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SMOOTH COMPACTLY SUPPORTED SOLUTIONS OF SOME UNDERDETERMINED ELLIPTIC PDE WITH

19 pages
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.

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SMOOTHCOMPACTLYSUPPORTEDSOLUTIONSOFSOMEUNDERDETERMINEDELLIPTICPDE,WITHGLUINGAPPLICATIONSERWANNDELAYAbstract.Wegivesufficientconditionsforsomeunderdeterminedel-lipticPDEofanyordertoconstructsmoothcompactlysupportedso-lutions.InparticularweshowthattwosmoothelementsinthekernelofcertainunderdeterminedlinearellipticoperatorsPcanbegluedinachosenregioninordertoobtainanewsmoothsolution.Thisnewsolu-tionisexactlyequaltothestartingelementsoutsidethegluingregion.Thisresultcompletelycontrastswiththeusualuniquecontinuationfordeterminedoroverdeterminedellipticoperators.Asacorollaryweob-taincompactlysupportedsolutionsinthekernelofPandalsosolutionsvanishinginachosenrelativelycompactopenregion.Weapplythere-sultfornaturalgeometricandphysicscontextssuchasdivergencefreefieldsorTT-tensors.Keywords:UndeterminedellipticPDE,compactlysupportedsolutions,gluing.MSC2010:35J99,58J99,35Q35,35Q60,35Q75Contents1.Introduction2.Weightedspaces3.Isomorphismproperties4.Regularity5.Compactlysupportedsolutions6.Thegluing7.Thebasicexample8.Applications8.1.Divergencefreevectorelds8.2.(Multi-)divergencefreetensors8.3.Divergencefreesymmetrictwotensors8.4.TT-tensors8.5.Linearizedscalarcurvatureoperator8.6.AnonlinearapplicationReferencesDate:January24,2011.124577891111212131617181