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.

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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.Divergencefreevectorﬁelds8.2.(Multi-)divergencefreetensors8.3.Divergencefreesymmetrictwotensors8.4.TT-tensors8.5.Linearizedscalarcurvatureoperator8.6.AnonlinearapplicationReferencesDate:January24,2011.124577891111212131617181

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