pointwise regularity for solutions of double obstacle problems on metric spaces
Open this publication in new window or tab >>Pointwise regularity for solutions of double obstacle problems on metric spaces
2011 (English)In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 109, no 2, 185-200 p.Article in journal (Refereed) Published
Abstract [en]

We study continuity at a given point for solutions of double obstacle problems. We obtain pointwise continuity of the solutions for discontinuous obstacles. We also show Hölder continuity for solutions of the double obstacle problems if the obstacles are Hölder continuous.

Place, publisher, year, edition, pages
Kobenhavn, 2011
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-51586 (URN)000298339300002 ()
Available from: 2009-11-09 Created: 2009-11-09 Last updated: 2017-12-12Bibliographically approved
3. Continuous dependence on obstacles for the double obstacle problem on metric spaces
Open this publication in new window or tab >>Continuous dependence on obstacles for the double obstacle problem on metric spaces
2010 (English)In: NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, ISSN 0362-546X, Vol. 73, no 9, 2819-2830 p.Article in journal (Refereed) Published
Abstract [en]

Let X be a complete metric space equipped with a doubling Borel measure supporting a p-Poincaré inequality. We obtain various convergence results for solutions of double obstacle problems on open subsets of X. In particular, we consider a sequence of double obstacle problems with converging obstacles and show that the corresponding solutions converge as well. We use the convergence properties to denfie and study a generalized solution of the double obstacle problem.

Place, publisher, year, edition, pages
Elsevier Science B.V., Amsterdam, 2010
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-51587 (URN)10.1016/j.na.2010.06.023 (DOI)000281062800005 ()
Available from: 2009-11-09 Created: 2009-11-09 Last updated: 2010-09-03Bibliographically approved
4. Convergence results for the obstacle problem in metric spaces
Open this publication in new window or tab >>Convergence results for the obstacle problem in metric spaces
Manuscript (Other academic)
Identifiers
urn:nbn:se:liu:diva-12897 (URN)
Available from: 2008-01-22 Created: 2008-01-22 Last updated: 2010-01-13

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