Applications of pseudodifferential operators with global symbols.

• Grant number: 12/18198-9
• Support Opportunities: Scholarships in Brazil - Post-Doctorate
• Effective date (Start): December 01, 2012
• Effective date (End): September 30, 2014
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Jorge Guillermo Hounie
• Grantee: Pedro Tavares Paes Lopes
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• Associated research grant: 12/03168-7 - Geometric theory of PDE and several complex variables, AP.TEM

Abstract

We propose a study of pseudodifferential operators with global symbols and their associated boundary value problems, in particular of some of their applications. Taking into account several recent results about these operators, we propose three different problems closely related. We intend to apply these operators in the Fredholm index theory, to the study of asymptotic properties of solutions of differential equations and to the construction of self-adjoint operators using the Weyl formula.More precisely, we would like to extend the Fredholm index formula for elliptic boundary value problems in the half-plane, obtained by Rempel and Schulze in their book, to SG pseudodifferential operators defined more recently by Schrohe, Schulze and Kapanadze and which were studied by the candidate in his Ph.D. Our aim is to obtain an index formula which does not require the strong assumptions made by Rempel and Schulze. In that way, we would obtain the analogous of Fedosov formula for elliptic problems on Rn as proved by Hörmander in his study of Weyl algebras.The second problem that we propose is the study of exponential decay for elliptic pseudodifferential equations with SG symbols on classes of non-compact manifolds that are compatible with this calculus. The study of the exponential decay for pseudodifferential equations with SG symbols on Rn was done recently by Rodino, Nicola e Capiello, among others, using techniques of several complex variables. Our aim is to study the same problem for the classes of SG compatible manifolds as defined by Schrohe, Maniccia, Schulze, among others. We would like also to understand the exponential decay for the associated boundary value problems. We start with the half-plane, using again the algebras that were studied in the Thesis of the candidate.Finally, we would like to study the Weyl formula for pseudodifferential operators of non-negative orders as it was done by Álvarez e Hounie, however using global symbols, like the SG and the Shubin ones. We would like to know if the operators that were constructed using this formula are still global pseudodifferential operators and under which conditions they are.

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

LOPES, PEDRO T. P.. Gelfand-Shilov regularity of SG boundary value problems. JOURNAL OF PSEUDO-DIFFERENTIAL OPERATORS AND APPLICATIONS, v. 8, n. 1, p. 55-81, MAR 2017. (12/18198-9)