Continuous solutions for divergence-type equations associated to elliptic systems of complex vector fields

Author(s)

Moonens, Laurent ^[1] ; Picon, Tiago ^[2]

Affiliation

^[1] Univ Paris Saclay, Lab Math UMR 8628, Univ Paris Sud, Batiment 307, F-91405 Orsay - France

^[2] Univ Sao Paulo, Fac Filosofia Ciencias & Letras Ribeirao Preto, Dept Computacao & Matemat, Ave Bandeirantes 3900, BR-14040901 Ribeirao Preto - Brazil

• Document type: Journal article
• Source: JOURNAL OF FUNCTIONAL ANALYSIS; v. 275, n. 5, p. 1073-1099, SEP 1 2018.
• Web of Science Citations: 0

Abstract

In this paper, we characterize all the distributions F is an element of D'(U) such that there exists a continuous weak solution upsilon is an element of C(U, C-n) (with U subset of Omega ) to the divergence-type equation L-1{*}upsilon(1) + ... + L-n{*}upsilon(n) = F-1 where [L-1 , ..., L-n] is an elliptic system of linearly independent vector fields with smooth complex coefficients defined on Omega subset of R-N. In case where (L-1 , ..., L-n) is the usual gradient field on R-N, we recover the classical result for the divergence equation proved by T. De Pauw and W. Pfeffer. Its proof is based on the closed range theorem and inspired by {[}3] and {[}6] in the classical case. Our method slightly differs from theirs by relying on the Banach-Grothendieck theorem and introducing tools from pseudodifferential operators, useful in our local setting of a system of complex vector fields with variable coefficients. (C) 2018 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 13/17636-5 - A priori estimates for elliptic complexes and applications
• Grantee: Tiago Henrique Picon
• Support Opportunities: Research Grants - Young Investigators Grants