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    STUDIA MATHEMATICA - Ediţia nr.2 din 2022  
         
  Articol:   POLYNOMIAL ESTIMATES FOR SOLUTIONS OF PARAMETRIC ELLIPTIC EQUATIONS ON COMPLETE MANIFOLDS.

Autori:  MIRELA KOHR, SIMON LABRUNIE, HASSAN MOHSEN, VICTOR NISTOR. 		 
       
         
  Rezumat:  
DOI: 10.24193/subbmath. 2022.2.13

Published Online: 2022-06-10
Published Print: 2022-06-30
pp. 369-382

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Abstract: Let P : C1(M;E) ! C1(M; F) be an order _ di_erential operator with coe_cients a and Pk := P : Hs0+k+_(M;E) ! Hs0+k(M; F). We prove polynomial norm estimates for the solution P􀀀1 0 f of the form kP􀀀1 0 fkHs0+k+_(M;E) _ C Xk q=0 kP􀀀1 0 kq+1 kakq Wjs0j+k kfkHs0+k􀀀q ; (thus in higher order Sobolev spaces, which amounts also to a parametric regularity result). The assumptions are that E; F ! M are Hermitian vector bundles and that M is a complete manifold satisfying the Fr_echet Finiteness Condition (FFC), which was introduced in (Kohr and Nistor, Annals of Global Analysis and Geometry, 2022). These estimates are useful for uncertainty quanti_cation, since the coe_cient a can be regarded as a vector valued random variable. We use these results to prove integrability of the norm kP􀀀1 k fk of the solution of Pku = f with respect to suitable Gaussian measures.

Key words: Parametric elliptic equations, complete manifolds, the Fréchet Finiteness Condition (FFC), Sobolev spaces, uncertainty quantification.

Mathematics Subject Classification (2010): 35R01, 35J75, 46E35, 65N75 		 
         
     
         
         
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