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Pré-Publication, Document De Travail Année : 2020

ON LOCAL CONTINUOUS SOLVABILITY OF EQUATIONS ASSOCIATED TO ELLIPTIC AND CANCELING LINEAR DIFFERENTIAL OPERATORS

Résumé

Consider A(x, D) ∶ C ∞ (Ω, E) → C ∞ (Ω, F) an elliptic and canceling linear differential operator of order ν with smooth complex coefficients in Ω ⊂ R^N from a finite dimension complex vector space E to a finite dimension complex vector space F and A * (x, D) its adjoint. In this work we characterize the (local) continuous solvability of the partial differential equation A * (x, D)v = f (in the distribution sense) for a given distribution f ; more precisely we show that any x 0 ∈ Ω is contained in a neighborhood U ⊂ Ω in which its continuous solvability is characterized by the following condition on f : for every ε > 0 and any compact set K ⊂⊂ U , there exists θ = θ(K, ε) > 0 such that the following holds for all smooth function φ supported in K: |f (φ)| ⩽ θ||φ|| W ν−1,1 + ε||A(x, D)φ|| L 1 , where W^{ν−1,1} stands for the homogenous Sobolev space of all L 1 functions whose derivatives of order ν − 1 belongs to L 1 (U). This characterization implies and extends results obtained before for operators associated to elliptic complex of vector fields (see [16]); we also provide local analogues, for a large range of differential operators, to global results obtained for the classical divergence operator in [4] and [9].
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hal-02542041 , version 1 (14-04-2020)

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Laurent Moonens, Tiago Picon. ON LOCAL CONTINUOUS SOLVABILITY OF EQUATIONS ASSOCIATED TO ELLIPTIC AND CANCELING LINEAR DIFFERENTIAL OPERATORS. 2020. ⟨hal-02542041⟩
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