Applications of pseudo-differential methods to boundary value problems

• Grant number: 16/07016-8
• Support Opportunities: Regular Research Grants
• Duration: August 01, 2016 - July 31, 2018
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Pedro Tavares Paes Lopes
• Grantee: Pedro Tavares Paes Lopes
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

Our project consists of three problems, where pseudo-differential methods applied to boundary value problems are essential.The first one is concerned with the Gelfand-Shilov regularity of SG operators. It is a subject that has attracted a lot of attention in the last 15 years with the works of mathematicians such as M. Cappiello, L. Rodino, T. Gamschev, F. Nicola and J. Toft, among others. We have also given our contribution in a recently accepted publication. Our main proposal is to obtain the analytic Gelfand-Shilov regularity for boundary value problems on unbounded domains.The second and third problems are related with a topic of intense research and many recent papers, with a interesting interaction with the study of evolution equations. Our aim is to study maximal regularity of boundary value problems. In this direction, we propose two problems. The first one is about boundary value problems that are not elliptic in the usual sense, that is, they do not satisfy the Lopatinski-Shapiro conditions. In particular, we are concerned with the class of problems studied by mathematician Kazuaki Taira. This author studies problems that are elliptic in the interior, but whose boundary conditions can change smoothly between the Dirichlet and Neumann conditions. A physical interpretation can be found in his recent book. The second problem related to maximal regularity is about elliptic boundary value problems on domains whose boundary are smooth outside a finite set, where there are conical singularities. On both problems, we are interested in the use of pseudo-differential techniques to study the realizations of operators on L^{p} spaces and the possible existence of a holomorphic functional calculus as the H infinity calculus defined by Alan McIntoshi. We believe that these results can have important applications, for instance, to the study of quasi-linear evolution equations. (AU)

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.; PEREIRA, MARCONE C.. Dynamical boundary conditions in a non-cylindrical domain for the Laplace equation. Journal of Mathematical Analysis and Applications, v. 465, n. 1, p. 379-402, SEP 1 2018. (16/07016-8, 17/02630-2)

LOPES, PEDRO T. P.; SCHROHE, ELMAR. Spectral Invariance of Pseudodifferential Boundary Value Problems on Manifolds with Conical Singularities. JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, v. 25, n. 3, p. 1147-1202, JUN 2019. (16/07016-8)