On local continuous solvability of equations associated to elliptic and canceling linear differential operators

Autor(es)

Moonens, Laurent ^{[1, 2]} ; Picon, Tiago ^[3]

Afiliação do(s) autor(es)

^[1] Univ Paris Saclay, Lab Math Orsay, CNRS UMR 8628, Batiment 307 IMO, Rue Michel Magat, F-91405 Orsay - France

^[2] PSL Univ, Ecole Normale Super, Dept Math & Applicat, CNRS, UMR 8553, 45 Rue Ulm, F-75230 Paris 5 - France

^[3] Univ Sao Paulo, Fac Filosofia Ciencias & Letras Ribeirao Preto, Dept Comp & Matemat, Ave Bandeirantes 3900, BR-14040901 Ribeirao Preto - Brazil

• Tipo de documento: Artigo Científico
• Fonte: JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES; v. 149, p. 47-72, MAY 2021.
• Citações Web of Science: 0

Resumo

Consider A (x, D) : C-infinity(Omega, E) -> C-infinity(Omega, F) an elliptic and canceling linear differential operator of order v with smooth complex coefficients in Omega subset of 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) is an element of Omega is contained in a neighborhood U subset of Omega in which its continuous solvability is characterized by the following condition on f: for every epsilon > 0 and any compact set K subset of subset of U, there exists theta = theta(K, epsilon) > 0 such that the following holds for all smooth function phi supported in K: vertical bar f(phi)vertical bar <= theta parallel to phi parallel to W-v-1,W-1 + epsilon parallel to A(x, D)phi parallel to(L1), where W-v-1,W-1 stands for the homogenous Sobolev space of all L-1 functions whose derivatives of order v - 1 belongs to L-1 (U). This characterization implies and extends results obtained before for operators associated to elliptic complexes of vector fields (see {[}1]); we also provide local analogues, for a large range of differential operators, to global results obtained for the classical divergence operator by Bourgain and Brezis in {[}2] and by De Pauw and Pfeffer in {[}3]. (C) 2020 Elsevier Masson SAS. All rights reserved. (AU)

• Processo FAPESP: 17/17804-6 - Resolubilidade contínua local para EDPs associadas a operadores e complexos elípticos
• Beneficiário: Tiago Henrique Picon
• Modalidade de apoio: Auxílio à Pesquisa - Pesquisador Visitante - Internacional
• Processo FAPESP: 18/15484-7 - Estimativas a priori para operadores elípticos e aplicações
• Beneficiário: Tiago Henrique Picon
• Modalidade de apoio: Auxílio à Pesquisa - Jovens Pesquisadores - Fase 2
• Processo FAPESP: 19/21179-5 - Estimativas a priori para operadores elípticos e aplicações
• Beneficiário: Tiago Henrique Picon
• Modalidade de apoio: Bolsas no Exterior - Pesquisa