SMOOTH COMPACTLY SUPPORTED SOLUTIONS OF SOME UNDERDETERMINED ELLIPTIC PDE WITH

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SMOOTH COMPACTLY SUPPORTED SOLUTIONS OF SOME UNDERDETERMINED ELLIPTIC PDE, WITH GLUING APPLICATIONS ERWANN DELAY Abstract. We give sufficient conditions for some underdetermined el- liptic PDE of any order to construct smooth compactly supported so- lutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators P can be glued in a chosen region in order to obtain a new smooth solution. This new solu- tion is exactly equal to the starting elements outside the gluing region. This result completely contrasts with the usual unique continuation for determined or overdetermined elliptic operators. As a corollary we ob- tain compactly supported solutions in the kernel of P and also solutions vanishing in a chosen relatively compact open region. We apply the re- sult for natural geometric and physics contexts such as divergence free fields or TT-tensors. Keywords : Undetermined elliptic PDE, compactly supported solutions, gluing. MSC 2010 : 35J99, 58J99, 35Q35, 35Q60, 35Q75 Contents 1. Introduction 2 2. Weighted spaces 4 3. Isomorphism properties 5 4. Regularity 7 5. Compactly supported solutions 7 6. The gluing 8 7. The basic example 9 8. Applications 11 8.1. Divergence free vector fields 11 8.2. (Multi-)divergence free tensors 12 8.3. Divergence free symmetric two tensors 12 8.4. TT-tensors 13 8.5.

• let ?


• smooth boundary


• elliptic operators


• contexts such


• underdetermined elliptic


• divergence operator


• tensor fields


• compactly supported


• construct

Sujets

• Tensor field
• Support (mathematics)
• Construct

Informations

• Publié par: pefav
• Nombre de lectures: 20
• Langue: English

Extrait

SMOOTHCOMPACTLYSUPPORTEDSOLUTIONSOF SOMEUNDERDETERMINEDELLIPTICPDE,WITH GLUINGAPPLICATIONS ERWANNDELAY Abstract. Wegivesufficientconditionsforsomeunderdeterminedel- lipticPDEofanyordertoconstructsmoothcompactlysupportedso- lutions.Inparticularweshowthattwosmoothelementsinthekernel ofcertainunderdeterminedlinearellipticoperatorsPcanbegluedina chosenregioninordertoobtainanewsmoothsolution.Thisnewsolu- tionisexactlyequaltothestartingelementsoutsidethegluingregion. Thisresultcompletelycontrastswiththeusualuniquecontinuationfor determinedoroverdeterminedellipticoperators.Asacorollaryweob- taincompactlysupportedsolutionsinthekernelofPandalsosolutions vanishinginachosenrelativelycompactopenregion.Weapplythere- sultfornaturalgeometricandphysicscontextssuchasdivergencefree fieldsorTT-tensors. Keywords:UndeterminedellipticPDE,compactlysupportedsolutions, gluing. MSC2010:35J99,58J99,35Q35,35Q60,35Q75 Contents 1.Introduction 2.Weightedspaces 3.Isomorphismproperties 4.Regularity 5.Compactlysupportedsolutions 6.Thegluing 7.Thebasicexample 8.Applications 8.1.Divergencefreevectorfields 8.2.(Multi-)divergencefreetensors 8.3.Divergencefreesymmetrictwotensors 8.4.TT-tensors 8.5.Linearizedscalarcurvatureoperator 8.6.Anonlinearapplication References Date:January24,2011. 1 2 4 5 7 7 8 9 11 11 21 21 31 61 71 81