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The double obstacle problem on metric spaces
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
2009 (English)In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, 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.

Place, publisher, year, edition, pages
2009. Vol. 34, no 1, 261-277 p.
Keyword [en]
Double obstacle problem, doubling measure, metric space, nonlinear, p-harmonic, Poincare inequality, potential theory, regularity
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-17169OAI: oai:DiVA.org:liu-17169DiVA: diva2:202160
Available from: 2009-03-07 Created: 2009-03-07 Last updated: 2009-11-09Bibliographically approved
In thesis
1. The Double Obstacle Problem on Metric Spaces
Open this publication in new window or tab >>The Double Obstacle Problem on Metric Spaces
2008 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

During the last decade, potential theory and p-harmonic functions have been developed in the setting of doubling metric measure spaces supporting a p-Poincar´e inequality. This theory unifies, and has applications in several areas of analysis, such as weighted Sobolev spaces, calculus on Riemannian manifolds and Carnot groups, subelliptic differential operators and potential theory on graphs.

In this thesis we investigate the double obstacle problem for p-harmonic functions on metric spaces. We show the existence and uniqueness of solutions and their continuity when the obstacles are continuous. Moreover the solution is p-harmonic in the open set where it does not touch the continuous obstacles. The boundary regularity of the solutions is also studied.

Furthermore we study two kinds of convergence problems for the solutions. First we let the obstacles vary and fix the boundary values and show the convergence of the solutions. 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.

Abstract [sv]

Låt oss börja med att betrakta följande situation: Vi vill förflytta oss från en plats vid ena sidan av en äng till en viss punkt på andra sidan ängen. På båda sidor om ängen finns skogsområden som vi inte får gå in i. Ängen är tyvärr inte homogen utan består av olika sorters mark som vi har noggrant beskrivet på en karta. Vi vill göra förflyttningen på smidigast sätt, men då ängen inte är homogen ska vi förmodligen inte gå rakaste vägen utan ska anpassa vägen optimalt efter terrängen. Detta är ett exempel på ett dubbelhinderproblem där hindren är skogsområdena på sidorna som vi måste hålla oss utanför.

Mer abstrakt vill man minimiera energin hos funktioner som tar vissa givna randvärden (de givna start- och slutpunkterna i exemplet ovan) och som håller sig mellan ett undre och ett övre hinder. I denna avhandling studeras detta dubbelhinderproblem i väldigt allmänna situationer.

För att kunna lösa hinderproblemet krävs det att vi tillåter ickekontinuerliga lösningar och då visas i avhandlingen att hinderproblemet är entydigt lösbart. Ett huvudresultat i avhandlingen är att om våra hinder är kontinuerliga så blir även lösningen kontinuerlig. Vidare visas diverse konvergenssatser som visar hur lösningarna varierar när hindren eller området i vilket problemet löses varierar.

Hinderproblem har utöver eget intresse viktiga tillämpningar i potentialteorin, bland annat för att studera motsvarande energiminimeringsproblem utan hinder.

Place, publisher, year, edition, pages
Matematiska institutionen, 2008. 4 p.
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1342
Keyword
metric space, nonlinear, obstacle problem, p-harmonic, potential theory, regularity, stability
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-10621 (URN)978-91-85831-00-5 (ISBN)
Presentation
2008-02-11, Glashuset, Hus B, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2008-01-22 Created: 2008-01-22 Last updated: 2016-05-04
2. The Double Obstacle Problem on Metric Spaces
Open this publication in new window or tab >>The Double Obstacle Problem on Metric Spaces
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:nbn:se:liu:diva-51588 (URN)978-91-85831-00-5 (ISBN)
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

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